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Higher Order Rewriting Union Beta pair #487093621
details
property
value
status
complete
benchmark
prenex_modif1.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n138.star.cs.uiowa.edu
space
Mixed_HO_12
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.121863 seconds
cpu usage
0.122012
user time
0.10713
system time
0.014882
max virtual memory
113188.0
max residence set size
3952.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: and : [form * form] --> form exists : [term -> form] --> form forall : [term -> form] --> form not : [form] --> form or : [form * form] --> form Rules: and(x, forall(/\y.f y)) => forall(/\z.and(x, f z)) and(forall(/\x.f x), y) => forall(/\z.and(f z, y)) and(x, exists(/\y.f y)) => exists(/\z.and(x, f z)) and(exists(/\x.f x), y) => exists(/\z.and(f z, y)) or(x, forall(/\y.f y)) => forall(/\z.or(x, f z)) or(forall(/\x.f x), y) => forall(/\z.or(f z, y)) or(x, exists(/\y.f y)) => exists(/\z.or(x, f z)) or(exists(/\x.f x), y) => exists(/\z.or(f z, y)) not(forall(/\x.f x)) => exists(/\y.not(f y)) not(exists(/\x.f x)) => forall(/\y.not(f y)) 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: and : [form * form] --> form exists : [term -> form] --> form forall : [term -> form] --> form not : [form] --> form or : [form * form] --> form ~AP1 : [term -> form * term] --> form Rules: and(X, forall(/\x.~AP1(F, x))) => forall(/\y.and(X, ~AP1(F, y))) and(forall(/\x.~AP1(F, x)), X) => forall(/\y.and(~AP1(F, y), X)) and(X, exists(/\x.~AP1(F, x))) => exists(/\y.and(X, ~AP1(F, y))) and(exists(/\x.~AP1(F, x)), X) => exists(/\y.and(~AP1(F, y), X)) or(X, forall(/\x.~AP1(F, x))) => forall(/\y.or(X, ~AP1(F, y))) or(forall(/\x.~AP1(F, x)), X) => forall(/\y.or(~AP1(F, y), X)) or(X, exists(/\x.~AP1(F, x))) => exists(/\y.or(X, ~AP1(F, y))) or(exists(/\x.~AP1(F, x)), X) => exists(/\y.or(~AP1(F, y), X)) not(forall(/\x.~AP1(F, x))) => exists(/\y.not(~AP1(F, y))) not(exists(/\x.~AP1(F, x))) => forall(/\y.not(~AP1(F, y))) ~AP1(F, X) => F X Symbol ~AP1 is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. This gives: Alphabet: and : [form * form] --> form exists : [term -> form] --> form forall : [term -> form] --> form not : [form] --> form or : [form * form] --> form Rules: and(X, forall(/\x.Y(x))) => forall(/\y.and(X, Y(y))) and(forall(/\x.X(x)), Y) => forall(/\y.and(X(y), Y)) and(X, exists(/\x.Y(x))) => exists(/\y.and(X, Y(y))) and(exists(/\x.X(x)), Y) => exists(/\y.and(X(y), Y)) or(X, forall(/\x.Y(x))) => forall(/\y.or(X, Y(y))) or(forall(/\x.X(x)), Y) => forall(/\y.or(X(y), Y)) or(X, exists(/\x.Y(x))) => exists(/\y.or(X, Y(y))) or(exists(/\x.X(x)), Y) => exists(/\y.or(X(y), Y)) not(forall(/\x.X(x))) => exists(/\y.not(X(y))) not(exists(/\x.X(x))) => forall(/\y.not(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]): and(X, forall(/\x.Y(x))) >? forall(/\y.and(X, Y(y))) and(forall(/\x.X(x)), Y) >? forall(/\y.and(X(y), Y)) and(X, exists(/\x.Y(x))) >? exists(/\y.and(X, Y(y))) and(exists(/\x.X(x)), Y) >? exists(/\y.and(X(y), Y)) or(X, forall(/\x.Y(x))) >? forall(/\y.or(X, Y(y))) or(forall(/\x.X(x)), Y) >? forall(/\y.or(X(y), Y)) or(X, exists(/\x.Y(x))) >? exists(/\y.or(X, Y(y))) or(exists(/\x.X(x)), Y) >? exists(/\y.or(X(y), Y)) not(forall(/\x.X(x))) >? exists(/\y.not(X(y))) not(exists(/\x.X(x))) >? forall(/\y.not(X(y))) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: and = \y0y1.y0 + 2y1 exists = \G0.1 + G0(0) forall = \G0.1 + G0(0) not = \y0.y0 or = \y0y1.y0 + 2y1 Using this interpretation, the requirements translate to:
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