/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. We are asked to determine termination of the following first-order TRS. 0 : [] --> o U11 : [o * o * o] --> o U12 : [o * o * o] --> o active : [o] --> o mark : [o] --> o plus : [o * o] --> o s : [o] --> o tt : [] --> o active(U11(tt, X, Y)) => mark(U12(tt, X, Y)) active(U12(tt, X, Y)) => mark(s(plus(Y, X))) active(plus(X, 0)) => mark(X) active(plus(X, s(Y))) => mark(U11(tt, Y, X)) mark(U11(X, Y, Z)) => active(U11(mark(X), Y, Z)) mark(tt) => active(tt) mark(U12(X, Y, Z)) => active(U12(mark(X), Y, Z)) mark(s(X)) => active(s(mark(X))) mark(plus(X, Y)) => active(plus(mark(X), mark(Y))) mark(0) => active(0) U11(mark(X), Y, Z) => U11(X, Y, Z) U11(X, mark(Y), Z) => U11(X, Y, Z) U11(X, Y, mark(Z)) => U11(X, Y, Z) U11(active(X), Y, Z) => U11(X, Y, Z) U11(X, active(Y), Z) => U11(X, Y, Z) U11(X, Y, active(Z)) => U11(X, Y, Z) U12(mark(X), Y, Z) => U12(X, Y, Z) U12(X, mark(Y), Z) => U12(X, Y, Z) U12(X, Y, mark(Z)) => U12(X, Y, Z) U12(active(X), Y, Z) => U12(X, Y, Z) U12(X, active(Y), Z) => U12(X, Y, Z) U12(X, Y, active(Z)) => U12(X, Y, Z) s(mark(X)) => s(X) s(active(X)) => s(X) plus(mark(X), Y) => plus(X, Y) plus(X, mark(Y)) => plus(X, Y) plus(active(X), Y) => plus(X, Y) plus(X, active(Y)) => plus(X, Y) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): active(U11(tt, X, Y)) >? mark(U12(tt, X, Y)) active(U12(tt, X, Y)) >? mark(s(plus(Y, X))) active(plus(X, 0)) >? mark(X) active(plus(X, s(Y))) >? mark(U11(tt, Y, X)) mark(U11(X, Y, Z)) >? active(U11(mark(X), Y, Z)) mark(tt) >? active(tt) mark(U12(X, Y, Z)) >? active(U12(mark(X), Y, Z)) mark(s(X)) >? active(s(mark(X))) mark(plus(X, Y)) >? active(plus(mark(X), mark(Y))) mark(0) >? active(0) U11(mark(X), Y, Z) >? U11(X, Y, Z) U11(X, mark(Y), Z) >? U11(X, Y, Z) U11(X, Y, mark(Z)) >? U11(X, Y, Z) U11(active(X), Y, Z) >? U11(X, Y, Z) U11(X, active(Y), Z) >? U11(X, Y, Z) U11(X, Y, active(Z)) >? U11(X, Y, Z) U12(mark(X), Y, Z) >? U12(X, Y, Z) U12(X, mark(Y), Z) >? U12(X, Y, Z) U12(X, Y, mark(Z)) >? U12(X, Y, Z) U12(active(X), Y, Z) >? U12(X, Y, Z) U12(X, active(Y), Z) >? U12(X, Y, Z) U12(X, Y, active(Z)) >? U12(X, Y, Z) s(mark(X)) >? s(X) s(active(X)) >? s(X) plus(mark(X), Y) >? plus(X, Y) plus(X, mark(Y)) >? plus(X, Y) plus(active(X), Y) >? plus(X, Y) plus(X, active(Y)) >? plus(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 1 U11 = \y0y1y2.2y0 + 2y1 + 2y2 U12 = \y0y1y2.y0 + 2y1 + 2y2 active = \y0.y0 mark = \y0.y0 plus = \y0y1.2y0 + 2y1 s = \y0.y0 tt = 0 Using this interpretation, the requirements translate to: [[active(U11(tt, _x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(U12(tt, _x0, _x1))]] [[active(U12(tt, _x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(s(plus(_x1, _x0)))]] [[active(plus(_x0, 0))]] = 2 + 2x0 > x0 = [[mark(_x0)]] [[active(plus(_x0, s(_x1)))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[mark(U11(tt, _x1, _x0))]] [[mark(U11(_x0, _x1, _x2))]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[active(U11(mark(_x0), _x1, _x2))]] [[mark(tt)]] = 0 >= 0 = [[active(tt)]] [[mark(U12(_x0, _x1, _x2))]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[active(U12(mark(_x0), _x1, _x2))]] [[mark(s(_x0))]] = x0 >= x0 = [[active(s(mark(_x0)))]] [[mark(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[active(plus(mark(_x0), mark(_x1)))]] [[mark(0)]] = 1 >= 1 = [[active(0)]] [[U11(mark(_x0), _x1, _x2)]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, mark(_x1), _x2)]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, mark(_x2))]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(active(_x0), _x1, _x2)]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, active(_x1), _x2)]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, active(_x2))]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U12(mark(_x0), _x1, _x2)]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, mark(_x1), _x2)]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, mark(_x2))]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(active(_x0), _x1, _x2)]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, active(_x1), _x2)]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, active(_x2))]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[s(mark(_x0))]] = x0 >= x0 = [[s(_x0)]] [[s(active(_x0))]] = x0 >= x0 = [[s(_x0)]] [[plus(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, mark(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(active(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, active(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[plus(_x0, _x1)]] We can thus remove the following rules: active(plus(X, 0)) => mark(X) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): active(U11(tt, X, Y)) >? mark(U12(tt, X, Y)) active(U12(tt, X, Y)) >? mark(s(plus(Y, X))) active(plus(X, s(Y))) >? mark(U11(tt, Y, X)) mark(U11(X, Y, Z)) >? active(U11(mark(X), Y, Z)) mark(tt) >? active(tt) mark(U12(X, Y, Z)) >? active(U12(mark(X), Y, Z)) mark(s(X)) >? active(s(mark(X))) mark(plus(X, Y)) >? active(plus(mark(X), mark(Y))) mark(0) >? active(0) U11(mark(X), Y, Z) >? U11(X, Y, Z) U11(X, mark(Y), Z) >? U11(X, Y, Z) U11(X, Y, mark(Z)) >? U11(X, Y, Z) U11(active(X), Y, Z) >? U11(X, Y, Z) U11(X, active(Y), Z) >? U11(X, Y, Z) U11(X, Y, active(Z)) >? U11(X, Y, Z) U12(mark(X), Y, Z) >? U12(X, Y, Z) U12(X, mark(Y), Z) >? U12(X, Y, Z) U12(X, Y, mark(Z)) >? U12(X, Y, Z) U12(active(X), Y, Z) >? U12(X, Y, Z) U12(X, active(Y), Z) >? U12(X, Y, Z) U12(X, Y, active(Z)) >? U12(X, Y, Z) s(mark(X)) >? s(X) s(active(X)) >? s(X) plus(mark(X), Y) >? plus(X, Y) plus(X, mark(Y)) >? plus(X, Y) plus(active(X), Y) >? plus(X, Y) plus(X, active(Y)) >? plus(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 0 U11 = \y0y1y2.2 + 2y0 + 2y1 + 2y2 U12 = \y0y1y2.1 + 2y0 + 2y1 + 2y2 active = \y0.y0 mark = \y0.y0 plus = \y0y1.2y0 + 2y1 s = \y0.1 + y0 tt = 0 Using this interpretation, the requirements translate to: [[active(U11(tt, _x0, _x1))]] = 2 + 2x0 + 2x1 > 1 + 2x0 + 2x1 = [[mark(U12(tt, _x0, _x1))]] [[active(U12(tt, _x0, _x1))]] = 1 + 2x0 + 2x1 >= 1 + 2x0 + 2x1 = [[mark(s(plus(_x1, _x0)))]] [[active(plus(_x0, s(_x1)))]] = 2 + 2x0 + 2x1 >= 2 + 2x0 + 2x1 = [[mark(U11(tt, _x1, _x0))]] [[mark(U11(_x0, _x1, _x2))]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[active(U11(mark(_x0), _x1, _x2))]] [[mark(tt)]] = 0 >= 0 = [[active(tt)]] [[mark(U12(_x0, _x1, _x2))]] = 1 + 2x0 + 2x1 + 2x2 >= 1 + 2x0 + 2x1 + 2x2 = [[active(U12(mark(_x0), _x1, _x2))]] [[mark(s(_x0))]] = 1 + x0 >= 1 + x0 = [[active(s(mark(_x0)))]] [[mark(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[active(plus(mark(_x0), mark(_x1)))]] [[mark(0)]] = 0 >= 0 = [[active(0)]] [[U11(mark(_x0), _x1, _x2)]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, mark(_x1), _x2)]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, mark(_x2))]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(active(_x0), _x1, _x2)]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, active(_x1), _x2)]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, active(_x2))]] = 2 + 2x0 + 2x1 + 2x2 >= 2 + 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U12(mark(_x0), _x1, _x2)]] = 1 + 2x0 + 2x1 + 2x2 >= 1 + 2x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, mark(_x1), _x2)]] = 1 + 2x0 + 2x1 + 2x2 >= 1 + 2x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, mark(_x2))]] = 1 + 2x0 + 2x1 + 2x2 >= 1 + 2x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(active(_x0), _x1, _x2)]] = 1 + 2x0 + 2x1 + 2x2 >= 1 + 2x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, active(_x1), _x2)]] = 1 + 2x0 + 2x1 + 2x2 >= 1 + 2x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, active(_x2))]] = 1 + 2x0 + 2x1 + 2x2 >= 1 + 2x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[s(mark(_x0))]] = 1 + x0 >= 1 + x0 = [[s(_x0)]] [[s(active(_x0))]] = 1 + x0 >= 1 + x0 = [[s(_x0)]] [[plus(mark(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, mark(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(active(_x0), _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, active(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[plus(_x0, _x1)]] We can thus remove the following rules: active(U11(tt, X, Y)) => mark(U12(tt, X, Y)) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): active(U12(tt, X, Y)) >? mark(s(plus(Y, X))) active(plus(X, s(Y))) >? mark(U11(tt, Y, X)) mark(U11(X, Y, Z)) >? active(U11(mark(X), Y, Z)) mark(tt) >? active(tt) mark(U12(X, Y, Z)) >? active(U12(mark(X), Y, Z)) mark(s(X)) >? active(s(mark(X))) mark(plus(X, Y)) >? active(plus(mark(X), mark(Y))) mark(0) >? active(0) U11(mark(X), Y, Z) >? U11(X, Y, Z) U11(X, mark(Y), Z) >? U11(X, Y, Z) U11(X, Y, mark(Z)) >? U11(X, Y, Z) U11(active(X), Y, Z) >? U11(X, Y, Z) U11(X, active(Y), Z) >? U11(X, Y, Z) U11(X, Y, active(Z)) >? U11(X, Y, Z) U12(mark(X), Y, Z) >? U12(X, Y, Z) U12(X, mark(Y), Z) >? U12(X, Y, Z) U12(X, Y, mark(Z)) >? U12(X, Y, Z) U12(active(X), Y, Z) >? U12(X, Y, Z) U12(X, active(Y), Z) >? U12(X, Y, Z) U12(X, Y, active(Z)) >? U12(X, Y, Z) s(mark(X)) >? s(X) s(active(X)) >? s(X) plus(mark(X), Y) >? plus(X, Y) plus(X, mark(Y)) >? plus(X, Y) plus(active(X), Y) >? plus(X, Y) plus(X, active(Y)) >? plus(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 0 U11 = \y0y1y2.y0 + y1 + y2 U12 = \y0y1y2.1 + y0 + 2y1 + 2y2 active = \y0.y0 mark = \y0.y0 plus = \y0y1.y0 + y1 s = \y0.2y0 tt = 0 Using this interpretation, the requirements translate to: [[active(U12(tt, _x0, _x1))]] = 1 + 2x0 + 2x1 > 2x0 + 2x1 = [[mark(s(plus(_x1, _x0)))]] [[active(plus(_x0, s(_x1)))]] = x0 + 2x1 >= x0 + x1 = [[mark(U11(tt, _x1, _x0))]] [[mark(U11(_x0, _x1, _x2))]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[active(U11(mark(_x0), _x1, _x2))]] [[mark(tt)]] = 0 >= 0 = [[active(tt)]] [[mark(U12(_x0, _x1, _x2))]] = 1 + x0 + 2x1 + 2x2 >= 1 + x0 + 2x1 + 2x2 = [[active(U12(mark(_x0), _x1, _x2))]] [[mark(s(_x0))]] = 2x0 >= 2x0 = [[active(s(mark(_x0)))]] [[mark(plus(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[active(plus(mark(_x0), mark(_x1)))]] [[mark(0)]] = 