We consider the system zipWith. Alphabet: 0 : [] --> nat cons : [nat * list] --> list false : [] --> bool gcd : [nat * nat] --> nat gcdlists : [list * list] --> list if : [bool * nat * nat] --> nat le : [nat * nat] --> bool minus : [nat * nat] --> nat nil : [] --> list s : [nat] --> nat true : [] --> bool zipWith : [nat -> nat -> nat * list * list] --> list Rules: le(0, x) => true le(s(x), 0) => false le(s(x), s(y)) => le(x, y) minus(x, 0) => x minus(s(x), s(y)) => minus(x, y) gcd(0, x) => 0 gcd(s(x), 0) => 0 gcd(s(x), s(y)) => if(le(y, x), s(x), s(y)) if(true, s(x), s(y)) => gcd(minus(x, y), s(y)) if(false, s(x), s(y)) => gcd(minus(y, x), s(x)) zipWith(f, x, nil) => nil zipWith(f, nil, x) => nil zipWith(f, cons(x, y), cons(z, u)) => cons(f x z, zipWith(f, y, u)) gcdlists(x, y) => zipWith(/\z./\u.gcd(z, u), x, y) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]). We thus obtain the following dependency pair problem (P_0, R_0, static, formative): Dependency Pairs P_0: 0] le#(s(X), s(Y)) =#> le#(X, Y) 1] minus#(s(X), s(Y)) =#> minus#(X, Y) 2] gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) 3] gcd#(s(X), s(Y)) =#> le#(Y, X) 4] if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) 5] if#(true, s(X), s(Y)) =#> minus#(X, Y) 6] if#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) 7] if#(false, s(X), s(Y)) =#> minus#(Y, X) 8] zipWith#(F, cons(X, Y), cons(Z, U)) =#> zipWith#(F, Y, U) 9] gcdlists#(X, Y) =#> zipWith#(/\x./\y.gcd(x, y), X, Y) 10] gcdlists#(X, Y) =#> gcd#(Z, U) Rules R_0: le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) minus(X, 0) => X minus(s(X), s(Y)) => minus(X, Y) gcd(0, X) => 0 gcd(s(X), 0) => 0 gcd(s(X), s(Y)) => if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) => gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) => gcd(minus(Y, X), s(X)) zipWith(F, X, nil) => nil zipWith(F, nil, X) => nil zipWith(F, cons(X, Y), cons(Z, U)) => cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) => zipWith(/\x./\y.gcd(x, y), X, Y) Thus, the original system is terminating if (P_0, R_0, static, formative) is finite. We consider the dependency pair problem (P_0, R_0, static, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: le#(s(X), s(Y)) >? le#(X, Y) minus#(s(X), s(Y)) >? minus#(X, Y) gcd#(s(X), s(Y)) >? if#(le(Y, X), s(X), s(Y)) gcd#(s(X), s(Y)) >? le#(Y, X) if#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) if#(true, s(X), s(Y)) >? minus#(X, Y) if#(false, s(X), s(Y)) >? gcd#(minus(Y, X), s(X)) if#(false, s(X), s(Y)) >? minus#(Y, X) zipWith#(F, cons(X, Y), cons(Z, U)) >? zipWith#(F, Y, U) gcdlists#(X, Y) >? zipWith#(/\x./\y.gcd(x, y), X, Y) gcdlists#(X, Y) >? gcd#(Z, U) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) minus(X, 0) >= X minus(s(X), s(Y)) >= minus(X, Y) gcd(0, X) >= 0 gcd(s(X), 0) >= 0 gcd(s(X), s(Y)) >= if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) zipWith(F, X, nil) >= nil zipWith(F, nil, X) >= nil zipWith(F, cons(X, Y), cons(Z, U)) >= cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) >= zipWith(/\x./\y.gcd(x, y), X, Y) We apply [Kop12, Thm. 6.75] and use the following argument functions: pi( gcdlists(X, Y) ) = #argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: #argfun-gcdlists# = \y0.3 + y0 0 = 0 cons = \y0y1.y0 false = 0 gcd = \y0y1.0 gcd# = \y0y1.0 gcdlists = \y0y1.0 gcdlists# = \y0y1.3 + 2y0 + 2y1 if = \y0y1y2.0 