/export/starexec/sandbox2/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. Alphabet: 0 : [] --> a add : [a * a] --> a fact : [] --> a -> a mult : [] --> a -> a -> a rec : [a -> a -> a * a] --> a -> a s : [a] --> a Rules: add(0, x) => x add(s(x), y) => s(add(x, y)) mult 0 x => 0 mult s(x) y => add(mult x y, y) rec(f, x) 0 => x rec(f, x) s(y) => f s(y) (rec(f, x) y) fact => rec(mult, s(0)) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 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]): add(0, X) >? X add(s(X), Y) >? s(add(X, Y)) mult 0 X >? 0 mult s(X) Y >? add(mult X Y, Y) rec(F, X) 0 >? X rec(F, X) s(Y) >? F s(Y) (rec(F, X) Y) fact >? rec(mult, s(0)) about to try horpo We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ We choose Lex = {} and Mul = {@_{o -> o -> o}, @_{o -> o}, add, fact, mult, rec, s}, and the following precedence: fact > mult > add > s > rec > @_{o -> o -> o} > @_{o -> o} Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: add(_|_, X) >= X add(s(X), Y) > s(add(X, Y)) @_{o -> o}(@_{o -> o -> o}(mult, _|_), X) >= _|_ @_{o -> o}(@_{o -> o -> o}(mult, s(X)), Y) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) @_{o -> o}(rec(F, X), _|_) > X @_{o -> o}(rec(F, X), s(Y)) > @_{o -> o}(@_{o -> o -> o}(F, s(Y)), @_{o -> o}(rec(F, X), Y)) fact > rec(mult, s(_|_)) With these choices, we have: 1] add(_|_, X) >= X because [2], by (Star) 2] add*(_|_, X) >= X because [3], by (Select) 3] X >= X by (Meta) 4] add(s(X), Y) > s(add(X, Y)) because [5], by definition 5] add*(s(X), Y) >= s(add(X, Y)) because add > s and [6], by (Copy) 6] add*(s(X), Y) >= add(X, Y) because add in Mul, [7] and [10], by (Stat) 7] s(X) > X because [8], by definition 8] s*(X) >= X because [9], by (Select) 9] X >= X by (Meta) 10] Y >= Y by (Meta) 11] @_{o -> o}(@_{o -> o -> o}(mult, _|_), X) >= _|_ by (Bot) 12] @_{o -> o}(@_{o -> o -> o}(mult, s(X)), Y) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because [13], by (Star) 13] @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because [14], by (Select) 14] @_{o -> o -> o}(mult, s(X)) @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because [15] 15] @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because [16], by (Select) 16] @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because [17], by (Select) 17] @_{o -> o -> o}(mult, s(X)) @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because [18] 18] @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because [19], by (Select) 19] mult @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)) @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because [20] 20] mult*(@_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)), @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y))) >= add(@_{o -> o}(@_{o -> o -> o}(mult, X), Y), Y) because mult > add, [21] and [31], by (Copy) 21] mult*(@_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)), @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y))) >= @_{o -> o}(@_{o -> o -> o}(mult, X), Y) because [22], by (Select) 22] @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)) >= @_{o -> o}(@_{o -> o -> o}(mult, X), Y) because @_{o -> o -> o} > @_{o -> o}, [23] and [28], by (Copy) 23] @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)) >= @_{o -> o -> o}(mult, X) because @_{o -> o -> o} in Mul, [24] and [25], by (Stat) 24] mult >= mult by (Fun) 25] s(X) > X because [26], by definition 26] s*(X) >= X because [27], by (Select) 27] X >= X by (Meta) 28] @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)) >= Y because [29], by (Select) 29] @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y) >= Y because [30], by (Select) 30] Y >= Y by (Meta) 31] mult*(@_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y)), @_{o -> o -> o}*(mult, s(X), @_{o -> o}*(@_{o -> o -> o}(mult, s(X)), Y))) >= Y because [28], by (Select) 32] @_{o -> o}(rec(F, X), _|_) > X because [33], by definition 33] @_{o -> o}*(rec(F, X), _|_) >= X because [34], by (Select) 34] rec(F, X) @_{o -> o}*(rec(F, X), _|_) >= X because [35] 35] rec*(F, X, @_{o -> o}*(rec(F, X), _|_)) >= X because [36], by (Select) 36] X >= X by (Meta) 37] @_{o -> o}(rec(F, X), s(Y)) > @_{o -> o}(@_{o -> o -> o}(F, s(Y)), @_{o -> o}(rec(F, X), Y)) because [38], by definition 38] @_{o -> o}*(rec(F, X), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), @_{o -> o}(rec(F, X), Y)) because [39], by (Select) 39] rec(F, X) @_{o -> o}*(rec(F, X), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), @_{o -> o}(rec(F, X), Y)) because [40] 40] rec*(F, X, @_{o -> o}*(rec(F, X), s(Y))) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), @_{o -> o}(rec(F, X), Y)) because rec > @_{o -> o}, [41] and [48], by (Copy) 41] rec*(F, X, @_{o -> o}*(rec(F, X), s(Y))) >= @_{o -> o -> o}(F, s(Y)) because rec > @_{o -> o -> o}, [42] and [44], by (Copy) 42] rec*(F, X, @_{o -> o}*(rec(F, X), s(Y))) >= F because [43], by (Select) 43] F >= F by (Meta) 44] rec*(F, X, @_{o -> o}*(rec(F, X), s(Y))) >= s(Y) because [45], by (Select) 45] @_{o -> o}*(rec(F, X), s(Y)) >= s(Y) because [46], by (Select) 46] s(Y) >= s(Y) because s in Mul and [47], by (Fun) 47] Y >= Y by (Meta) 48] rec*(F, X, @_{o -> o}*(rec(F, X), s(Y))) >= @_{o -> o}(rec(F, X), Y) because [49], by (Select) 49] @_{o -> o}*(rec(F, X), s(Y)) >= @_{o -> o}(rec(F, X), Y) because @_{o -> o} in Mul, [50] and [53], by (Stat) 50] rec(F, X) >= rec(F, X) because rec in Mul, [51] and [52], by (Fun) 51] F >= F by (Meta) 52] X >= X by (Meta) 53] s(Y) > Y because [54], by definition 54] s*(Y) >= Y because [47], by (Select) 55] fact > rec(mult, s(_|_)) because [56], by definition 56] fact* >= rec(mult, s(_|_)) because fact > rec, [57] and [58], by (Copy) 57] fact* >= mult because fact > mult, by (Copy) 58] fact* >= s(_|_) because fact > s and [59], by (Copy) 59] fact* >= _|_ by (Bot) We can thus remove the following rules: add(s(X), Y) => s(add(X, Y)) rec(F, X) 0 => X rec(F, X) s(Y) => F s(Y) (rec(F, X) Y) fact => rec(mult, s(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]): add(0, X) >? X mult(0, X) >? 0 mult(s(X), Y) >? add(mult(X, Y), Y) about to try horpo We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ We choose Lex = {} and Mul = {add, mult, s}, and the following precedence: mult > add > s Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: add(_|_, X) >= X mult(_|_, X) >= _|_ mult(s(X), Y) > add(mult(X, Y), Y) With these choices, we have: 1] add(_|_, X) >= X because [2], by (Star) 2] add*(_|_, X) >= X because [3], by (Select) 3] X >= X by (Meta) 4] mult(_|_, X) >= _|_ by (Bot) 5] mult(s(X), Y) > add(mult(X, Y), Y) because [6], by definition 6] mult*(s(X), Y) >= add(mult(X, Y), Y) because mult > add, [7] and [12], by (Copy) 7] mult*(s(X), Y) >= mult(X, Y) because mult in Mul, [8] and [11], by (Stat) 8] s(X) > X because [9], by definition 9] s*(X) >= X because [10], by (Select) 10] X >= X by (Meta) 11] Y >= Y by (Meta) 12] mult*(s(X), Y) >= Y because [11], by (Select) We can thus remove the following rules: mult(s(X), Y) => add(mult(X, Y), 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]): add(0, X) >? X mult(0, X) >? 0 We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: 0 = 0 add = \y0y1.3 + y0 + y1 mult = \y0y1.3 + y1 + 3y0 Using this interpretation, the requirements translate to: [[add(0, _x0)]] = 3 + x0 > x0 = [[_x0]] [[mult(0, _x0)]] = 3 + x0 > 0 = [[0]] We can thus remove the following rules: add(0, X) => X mult(0, X) => 0 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.