0 >= 0 = [[active(0)]] [[U11(mark(_x0), _x1, _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, mark(_x1), _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, mark(_x2))]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(active(_x0), _x1, _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, active(_x1), _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, active(_x2))]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U12(mark(_x0), _x1, _x2)]] = 1 + x0 + 2x1 + 2x2 >= 1 + x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, mark(_x1), _x2)]] = 1 + x0 + 2x1 + 2x2 >= 1 + x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, mark(_x2))]] = 1 + x0 + 2x1 + 2x2 >= 1 + x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(active(_x0), _x1, _x2)]] = 1 + x0 + 2x1 + 2x2 >= 1 + x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, active(_x1), _x2)]] = 1 + x0 + 2x1 + 2x2 >= 1 + x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, active(_x2))]] = 1 + x0 + 2x1 + 2x2 >= 1 + x0 + 2x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[s(mark(_x0))]] = 2x0 >= 2x0 = [[s(_x0)]] [[s(active(_x0))]] = 2x0 >= 2x0 = [[s(_x0)]] [[plus(mark(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] [[plus(_x0, mark(_x1))]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] [[plus(active(_x0), _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] [[plus(_x0, active(_x1))]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] We can thus remove the following rules: active(U12(tt, X, Y)) => mark(s(plus(Y, X))) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): active(plus(X, s(Y))) >? mark(U11(tt, Y, X)) mark(U11(X, Y, Z)) >? active(U11(mark(X), Y, Z)) mark(tt) >? active(tt) mark(U12(X, Y, Z)) >? active(U12(mark(X), Y, Z)) mark(s(X)) >? active(s(mark(X))) mark(plus(X, Y)) >? active(plus(mark(X), mark(Y))) mark(0) >? active(0) U11(mark(X), Y, Z) >? U11(X, Y, Z) U11(X, mark(Y), Z) >? U11(X, Y, Z) U11(X, Y, mark(Z)) >? U11(X, Y, Z) U11(active(X), Y, Z) >? U11(X, Y, Z) U11(X, active(Y), Z) >? U11(X, Y, Z) U11(X, Y, active(Z)) >? U11(X, Y, Z) U12(mark(X), Y, Z) >? U12(X, Y, Z) U12(X, mark(Y), Z) >? U12(X, Y, Z) U12(X, Y, mark(Z)) >? U12(X, Y, Z) U12(active(X), Y, Z) >? U12(X, Y, Z) U12(X, active(Y), Z) >? U12(X, Y, Z) U12(X, Y, active(Z)) >? U12(X, Y, Z) s(mark(X)) >? s(X) s(active(X)) >? s(X) plus(mark(X), Y) >? plus(X, Y) plus(X, mark(Y)) >? plus(X, Y) plus(active(X), Y) >? plus(X, Y) plus(X, active(Y)) >? plus(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 0 U11 = \y0y1y2.y1 + y2 + 2y0 U12 = \y0y1y2.y0 + y1 + y2 active = \y0.y0 mark = \y0.y0 plus = \y0y1.1 + y0 + y1 s = \y0.2y0 tt = 0 Using this interpretation, the requirements translate to: [[active(plus(_x0, s(_x1)))]] = 1 + x0 + 2x1 > x0 + x1 = [[mark(U11(tt, _x1, _x0))]] [[mark(U11(_x0, _x1, _x2))]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[active(U11(mark(_x0), _x1, _x2))]] [[mark(tt)]] = 0 >= 0 = [[active(tt)]] [[mark(U12(_x0, _x1, _x2))]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[active(U12(mark(_x0), _x1, _x2))]] [[mark(s(_x0))]] = 2x0 >= 2x0 = [[active(s(mark(_x0)))]] [[mark(plus(_x0, _x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[active(plus(mark(_x0), mark(_x1)))]] [[mark(0)]] = 0 >= 0 = [[active(0)]] [[U11(mark(_x0), _x1, _x2)]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, mark(_x1), _x2)]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, mark(_x2))]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U11(_x0, _x1, _x2)]] [[U11(active(_x0), _x1, _x2)]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, active(_x1), _x2)]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, active(_x2))]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U11(_x0, _x1, _x2)]] [[U12(mark(_x0), _x1, _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, mark(_x1), _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, mark(_x2))]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(active(_x0), _x1, _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, active(_x1), _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, active(_x2))]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[s(mark(_x0))]] = 2x0 >= 2x0 = [[s(_x0)]] [[s(active(_x0))]] = 2x0 >= 2x0 = [[s(_x0)]] [[plus(mark(_x0), _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[plus(_x0, _x1)]] [[plus(_x0, mark(_x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[plus(_x0, _x1)]] [[plus(active(_x0), _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[plus(_x0, _x1)]] [[plus(_x0, active(_x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[plus(_x0, _x1)]] We can thus remove the following rules: active(plus(X, s(Y))) => mark(U11(tt, Y, X)) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): mark(U11(X, Y, Z)) >? active(U11(mark(X), Y, Z)) mark(tt) >? active(tt) mark(U12(X, Y, Z)) >? active(U12(mark(X), Y, Z)) mark(s(X)) >? active(s(mark(X))) mark(plus(X, Y)) >? active(plus(mark(X), mark(Y))) mark(0) >? active(0) U11(mark(X), Y, Z) >? U11(X, Y, Z) U11(X, mark(Y), Z) >? U11(X, Y, Z) U11(X, Y, mark(Z)) >? U11(X, Y, Z) U11(active(X), Y, Z) >? U11(X, Y, Z) U11(X, active(Y), Z) >? U11(X, Y, Z) U11(X, Y, active(Z)) >? U11(X, Y, Z) U12(mark(X), Y, Z) >? U12(X, Y, Z) U12(X, mark(Y), Z) >? U12(X, Y, Z) U12(X, Y, mark(Z)) >? U12(X, Y, Z) U12(active(X), Y, Z) >? U12(X, Y, Z) U12(X, active(Y), Z) >? U12(X, Y, Z) U12(X, Y, active(Z)) >? U12(X, Y, Z) s(mark(X)) >? s(X) s(active(X)) >? s(X) plus(mark(X), Y) >? plus(X, Y) plus(X, mark(Y)) >? plus(X, Y) plus(active(X), Y) >? plus(X, Y) plus(X, active(Y)) >? plus(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 2 U11 = \y0y1y2.y0 + 2y1 + 2y2 U12 = \y0y1y2.1 + y0 + y1 + y2 active = \y0.y0 mark = \y0.2y0 plus = \y0y1.1 + y0 + 2y1 s = \y0.y0 tt = 1 Using this interpretation, the requirements translate to: [[mark(U11(_x0, _x1, _x2))]] = 2x0 + 4x1 + 4x2 >= 2x0 + 2x1 + 2x2 = [[active(U11(mark(_x0), _x1, _x2))]] [[mark(tt)]] = 2 > 1 = [[active(tt)]] [[mark(U12(_x0, _x1, _x2))]] = 2 + 2x0 + 2x1 + 2x2 > 1 + x1 + x2 + 2x0 = [[active(U12(mark(_x0), _x1, _x2))]] [[mark(s(_x0))]] = 2x0 >= 2x0 = [[active(s(mark(_x0)))]] [[mark(plus(_x0, _x1))]] = 2 + 2x0 + 4x1 > 1 + 2x0 + 4x1 = [[active(plus(mark(_x0), mark(_x1)))]] [[mark(0)]] = 4 > 2 = [[active(0)]] [[U11(mark(_x0), _x1, _x2)]] = 2x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, mark(_x1), _x2)]] = x0 + 2x2 + 4x1 >= x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, mark(_x2))]] = x0 + 2x1 + 4x2 >= x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(active(_x0), _x1, _x2)]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, active(_x1), _x2)]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, active(_x2))]] = x0 + 2x1 + 2x2 >= x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U12(mark(_x0), _x1, _x2)]] = 1 + x1 + x2 + 2x0 >= 1 + x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, mark(_x1), _x2)]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, mark(_x2))]] = 1 + x0 + x1 + 2x2 >= 1 + x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(active(_x0), _x1, _x2)]] = 1 + x0 + x1 + x2 >= 1 + x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, active(_x1), _x2)]] = 1 + x0 + x1 + x2 >= 1 + x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, active(_x2))]] = 1 + x0 + x1 + x2 >= 1 + x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[s(mark(_x0))]] = 2x0 >= x0 = [[s(_x0)]] [[s(active(_x0))]] = x0 >= x0 = [[s(_x0)]] [[plus(mark(_x0), _x1)]] = 1 + 2x0 + 2x1 >= 1 + x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, mark(_x1))]] = 1 + x0 + 4x1 >= 1 + x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(active(_x0), _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, active(_x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[plus(_x0, _x1)]] We can thus remove the following rules: mark(tt) => active(tt) mark(U12(X, Y, Z)) => active(U12(mark(X), Y, Z)) mark(plus(X, Y)) => active(plus(mark(X), mark(Y))) mark(0) => active(0) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): mark(U11(X, Y, Z)) >? active(U11(mark(X), Y, Z)) mark(s(X)) >? active(s(mark(X))) U11(mark(X), Y, Z) >? U11(X, Y, Z) U11(X, mark(Y), Z) >? U11(X, Y, Z) U11(X, Y, mark(Z)) >? U11(X, Y, Z) U11(active(X), Y, Z) >? U11(X, Y, Z) U11(X, active(Y), Z) >? U11(X, Y, Z) U11(X, Y, active(Z)) >? U11(X, Y, Z) U12(mark(X), Y, Z) >? U12(X, Y, Z) U12(X, mark(Y), Z) >? U12(X, Y, Z) U12(X, Y, mark(Z)) >? U12(X, Y, Z) U12(active(X), Y, Z) >? U12(X, Y, Z) U12(X, active(Y), Z) >? U12(X, Y, Z) U12(X, Y, active(Z)) >? U12(X, Y, Z) s(mark(X)) >? s(X) s(active(X)) >? s(X) plus(mark(X), Y) >? plus(X, Y) plus(X, mark(Y)) >? plus(X, Y) plus(active(X), Y) >? plus(X, Y) plus(X, active(Y)) >? plus(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U11 = \y0y1y2.2y0 + 2y1 + 2y2 U12 = \y0y1y2.y0 + y2 + 2y1 active = \y0.y0 mark = \y0.2y0 plus = \y0y1.y0 + 2y1 s = \y0.2 + 2y0 Using this interpretation, the requirements translate to: [[mark(U11(_x0, _x1, _x2))]] = 4x0 + 4x1 + 4x2 >= 2x1 + 2x2 + 4x0 = [[active(U11(mark(_x0), _x1, _x2))]] [[mark(s(_x0))]] = 4 + 4x0 > 2 + 4x0 = [[active(s(mark(_x0)))]] [[U11(mark(_x0), _x1, _x2)]] = 2x1 + 2x2 + 4x0 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, mark(_x1), _x2)]] = 2x0 + 2x2 + 4x1 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, mark(_x2))]] = 2x0 + 2x1 + 4x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(active(_x0), _x1, _x2)]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, active(_x1), _x2)]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, active(_x2))]] = 2x0 + 2x1 + 2x2 >= 2x0 + 2x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U12(mark(_x0), _x1, _x2)]] = x2 + 2x0 + 2x1 >= x0 + x2 + 2x1 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, mark(_x1), _x2)]] = x0 + x2 + 4x1 >= x0 + x2 + 2x1 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, mark(_x2))]] = x0 + 2x1 + 2x2 >= x0 + x2 + 2x1 = [[U12(_x0, _x1, _x2)]] [[U12(active(_x0), _x1, _x2)]] = x0 + x2 + 2x1 >= x0 + x2 + 2x1 