Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270521

details

property	value
status	complete
benchmark	AotoYamada_05__010.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n061.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	0.637541 seconds
cpu usage	0.638242
user time	0.560713
system time	0.077529
max virtual memory	131756.0
max residence set size	16308.0

stage attributes

key	value
starexec-result	YES

output

0.54/0.59	YES
0.61/0.63	We consider the system theBenchmark.
0.61/0.63	
0.61/0.63	  Alphabet:
0.61/0.63	
0.61/0.63	    0 : [] --> b 
0.61/0.63	    cons : [b * a] --> a 
0.61/0.63	    curry : [b -> b -> b * b] --> b -> b 
0.61/0.63	    double : [] --> a -> a 
0.61/0.63	    inc : [] --> a -> a 
0.61/0.63	    map : [b -> b] --> a -> a 
0.61/0.63	    nil : [] --> a 
0.61/0.63	    plus : [] --> b -> b -> b 
0.61/0.63	    s : [b] --> b 
0.61/0.63	    times : [] --> b -> b -> b 
0.61/0.63	
0.61/0.63	  Rules:
0.61/0.63	
0.61/0.63	    plus 0 x => x 
0.61/0.63	    plus s(x) y => s(plus x y) 
0.61/0.63	    times 0 x => 0 
0.61/0.63	    times s(x) y => plus (times x y) y 
0.61/0.63	    curry(f, x) y => f x y 
0.61/0.63	    map(f) nil => nil 
0.61/0.63	    map(f) cons(x, y) => cons(f x, map(f) y) 
0.61/0.63	    inc => map(curry(plus, s(0))) 
0.61/0.63	    double => map(curry(times, s(s(0)))) 
0.61/0.63	
0.61/0.63	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.61/0.63	
0.61/0.63	Symbol curry 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:
0.61/0.63	
0.61/0.63	  Alphabet:
0.61/0.63	
0.61/0.63	    0 : [] --> b 
0.61/0.63	    cons : [b * a] --> a 
0.61/0.63	    double : [] --> a -> a 
0.61/0.63	    inc : [] --> a -> a 
0.61/0.63	    map : [b -> b] --> a -> a 
0.61/0.63	    nil : [] --> a 
0.61/0.63	    plus : [b] --> b -> b 
0.61/0.63	    s : [b] --> b 
0.61/0.63	    times : [b] --> b -> b 
0.61/0.63	
0.61/0.63	  Rules:
0.61/0.63	
0.61/0.63	    plus(0) X => X 
0.61/0.63	    plus(s(X)) Y => s(plus(X) Y) 
0.61/0.63	    times(0) X => 0 
0.61/0.63	    times(s(X)) Y => plus(times(X) Y) Y 
0.61/0.63	    map(F) nil => nil 
0.61/0.63	    map(F) cons(X, Y) => cons(F X, map(F) Y) 
0.61/0.63	    inc => map(plus(s(0))) 
0.61/0.63	    double => map(times(s(s(0)))) 
0.61/0.63	
0.61/0.63	We use rule removal, following [Kop12, Theorem 2.23].
0.61/0.63	
0.61/0.63	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.61/0.63	
0.61/0.63	  plus(0) X >? X 
0.61/0.63	  plus(s(X)) Y >? s(plus(X) Y) 
0.61/0.63	  times(0) X >? 0 
0.61/0.63	  times(s(X)) Y >? plus(times(X) Y) Y 
0.61/0.63	  map(F) nil >? nil 
0.61/0.63	  map(F) cons(X, Y) >? cons(F X, map(F) Y) 
0.61/0.63	  inc >? map(plus(s(0))) 
0.61/0.63	  double >? map(times(s(s(0)))) 
0.61/0.63	
0.61/0.63	We use a recursive path ordering as defined in [Kop12, Chapter 5].
0.61/0.63	
0.61/0.63	Argument functions:
0.61/0.63	
0.61/0.63	  [[0]] = _|_ 
0.61/0.63	  [[nil]] = _|_ 
0.61/0.63	
0.61/0.63	We choose Lex = {} and Mul = {@_{o -> o}, cons, double, inc, map, plus, s, times}, and the following precedence: double > inc > times > plus > @_{o -> o} > cons > map > s
0.61/0.63	
0.61/0.63	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
0.61/0.63	
0.61/0.63	  @_{o -> o}(plus(_|_), X) >= X 
0.61/0.63	  @_{o -> o}(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y)) 
0.61/0.63	  @_{o -> o}(times(_|_), X) >= _|_ 
0.61/0.63	  @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) 
0.61/0.63	  @_{o -> o}(map(F), _|_) >= _|_ 
0.61/0.63	  @_{o -> o}(map(F), cons(X, Y)) > cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y)) 
0.61/0.63	  inc > map(plus(s(_|_))) 
0.61/0.63	  double > map(times(s(s(_|_)))) 
0.61/0.63	
0.61/0.63	With these choices, we have:
0.61/0.63	
0.61/0.63	  1] @_{o -> o}(plus(_|_), X) >= X  because [2], by (Star) 
0.61/0.63	  2] @_{o -> o}*(plus(_|_), X) >= X  because [3], by (Select) 
0.61/0.63	  3] X >= X  by (Meta) 
0.61/0.63	
0.61/0.63	  4] @_{o -> o}(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y))  because [5], by (Star) 
0.61/0.63	  5] @_{o -> o}*(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y))  because @_{o -> o} > s and [6], by (Copy) 
0.61/0.63	  6] @_{o -> o}*(plus(s(X)), Y) >= @_{o -> o}(plus(X), Y)  because [7], by (Select) 
0.61/0.63	  7] plus(s(X)) @_{o -> o}*(plus(s(X)), Y) >= @_{o -> o}(plus(X), Y)  because [8] 
0.61/0.63	  8] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= @_{o -> o}(plus(X), Y)  because plus > @_{o -> o}, [9] and [13], by (Copy) 
0.61/0.63	  9] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= plus(X)  because plus in Mul and [10], by (Stat)

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