if# = \y0y1y2.0 le = \y0y1.0 le# = \y0y1.0 minus = \y0y1.y0 minus# = \y0y1.0 nil = 1 s = \y0.y0 true = 0 zipWith = \G0y1y2.2y1 + 2y2 + 2G0(y1,y2) zipWith# = \G0y1y2.0 Using this interpretation, the requirements translate to: [[le#(s(_x0), s(_x1))]] = 0 >= 0 = [[le#(_x0, _x1)]] [[minus#(s(_x0), s(_x1))]] = 0 >= 0 = [[minus#(_x0, _x1)]] [[gcd#(s(_x0), s(_x1))]] = 0 >= 0 = [[if#(le(_x1, _x0), s(_x0), s(_x1))]] [[gcd#(s(_x0), s(_x1))]] = 0 >= 0 = [[le#(_x1, _x0)]] [[if#(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[if#(true, s(_x0), s(_x1))]] = 0 >= 0 = [[minus#(_x0, _x1)]] [[if#(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd#(minus(_x1, _x0), s(_x0))]] [[if#(false, s(_x0), s(_x1))]] = 0 >= 0 = [[minus#(_x1, _x0)]] [[zipWith#(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 0 >= 0 = [[zipWith#(_F0, _x2, _x4)]] [[gcdlists#(_x0, _x1)]] = 3 + 2x0 + 2x1 > 0 = [[zipWith#(/\x./\y.gcd(x, y), _x0, _x1)]] [[gcdlists#(_x0, _x1)]] = 3 + 2x0 + 2x1 > 0 = [[gcd#(_x2, _x3)]] [[le(0, _x0)]] = 0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 0 >= 0 = [[false]] [[le(s(_x0), s(_x1))]] = 0 >= 0 = [[le(_x0, _x1)]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(s(_x0), s(_x1))]] = x0 >= x0 = [[minus(_x0, _x1)]] [[gcd(0, _x0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), 0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), s(_x1))]] = 0 >= 0 = [[if(le(_x1, _x0), s(_x0), s(_x1))]] [[if(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x0, _x1), s(_x1))]] [[if(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x1, _x0), s(_x0))]] [[zipWith(_F0, _x1, nil)]] = 2 + 2x1 + 2F0(x1,1) >= 1 = [[nil]] [[zipWith(_F0, nil, _x1)]] = 2 + 2x1 + 2F0(1,x1) >= 1 = [[nil]] [[zipWith(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 2x1 + 2x3 + 2F0(x1,x3) >= F0(x1,x3) = [[cons(_F0 _x1 _x3, zipWith(_F0, _x2, _x4))]] [[#argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), _x0, _x1))]] = 3 + 2x0 + 2x1 >= 2x0 + 2x1 = [[zipWith(/\x./\y.gcd(x, y), _x0, _x1)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_0, static, formative) by (P_1, R_0, static, formative), where P_1 consists of: le#(s(X), s(Y)) =#> le#(X, Y) minus#(s(X), s(Y)) =#> minus#(X, Y) gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) gcd#(s(X), s(Y)) =#> le#(Y, X) if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) if#(true, s(X), s(Y)) =#> minus#(X, Y) if#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) if#(false, s(X), s(Y)) =#> minus#(Y, X) zipWith#(F, cons(X, Y), cons(Z, U)) =#> zipWith#(F, Y, U) Thus, the original system is terminating if (P_1, R_0, static, formative) is finite. We consider the dependency pair problem (P_1, R_0, static, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: le#(s(X), s(Y)) >? le#(X, Y) minus#(s(X), s(Y)) >? minus#(X, Y) gcd#(s(X), s(Y)) >? if#(le(Y, X), s(X), s(Y)) gcd#(s(X), s(Y)) >? le#(Y, X) if#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) if#(true, s(X), s(Y)) >? minus#(X, Y) if#(false, s(X), s(Y)) >? gcd#(minus(Y, X), s(X)) if#(false, s(X), s(Y)) >? minus#(Y, X) zipWith#(F, cons(X, Y), cons(Z, U)) >? zipWith#(F, Y, U) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) minus(X, 0) >= X minus(s(X), s(Y)) >= minus(X, Y) gcd(0, X) >= 0 gcd(s(X), 0) >= 0 gcd(s(X), s(Y)) >= if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) zipWith(F, X, nil) >= nil zipWith(F, nil, X) >= nil