124.16/123.48 MAYBE 124.16/123.49 We consider the system theBenchmark. 124.16/123.49 124.16/123.49 Alphabet: 124.16/123.49 124.16/123.49 0 : [] --> N 124.16/123.49 false : [] --> B 124.16/123.49 g : [] --> N -> B 124.16/123.49 g2 : [] --> N -> B 124.16/123.49 geq : [] --> N -> N -> B 124.16/123.49 h : [] --> N -> (N -> B) -> N -> B 124.16/123.49 h2 : [] --> N -> (N -> B) -> N -> B 124.16/123.49 iszero : [] --> N -> N -> B 124.16/123.49 pred : [] --> N -> N 124.16/123.49 rec : [] --> (N -> (N -> B) -> N -> B) -> B -> N -> B 124.16/123.49 s : [] --> N -> N 124.16/123.49 true : [] --> B 124.16/123.49 124.16/123.49 Rules: 124.16/123.49 124.16/123.49 rec f (i 0) => i 124.16/123.49 rec f (i (s x)) => f x (rec f (i x)) 124.16/123.49 g x => true 124.16/123.49 h x f y => false 124.16/123.49 iszero x y => rec h (g x) y 124.16/123.49 pred 0 => 0 124.16/123.49 pred (s x) => x 124.16/123.49 g2 x => iszero x 0 124.16/123.49 h2 x f y => f (pred y) 124.16/123.49 geq x y => rec h2 (g2 x) y 124.16/123.49 124.16/123.49 Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. 124.16/123.49 124.16/123.49 We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: 124.16/123.49 124.16/123.49 Alphabet: 124.16/123.49 124.16/123.49 0 : [] --> N 124.16/123.49 false : [] --> B 124.16/123.49 g : [] --> N -> B 124.16/123.49 g2 : [] --> N -> B 124.16/123.49 geq : [N] --> N -> B 124.16/123.49 h : [] --> N -> (N -> B) -> N -> B 124.16/123.49 h2 : [] --> N -> (N -> B) -> N -> B 124.16/123.49 iszero : [N] --> N -> B 124.16/123.49 pred : [N] --> N 124.16/123.49 rec : [N -> (N -> B) -> N -> B * B] --> N -> B 124.16/123.49 s : [N] --> N 124.16/123.49 true : [] --> B 124.16/123.49 ~AP1 : [N -> B * N] --> B 124.16/123.49 124.16/123.49 Rules: 124.16/123.49 124.16/123.49 rec(F, ~AP1(G, 0)) => G 124.16/123.49 rec(F, ~AP1(G, s(X))) => F X rec(F, ~AP1(G, X)) 124.16/123.49 g X => true 124.16/123.49 h X F Y => false 124.16/123.49 iszero(X) Y => ~AP1(rec(h, g X), Y) 124.16/123.49 pred(0) => 0 124.16/123.49 pred(s(X)) => X 124.16/123.49 g2 X => iszero(X) 0 124.16/123.49 h2 X F Y => ~AP1(F, pred(Y)) 124.16/123.49 geq(X) Y => ~AP1(rec(h2, g2 X), Y) 124.16/123.49 rec(F, g 0) => g 124.16/123.49 rec(F, g2 0) => g2 124.16/123.49 rec(F, geq(X) 0) => geq(X) 124.16/123.49 rec(F, h X G 0) => h X G 124.16/123.49 rec(F, h2 X G 0) => h2 X G 124.16/123.49 rec(F, iszero(X) 0) => iszero(X) 124.16/123.49 rec(F, g s(X)) => F X rec(F, g X) 124.16/123.49 rec(F, g2 s(X)) => F X rec(F, g2 X) 124.16/123.49 rec(F, geq(X) s(Y)) => F Y rec(F, geq(X) Y) 124.16/123.49 rec(F, h X G s(Y)) => F Y rec(F, h X G Y) 124.16/123.49 rec(F, h2 X G s(Y)) => F Y rec(F, h2 X G Y) 124.16/123.49 rec(F, iszero(X) s(Y)) => F Y rec(F, iszero(X) Y) 124.16/123.49 ~AP1(F, X) => F X 124.16/123.49 124.16/123.49 124.16/123.49 +++ Citations +++ 124.16/123.49 124.16/123.49 [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. 124.81/124.13 EOF