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, active(_x1), _x2)]] = x0 + x2 + 2x1 >= x0 + x2 + 2x1 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, active(_x2))]] = x0 + x2 + 2x1 >= x0 + x2 + 2x1 = [[U12(_x0, _x1, _x2)]] [[s(mark(_x0))]] = 2 + 4x0 >= 2 + 2x0 = [[s(_x0)]] [[s(active(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[s(_x0)]] [[plus(mark(_x0), _x1)]] = 2x0 + 2x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, mark(_x1))]] = x0 + 4x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(active(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, active(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] We can thus remove the following rules: mark(s(X)) => active(s(mark(X))) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): mark(U11(X, Y, Z)) >? active(U11(mark(X), Y, Z)) U11(mark(X), Y, Z) >? U11(X, Y, Z) U11(X, mark(Y), Z) >? U11(X, Y, Z) U11(X, Y, mark(Z)) >? U11(X, Y, Z) U11(active(X), Y, Z) >? U11(X, Y, Z) U11(X, active(Y), Z) >? U11(X, Y, Z) U11(X, Y, active(Z)) >? U11(X, Y, Z) U12(mark(X), Y, Z) >? U12(X, Y, Z) U12(X, mark(Y), Z) >? U12(X, Y, Z) U12(X, Y, mark(Z)) >? U12(X, Y, Z) U12(active(X), Y, Z) >? U12(X, Y, Z) U12(X, active(Y), Z) >? U12(X, Y, Z) U12(X, Y, active(Z)) >? U12(X, Y, Z) s(mark(X)) >? s(X) s(active(X)) >? s(X) plus(mark(X), Y) >? plus(X, Y) plus(X, mark(Y)) >? plus(X, Y) plus(active(X), Y) >? plus(X, Y) plus(X, active(Y)) >? plus(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U11 = \y0y1y2.2 + y0 + y1 + 2y2 U12 = \y0y1y2.y0 + y1 + 2y2 active = \y0.y0 mark = \y0.2y0 plus = \y0y1.y0 + 2y1 s = \y0.y0 Using this interpretation, the requirements translate to: [[mark(U11(_x0, _x1, _x2))]] = 4 + 2x0 + 2x1 + 4x2 > 2 + x1 + 2x0 + 2x2 = [[active(U11(mark(_x0), _x1, _x2))]] [[U11(mark(_x0), _x1, _x2)]] = 2 + x1 + 2x0 + 2x2 >= 2 + x0 + x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, mark(_x1), _x2)]] = 2 + x0 + 2x1 + 2x2 >= 2 + x0 + x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, mark(_x2))]] = 2 + x0 + x1 + 4x2 >= 2 + x0 + x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(active(_x0), _x1, _x2)]] = 2 + x0 + x1 + 2x2 >= 2 + x0 + x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, active(_x1), _x2)]] = 2 + x0 + x1 + 2x2 >= 2 + x0 + x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, active(_x2))]] = 2 + x0 + x1 + 2x2 >= 2 + x0 + x1 + 2x2 = [[U11(_x0, _x1, _x2)]] [[U12(mark(_x0), _x1, _x2)]] = x1 + 2x0 + 2x2 >= x0 + x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, mark(_x1), _x2)]] = x0 + 2x1 + 2x2 >= x0 + x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, mark(_x2))]] = x0 + x1 + 4x2 >= x0 + x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(active(_x0), _x1, _x2)]] = x0 + x1 + 2x2 >= x0 + x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, active(_x1), _x2)]] = x0 + x1 + 2x2 >= x0 + x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, active(_x2))]] = x0 + x1 + 2x2 >= x0 + x1 + 2x2 = [[U12(_x0, _x1, _x2)]] [[s(mark(_x0))]] = 2x0 >= x0 = [[s(_x0)]] [[s(active(_x0))]] = x0 >= x0 = [[s(_x0)]] [[plus(mark(_x0), _x1)]] = 2x0 + 2x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, mark(_x1))]] = x0 + 4x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(active(_x0), _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] [[plus(_x0, active(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] We