zipWith(F, cons(X, Y), cons(Z, U)) >= cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) >= zipWith(/\x./\y.gcd(x, y), X, Y) We apply [Kop12, Thm. 6.75] and use the following argument functions: pi( gcdlists(X, Y) ) = #argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: #argfun-gcdlists# = \y0.3 + y0 0 = 0 cons = \y0y1.3 + y1 false = 0 gcd = \y0y1.0 gcd# = \y0y1.0 gcdlists = \y0y1.0 if = \y0y1y2.0 if# = \y0y1y2.0 le = \y0y1.0 le# = \y0y1.0 minus = \y0y1.y0 minus# = \y0y1.0 nil = 1 s = \y0.y0 true = 0 zipWith = \G0y1y2.2y2 + 3y1 zipWith# = \G0y1y2.y2 Using this interpretation, the requirements translate to: [[le#(s(_x0), s(_x1))]] = 0 >= 0 = [[le#(_x0, _x1)]] [[minus#(s(_x0), s(_x1))]] = 0 >= 0 = [[minus#(_x0, _x1)]] [[gcd#(s(_x0), s(_x1))]] = 0 >= 0 = [[if#(le(_x1, _x0), s(_x0), s(_x1))]] [[gcd#(s(_x0), s(_x1))]] = 0 >= 0 = [[le#(_x1, _x0)]] [[if#(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[if#(true, s(_x0), s(_x1))]] = 0 >= 0 = [[minus#(_x0, _x1)]] [[if#(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd#(minus(_x1, _x0), s(_x0))]] [[if#(false, s(_x0), s(_x1))]] = 0 >= 0 = [[minus#(_x1, _x0)]] [[zipWith#(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 3 + x4 > x4 = [[zipWith#(_F0, _x2, _x4)]] [[le(0, _x0)]] = 0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 0 >= 0 = [[false]] [[le(s(_x0), s(_x1))]] = 0 >= 0 = [[le(_x0, _x1)]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(s(_x0), s(_x1))]] = x0 >= x0 = [[minus(_x0, _x1)]] [[gcd(0, _x0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), 0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), s(_x1))]] = 0 >= 0 = [[if(le(_x1, _x0), s(_x0), s(_x1))]] [[if(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x0, _x1), s(_x1))]] [[if(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x1, _x0), s(_x0))]] [[zipWith(_F0, _x1, nil)]] = 2 + 3x1 >= 1 = [[nil]] [[zipWith(_F0, nil, _x1)]] = 3 + 2x1 >= 1 = [[nil]] [[zipWith(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 15 + 2x4 + 3x2 >= 3 + 2x4 + 3x2 = [[cons(_F0 _x1 _x3, zipWith(_F0, _x2, _x4))]] [[#argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), _x0, _x1))]] = 3 + 2x1 + 3x0 >= 2x1 + 3x0 = [[zipWith(/\x./\y.gcd(x, y), _x0, _x1)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, static, formative) by (P_2, R_0, static, formative), where P_2 consists of: le#(s(X), s(Y)) =#> le#(X, Y) minus#(s(X), s(Y)) =#> minus#(X, Y) gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) gcd#(s(X), s(Y)) =#> le#(Y, X) if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) if#(true, s(X), s(Y)) =#> minus#(X, Y) if#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) if#(false, s(X), s(Y)) =#> minus#(Y, X) Thus, the original system is terminating if (P_2, R_0, static, formative) is finite. We consider the dependency pair problem (P_2, R_0, static, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: le#(s(X), s(Y)) >? le#(X, Y) minus#(s(X), s(Y)) >? minus#(X, Y) gcd#(s(X), s(Y)) >? if#(le(Y, X), s(X), s(Y)) gcd#(s(X), s(Y)) >? le#(Y, X) if#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) if#(true, s(X), s(Y)) >? minus#(X, Y) if#(false, s(X), s(Y)) >? gcd#(minus(Y, X), s(X)) if#(false, s(X), s(Y)) >? minus#(Y, X) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) minus(X, 0) >= X minus(s(X), s(Y)) >= minus(X, Y) gcd(0, X) >= 0 gcd(s(X), 0) >= 0 