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Higher Order Rewriting Union Beta pair #487094115
details
property
value
status
complete
benchmark
SystemT.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n145.star.cs.uiowa.edu
space
Hamana_17
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.202485 seconds
cpu usage
0.106934
user time
0.086567
system time
0.020367
max virtual memory
113188.0
max residence set size
4816.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: app : [] --> arrab -> a -> b lam : [] --> (a -> b) -> arrab rec : [] --> Nat -> a -> (Nat -> a -> a) -> a succ : [] --> Nat -> Nat zero : [] --> Nat Rules: app (lam (/\x.f x)) y => f y lam (/\x.app y x) => y rec zero x (/\y.f y) => x rec (succ x) y (/\z.f z) => f x (rec x y (/\u.f u)) 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. 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: Alphabet: app : [arrab * a] --> b lam : [a -> b] --> arrab rec : [Nat * a * Nat -> a -> a] --> a succ : [Nat] --> Nat zero : [] --> Nat ~AP1 : [a -> b * a] --> b ~AP2 : [Nat -> a -> a * Nat] --> a -> a Rules: app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) lam(/\x.app(X, x)) => X rec(zero, X, /\x.~AP2(F, x)) => X rec(succ(X), Y, /\x.~AP2(F, x)) => ~AP2(F, X) rec(X, Y, /\y.~AP2(F, y)) app(lam(/\x.app(X, x)), Y) => app(X, Y) ~AP1(F, X) => F X ~AP2(F, X) => F X Symbols ~AP1, and ~AP2 are encodings for application that are only used in innocuous ways. We can simplify the program (without losing non-termination) by removing them. Additionally, we can remove some (now-)redundant rules. This gives: Alphabet: app : [arrab * a] --> b lam : [a -> b] --> arrab rec : [Nat * a * Nat -> a -> a] --> a succ : [Nat] --> Nat zero : [] --> Nat Rules: app(lam(/\x.X(x)), Y) => X(Y) lam(/\x.app(X, x)) => X rec(zero, X, /\x.F(x)) => X rec(succ(X), Y, /\x.F(x)) => F(X) rec(X, Y, /\y.F(y)) 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 [FuhKop19]). We thus obtain the following dependency pair problem (P_0, R_0, computable, all): Dependency Pairs P_0: 0] rec#(succ(X), Y, /\x.F(x)) =#> rec#(X, Y, /\y.F(y)) Rules R_0: app(lam(/\x.X(x)), Y) => X(Y) lam(/\x.app(X, x)) => X rec(zero, X, /\x.F(x)) => X rec(succ(X), Y, /\x.F(x)) => F(X) rec(X, Y, /\y.F(y)) Thus, the original system is terminating if (P_0, R_0, computable, all) is finite. We consider the dependency pair problem (P_0, R_0, computable, all). We apply the subterm criterion with the following projection function: nu(rec#) = 1 Thus, we can orient the dependency pairs as follows: nu(rec#(succ(X), Y, /\x.F(x))) = succ(X) |> X = nu(rec#(X, Y, /\y.F(y))) By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_0, R_0, computable, f) by ({}, R_0, computable, f). 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 +++ [FuhKop19] C. Fuhs, and C. Kop. A static higher-order dependency pair framework. In Proceedings of ESOP 2019, 2019. [Kop11] C. Kop. Simplifying Algebraic Functional Systems. In Proceedings of CAI 2011, volume 6742 of LNCS. 201--215, Springer, 2011. [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. [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.
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