details

property	value
status	complete
benchmark	05height.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n139.star.cs.uiowa.edu
space	Blanqui_15

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.0188609 seconds
cpu usage	0.018959
user time	0.008923
system time	0.010036
max virtual memory	113188.0
max residence set size	1760.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: cons : [t * f] --> f heightf : [f] --> N heightt : [t] --> N leaf : [] --> t max : [N * N] --> N nil : [] --> f node : [f] --> t s : [N] --> N z : [] --> N Rules: heightf(nil) => z heightf(cons(x, y)) => max(heightt(x), heightf(y)) heightt(leaf) => z heightt(node(x)) => s(heightf(x)) 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]): heightf(nil) >? z heightf(cons(X, Y)) >? max(heightt(X), heightf(Y)) heightt(leaf) >? z heightt(node(X)) >? s(heightf(X)) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: cons = \y0y1.3 + 3y0 + 3y1 heightf = \y0.y0 heightt = \y0.y0 leaf = 3 max = \y0y1.y0 + y1 nil = 3 node = \y0.3 + 3y0 s = \y0.y0 z = 0 Using this interpretation, the requirements translate to: [[heightf(nil)]] = 3 > 0 = [[z]] [[heightf(cons(_x0, _x1))]] = 3 + 3x0 + 3x1 > x0 + x1 = [[max(heightt(_x0), heightf(_x1))]] [[heightt(leaf)]] = 3 > 0 = [[z]] [[heightt(node(_x0))]] = 3 + 3x0 > x0 = [[s(heightf(_x0))]] We can thus remove the following rules: heightf(nil) => z heightf(cons(X, Y)) => max(heightt(X), heightf(Y)) heightt(leaf) => z heightt(node(X)) => s(heightf(X)) 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. popout

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all output return to Higher Order Rewriting Union Beta