can thus remove the following rules: mark(U11(X, Y, Z)) => active(U11(mark(X), Y, Z)) We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): U11(mark(X), Y, Z) >? U11(X, Y, Z) U11(X, mark(Y), Z) >? U11(X, Y, Z) U11(X, Y, mark(Z)) >? U11(X, Y, Z) U11(active(X), Y, Z) >? U11(X, Y, Z) U11(X, active(Y), Z) >? U11(X, Y, Z) U11(X, Y, active(Z)) >? U11(X, Y, Z) U12(mark(X), Y, Z) >? U12(X, Y, Z) U12(X, mark(Y), Z) >? U12(X, Y, Z) U12(X, Y, mark(Z)) >? U12(X, Y, Z) U12(active(X), Y, Z) >? U12(X, Y, Z) U12(X, active(Y), Z) >? U12(X, Y, Z) U12(X, Y, active(Z)) >? U12(X, Y, Z) s(mark(X)) >? s(X) s(active(X)) >? s(X) plus(mark(X), Y) >? plus(X, Y) plus(X, mark(Y)) >? plus(X, Y) plus(active(X), Y) >? plus(X, Y) plus(X, active(Y)) >? plus(X, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: U11 = \y0y1y2.y0 + y1 + y2 U12 = \y0y1y2.y0 + y1 + y2 active = \y0.3 + 3y0 mark = \y0.3 + 3y0 plus = \y0y1.y0 + y1 s = \y0.y0 Using this interpretation, the requirements translate to: [[U11(mark(_x0), _x1, _x2)]] = 3 + x1 + x2 + 3x0 > x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, mark(_x1), _x2)]] = 3 + x0 + x2 + 3x1 > x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, mark(_x2))]] = 3 + x0 + x1 + 3x2 > x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(active(_x0), _x1, _x2)]] = 3 + x1 + x2 + 3x0 > x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, active(_x1), _x2)]] = 3 + x0 + x2 + 3x1 > x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U11(_x0, _x1, active(_x2))]] = 3 + x0 + x1 + 3x2 > x0 + x1 + x2 = [[U11(_x0, _x1, _x2)]] [[U12(mark(_x0), _x1, _x2)]] = 3 + x1 + x2 + 3x0 > x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, mark(_x1), _x2)]] = 3 + x0 + x2 + 3x1 > x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, mark(_x2))]] = 3 + x0 + x1 + 3x2 > x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(active(_x0), _x1, _x2)]] = 3 + x1 + x2 + 3x0 > x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, active(_x1), _x2)]] = 3 + x0 + x2 + 3x1 > x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[U12(_x0, _x1, active(_x2))]] = 3 + x0 + x1 + 3x2 > x0 + x1 + x2 = [[U12(_x0, _x1, _x2)]] [[s(mark(_x0))]] = 3 + 3x0 > x0 = [[s(_x0)]] [[s(active(_x0))]] = 3 + 3x0 > x0 = [[s(_x0)]] [[plus(mark(_x0), _x1)]] = 3 + x1 + 3x0 > x0 + x1 = [[plus(_x0, _x1)]] [[plus(_x0, mark(_x1))]] = 3 + x0 + 3x1 > x0 + x1 = [[plus(_x0, _x1)]] [[plus(active(_x0), _x1)]] = 3 + x1 + 3x0 > x0 + x1 = [[plus(_x0, _x1)]] [[plus(_x0, active(_x1))]] = 3 + x0 + 3x1 > x0 + x1 = [[plus(_x0, _x1)]] We can thus remove the following rules: U11(mark(X), Y, Z) => U11(X, Y, Z) U11(X, mark(Y), Z) => U11(X, Y, Z) U11(X, Y, mark(Z)) => U11(X, Y, Z) U11(active(X), Y, Z) => U11(X, Y, Z) U11(X, active(Y), Z) => U11(X, Y, Z) U11(X, Y, active(Z)) => U11(X, Y, Z) U12(mark(X), Y, Z) => U12(X, Y, Z) U12(X, mark(Y), Z) => U12(X, Y, Z) U12(X, Y, mark(Z)) => U12(X, Y, Z) U12(active(X), Y, Z) => U12(X, Y, Z) U12(X, active(Y), Z) => U12(X, Y, Z) U12(X, Y, active(Z)) => U12(X, Y, Z) s(mark(X)) => s(X) s(active(X)) => s(X) plus(mark(X), Y) => plus(X, Y) plus(X, mark(Y)) => plus(X, Y) plus(active(X), Y) => plus(X, Y) plus(X, active(Y)) => plus(X, Y) All rules were succesfully removed. Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold. +++ Citations +++ [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012.