gcd(s(X), s(Y)) >= if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) zipWith(F, X, nil) >= nil zipWith(F, nil, X) >= nil zipWith(F, cons(X, Y), cons(Z, U)) >= cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) >= zipWith(/\x./\y.gcd(x, y), X, Y) We apply [Kop12, Thm. 6.75] and use the following argument functions: pi( gcdlists(X, Y) ) = #argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: #argfun-gcdlists# = \y0.3 + y0 0 = 0 cons = \y0y1.0 false = 0 gcd = \y0y1.0 gcd# = \y0y1.1 gcdlists = \y0y1.0 if = \y0y1y2.0 if# = \y0y1y2.1 le = \y0y1.y1 le# = \y0y1.0 minus = \y0y1.y0 + 2y1 minus# = \y0y1.0 nil = 0 s = \y0.2y0 true = 0 zipWith = \G0y1y2.0 Using this interpretation, the requirements translate to: [[le#(s(_x0), s(_x1))]] = 0 >= 0 = [[le#(_x0, _x1)]] [[minus#(s(_x0), s(_x1))]] = 0 >= 0 = [[minus#(_x0, _x1)]] [[gcd#(s(_x0), s(_x1))]] = 1 >= 1 = [[if#(le(_x1, _x0), s(_x0), s(_x1))]] [[gcd#(s(_x0), s(_x1))]] = 1 > 0 = [[le#(_x1, _x0)]] [[if#(true, s(_x0), s(_x1))]] = 1 >= 1 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[if#(true, s(_x0), s(_x1))]] = 1 > 0 = [[minus#(_x0, _x1)]] [[if#(false, s(_x0), s(_x1))]] = 1 >= 1 = [[gcd#(minus(_x1, _x0), s(_x0))]] [[if#(false, s(_x0), s(_x1))]] = 1 > 0 = [[minus#(_x1, _x0)]] [[le(0, _x0)]] = x0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 0 >= 0 = [[false]] [[le(s(_x0), s(_x1))]] = 2x1 >= x1 = [[le(_x0, _x1)]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(s(_x0), s(_x1))]] = 2x0 + 4x1 >= x0 + 2x1 = [[minus(_x0, _x1)]] [[gcd(0, _x0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), 0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), s(_x1))]] = 0 >= 0 = [[if(le(_x1, _x0), s(_x0), s(_x1))]] [[if(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x0, _x1), s(_x1))]] [[if(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x1, _x0), s(_x0))]] [[zipWith(_F0, _x1, nil)]] = 0 >= 0 = [[nil]] [[zipWith(_F0, nil, _x1)]] = 0 >= 0 = [[nil]] [[zipWith(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 0 >= 0 = [[cons(_F0 _x1 _x3, zipWith(_F0, _x2, _x4))]] [[#argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), _x0, _x1))]] = 3 >= 0 = [[zipWith(/\x./\y.gcd(x, y), _x0, _x1)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_0, static, formative) by (P_3, R_0, static, formative), where P_3 consists of: le#(s(X), s(Y)) =#> le#(X, Y) minus#(s(X), s(Y)) =#> minus#(X, Y) gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) if#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) Thus, the original system is terminating if (P_3, R_0, static, formative) is finite. We consider the dependency pair problem (P_3, R_0, static, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: le#(s(X), s(Y)) >? le#(X, Y) minus#(s(X), s(Y)) >? minus#(X, Y) gcd#(s(X), s(Y)) >? if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) if#(false, s(X), s(Y)) >? gcd#(minus(Y, X), s(X)) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) minus(X, 0) >= X minus(s(X), s(Y)) >= minus(X, Y) gcd(0, X) >= 0 gcd(s(X), 0) >= 0 gcd(s(X), s(Y)) >= if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) zipWith(F, X, nil) >= nil zipWith(F, nil, X) >= nil zipWith(F, cons(X, Y), cons(Z, U)) >= cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) >= zipWith(/\x./\y.gcd(x, y), X, Y) We apply [Kop12, Thm. 6.75] and use the following argument functions: pi( gcdlists(X, Y) ) = #argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: #argfun-gcdlists# = \y0.3 + y0 0 = 0 cons = \y0y1.y1 false = 0 gcd = \y0y1.0 gcd# = \y0y1.0 gcdlists = \y0y1.0 if = \y0y1y2.0 if# = \y0y1y2.0 le = \y0y1.0 le# = \y0y1.y0 minus = \y0y1.2y0 + 2y1 minus# = \y0y1.0 nil = 0 s = \y0.2 + 2y0 true = 0 zipWith = \G0y1y2.0 Using this interpretation, the requirements translate to: [[le#(s(_x0), s(_x1))]] = 2 + 2x0 > x0 = [[le#(_x0, _x1)]] [[minus#(s(_x0), s(_x1))]] = 0 >= 0 = [[minus#(_x0, _x1)]] [[gcd#(s(_x0), s(_x1))]] = 0 >= 0 = [[if#(le(_x1, _x0), s(_x0), s(_x1))]] [[if#(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[if#(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd#(minus(_x1, _x0), s(_x0))]] [[le(0, _x0)]] = 0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 0 >= 0 = [[false]] [[le(s(_x0), s(_x1))]] = 0 >= 0 = [[le(_x0, _x1)]] [[minus(_x0, 0)]] = 2x0 >= x0 = [[_x0]] [[minus(s(_x0), s(_x1))]] = 8 + 4x0 + 4x1 >= 2x0 + 2x1 = [[minus(_x0, _x1)]] [[gcd(0, _x0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), 0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), s(_x1))]] = 0 >= 0 = [[if(le(_x1, _x0), s(_x0), s(_x1))]] [[if(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x0, _x1), s(_x1))]] [[if(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x1, _x0), s(_x0))]] [[zipWith(_F0, _x1, nil)]] = 0 >= 0 = [[nil]] [[zipWith(_F0, nil, _x1)]] = 0 >= 0 = [[nil]] [[zipWith(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 0 >= 0 = [[cons(_F0 _x1 _x3, zipWith(_F0, _x2, _x4))]] [[#argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), _x0, _x1))]] = 3 >= 0 = [[zipWith(/\x./\y.gcd(x, y), _x0, _x1)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_0, static, formative) by (P_4, R_0, static, formative), where P_4 consists of: minus#(s(X), s(Y)) =#> minus#(X, Y) gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) if#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) Thus, the original system is terminating if (P_4, R_0, static, formative) is finite. We consider the dependency pair problem (P_4, R_0, static, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: minus#(s(X), s(Y)) >? minus#(X, Y) gcd#(s(X), s(Y)) >? if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) if#(false, s(X), s(Y)) >? gcd#(minus(Y, X), s(X)) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) minus(X, 0) >= X minus(s(X), s(Y)) >= minus(X, Y) gcd(0, X) >= 0 gcd(s(X), 0) >= 0 gcd(s(X), s(Y)) >= if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) zipWith(F, X, nil) >= nil zipWith(F, nil, X) >= nil zipWith(F, cons(X, Y), cons(Z, U)) >= cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) >= zipWith(/\x./\y.gcd(x, y), X, Y) We apply [Kop12, Thm. 6.75] and use the following argument functions: pi( gcdlists(X, Y) ) = #argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: #argfun-gcdlists# = \y0.3 + y0 0 = 0 cons = \y0y1.0 false = 0 gcd = \y0y1.0 gcd# = \y0y1.0 gcdlists = \y0y1.0 if = \y0y1y2.0 if# = \y0y1y2.0 le = \y0y1.0 minus = \y0y1.y0 minus# = \y0y1.y0 nil = 0 s = \y0.2 + 2y0 true = 0 zipWith = \G0y1y2.0 Using this interpretation, the requirements translate to: [[minus#(s(_x0), s(_x1))]] = 2 + 2x0 > x0 = [[minus#(_x0, _x1)]] [[gcd#(s(_x0), s(_x1))]] = 0 >= 0 = [[if#(le(_x1, _x0), s(_x0), s(_x1))]] [[if#(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[if#(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd#(minus(_x1, _x0), s(_x0))]] [[le(0, _x0)]] = 0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 0 >= 0 = [[false]] [[le(s(_x0), s(_x1))]] = 0 >= 0 = [[le(_x0, _x1)]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(s(_x0), s(_x1))]] = 2 + 2x0 >= x0 = [[minus(_x0, _x1)]] [[gcd(0, _x0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), 0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), s(_x1))]] = 0 >= 0 = [[if(le(_x1, _x0), s(_x0), s(_x1))]] [[if(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x0, _x1), s(_x1))]] [[if(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x1, _x0), s(_x0))]] [[zipWith(_F0, _x1, nil)]] = 0 >= 0 = [[nil]] [[zipWith(_F0, nil, _x1)]] = 0 >= 0 = [[nil]] [[zipWith(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 0 >= 0 = [[cons(_F0 _x1 _x3, zipWith(_F0, _x2, _x4))]] [[#argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), _x0, _x1))]] = 3 >= 0 = [[zipWith(/\x./\y.gcd(x, y), _x0, _x1)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_4, R_0, static, formative) by (P_5, R_0, static, formative), where P_5 consists of: gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) if#(false, s(X), s(Y)) =#> gcd#(minus(Y, X), s(X)) Thus, the original system is terminating if (P_5, R_0, static, formative) is finite. We consider the dependency pair problem (P_5, R_0, static, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: gcd#(s(X), s(Y)) >? if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) if#(false, s(X), s(Y)) >? gcd#(minus(Y, X), s(X)) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) minus(X, 0) >= X minus(s(X), s(Y)) >= minus(X, Y) gcd(0, X) >= 0 gcd(s(X), 0) >= 0 gcd(s(X), s(Y)) >= if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) zipWith(F, X, nil) >= nil zipWith(F, nil, X) >= nil zipWith(F, cons(X, Y), cons(Z, U)) >= cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) >= zipWith(/\x./\y.gcd(x, y), X, Y) We apply [Kop12, Thm. 6.75] and use the following argument functions: pi( gcdlists(X, Y) ) = #argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: #argfun-gcdlists# = \y0.3 + y0 0 = 1 cons = \y0y1.0 false = 1 gcd = \y0y1.1 gcd# = \y0y1.y1 + 2y0 gcdlists = \y0y1.0 if = \y0y1y2.1 if# = \y0y1y2.y1 + y2 + 2y0 le = \y0y1.y1 minus = \y0y1.y0 nil = 0 s = \y0.2y0 true = 0 zipWith = \G0y1y2.2G0(y1,y1) Using this interpretation, the requirements translate to: [[gcd#(s(_x0), s(_x1))]] = 2x1 + 4x0 >= 2x1 + 4x0 = [[if#(le(_x1, _x0), s(_x0), s(_x1))]] [[if#(true, s(_x0), s(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[if#(false, s(_x0), s(_x1))]] = 2 + 2x0 + 2x1 > 2x0 + 2x1 = [[gcd#(minus(_x1, _x0), s(_x0))]] [[le(0, _x0)]] = x0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 1 >= 1 = [[false]] [[le(s(_x0), s(_x1))]] = 2x1 >= x1 = [[le(_x0, _x1)]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(s(_x0), s(_x1))]] = 2x0 >= x0 = [[minus(_x0, _x1)]] [[gcd(0, _x0)]] = 1 >= 1 = [[0]] [[gcd(s(_x0), 0)]] = 1 >= 1 = [[0]] [[gcd(s(_x0), s(_x1))]] = 1 >= 1 = [[if(le(_x1, _x0), s(_x0), s(_x1))]] [[if(true, s(_x0), s(_x1))]] = 1 >= 1 = [[gcd(minus(_x0, _x1), s(_x1))]] [[if(false, s(_x0), s(_x1))]] = 1 >= 1 = [[gcd(minus(_x1, _x0), s(_x0))]] [[zipWith(_F0, _x1, nil)]] = 2F0(x1,x1) >= 0 = [[nil]] [[zipWith(_F0, nil, _x1)]] = 2F0(0,0) >= 0 = [[nil]] [[zipWith(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 2F0(0,0) >= 0 = [[cons(_F0 _x1 _x3, zipWith(_F0, _x2, _x4))]] [[#argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), _x0, _x1))]] = 5 >= 2 = [[zipWith(/\x./\y.gcd(x, y), _x0, _x1)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_0, static, formative) by (P_6, R_0, static, formative), where P_6 consists of: gcd#(s(X), s(Y)) =#> if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) =#> gcd#(minus(X, Y), s(Y)) Thus, the original system is terminating if (P_6, R_0, static, formative) is finite. We consider the dependency pair problem (P_6, R_0, static, formative). We will use the reduction pair processor [Kop12, Thm. 7.16]. It suffices to find a standard reduction pair [Kop12, Def. 6.69]. Thus, we must orient: gcd#(s(X), s(Y)) >? if#(le(Y, X), s(X), s(Y)) if#(true, s(X), s(Y)) >? gcd#(minus(X, Y), s(Y)) le(0, X) >= true le(s(X), 0) >= false le(s(X), s(Y)) >= le(X, Y) minus(X, 0) >= X minus(s(X), s(Y)) >= minus(X, Y) gcd(0, X) >= 0 gcd(s(X), 0) >= 0 gcd(s(X), s(Y)) >= if(le(Y, X), s(X), s(Y)) if(true, s(X), s(Y)) >= gcd(minus(X, Y), s(Y)) if(false, s(X), s(Y)) >= gcd(minus(Y, X), s(X)) zipWith(F, X, nil) >= nil zipWith(F, nil, X) >= nil zipWith(F, cons(X, Y), cons(Z, U)) >= cons(F X Z, zipWith(F, Y, U)) gcdlists(X, Y) >= zipWith(/\x./\y.gcd(x, y), X, Y) We apply [Kop12, Thm. 6.75] and use the following argument functions: pi( gcdlists(X, Y) ) = #argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), X, Y)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: #argfun-gcdlists# = \y0.3 + y0 0 = 0 cons = \y0y1.y1 false = 0 gcd = \y0y1.0 gcd# = \y0y1.2y0 gcdlists = \y0y1.0 if = \y0y1y2.0 if# = \y0y1y2.y1 le = \y0y1.0 minus = \y0y1.y0 nil = 0 s = \y0.2 + 2y0 true = 0 zipWith = \G0y1y2.0 Using this interpretation, the requirements translate to: [[gcd#(s(_x0), s(_x1))]] = 4 + 4x0 > 2 + 2x0 = [[if#(le(_x1, _x0), s(_x0), s(_x1))]] [[if#(true, s(_x0), s(_x1))]] = 2 + 2x0 > 2x0 = [[gcd#(minus(_x0, _x1), s(_x1))]] [[le(0, _x0)]] = 0 >= 0 = [[true]] [[le(s(_x0), 0)]] = 0 >= 0 = [[false]] [[le(s(_x0), s(_x1))]] = 0 >= 0 = [[le(_x0, _x1)]] [[minus(_x0, 0)]] = x0 >= x0 = [[_x0]] [[minus(s(_x0), s(_x1))]] = 2 + 2x0 >= x0 = [[minus(_x0, _x1)]] [[gcd(0, _x0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), 0)]] = 0 >= 0 = [[0]] [[gcd(s(_x0), s(_x1))]] = 0 >= 0 = [[if(le(_x1, _x0), s(_x0), s(_x1))]] [[if(true, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x0, _x1), s(_x1))]] [[if(false, s(_x0), s(_x1))]] = 0 >= 0 = [[gcd(minus(_x1, _x0), s(_x0))]] [[zipWith(_F0, _x1, nil)]] = 0 >= 0 = [[nil]] [[zipWith(_F0, nil, _x1)]] = 0 >= 0 = [[nil]] [[zipWith(_F0, cons(_x1, _x2), cons(_x3, _x4))]] = 0 >= 0 = [[cons(_F0 _x1 _x3, zipWith(_F0, _x2, _x4))]] [[#argfun-gcdlists#(zipWith(/\x./\y.gcd(x, y), _x0, _x1))]] = 3 >= 0 = [[zipWith(/\x./\y.gcd(x, y), _x0, _x1)]] By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_6, R_0) by ({}, R_0). By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed. As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. +++ Citations +++ [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. [Kop13] C. Kop. Static Dependency Pairs with Accessibility. Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013. [KusIsoSakBla09] K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui. Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems. In volume 92(10) of IEICE Transactions on Information and Systems. 2007--2015, 2009.