Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270461

details

property	value
status	complete
benchmark	h49.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n180.star.cs.uiowa.edu
space	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	20.3779 seconds
cpu usage	20.376
user time	19.9602
system time	0.415843
max virtual memory	425596.0
max residence set size	247192.0

stage attributes

key	value
starexec-result	YES

output

20.20/20.27	YES
20.27/20.35	We consider the system theBenchmark.
20.27/20.35	
20.27/20.35	  Alphabet:
20.27/20.35	
20.27/20.35	    0 : [] --> nat 
20.27/20.35	    mult : [nat * nat] --> nat 
20.27/20.35	    plus : [nat * nat] --> nat 
20.27/20.35	    plus3 : [nat] --> nat -> nat -> nat 
20.27/20.35	    rec : [nat * nat * nat -> nat -> nat] --> nat 
20.27/20.35	    s : [nat] --> nat 
20.27/20.35	    succ2 : [] --> nat -> nat -> nat 
20.27/20.35	    xap : [nat -> nat -> nat * nat] --> nat -> nat 
20.27/20.35	    yap : [nat -> nat * nat] --> nat 
20.27/20.35	
20.27/20.35	  Rules:
20.27/20.35	
20.27/20.35	    rec(0, x, /\y./\z.yap(xap(f, y), z)) => x 
20.27/20.35	    rec(s(x), y, /\z./\u.yap(xap(f, z), u)) => yap(xap(f, x), rec(x, y, /\v./\w.yap(xap(f, v), w))) 
20.27/20.35	    succ2 x y => s(y) 
20.27/20.35	    plus(x, y) => rec(x, y, succ2) 
20.27/20.35	    plus3(x) y z => plus(x, plus(y, z)) 
20.27/20.35	    mult(x, y) => rec(x, 0, plus3(y)) 
20.27/20.35	    xap(f, x) => f x 
20.27/20.35	    yap(f, x) => f x 
20.27/20.35	
20.27/20.35	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
20.27/20.35	
20.27/20.35	Symbol xap 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:
20.27/20.35	
20.27/20.35	  Alphabet:
20.27/20.35	
20.27/20.35	    0 : [] --> nat 
20.27/20.35	    mult : [nat * nat] --> nat 
20.27/20.35	    plus : [nat * nat] --> nat 
20.27/20.35	    plus3 : [nat] --> nat -> nat -> nat 
20.27/20.35	    rec : [nat * nat * nat -> nat -> nat] --> nat 
20.27/20.35	    s : [nat] --> nat 
20.27/20.35	    succ2 : [] --> nat -> nat -> nat 
20.27/20.35	    yap : [nat -> nat * nat] --> nat 
20.27/20.35	
20.27/20.35	  Rules:
20.27/20.35	
20.27/20.35	    rec(0, X, /\x./\y.yap(F(x), y)) => X 
20.27/20.35	    rec(s(X), Y, /\x./\y.yap(F(x), y)) => yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 
20.27/20.35	    succ2 X Y => s(Y) 
20.27/20.35	    plus(X, Y) => rec(X, Y, succ2) 
20.27/20.35	    plus3(X) Y Z => plus(X, plus(Y, Z)) 
20.27/20.35	    mult(X, Y) => rec(X, 0, plus3(Y)) 
20.27/20.35	    yap(F, X) => F X 
20.27/20.35	
20.27/20.35	We use rule removal, following [Kop12, Theorem 2.23].
20.27/20.35	
20.27/20.35	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
20.27/20.35	
20.27/20.35	  rec(0, X, /\x./\y.yap(F(x), y)) >? X 
20.27/20.35	  rec(s(X), Y, /\x./\y.yap(F(x), y)) >? yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) 
20.27/20.35	  succ2 X Y >? s(Y) 
20.27/20.35	  plus(X, Y) >? rec(X, Y, succ2) 
20.27/20.35	  plus3(X) Y Z >? plus(X, plus(Y, Z)) 
20.27/20.35	  mult(X, Y) >? rec(X, 0, plus3(Y)) 
20.27/20.35	  yap(F, X) >? F X 
20.27/20.35	
20.27/20.35	We use a recursive path ordering as defined in [Kop12, Chapter 5].
20.27/20.35	
20.27/20.35	Argument functions:
20.27/20.35	
20.27/20.35	  [[0]] = _|_ 
20.27/20.35	
20.27/20.35	We choose Lex = {} and Mul = {@_{o -> o -> o}, @_{o -> o}, mult, plus, plus3, rec, s, succ2, yap}, and the following precedence: @_{o -> o -> o} > yap > s > mult = plus3 > plus > @_{o -> o} > succ2 > rec
20.27/20.35	
20.27/20.35	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
20.27/20.35	
20.27/20.35	  rec(_|_, X, /\x./\y.yap(F(x), y)) > X 
20.27/20.35	  rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) 
20.27/20.35	  @_{o -> o}(@_{o -> o -> o}(succ2, X), Y) >= s(Y) 
20.27/20.35	  plus(X, Y) >= rec(X, Y, succ2) 
20.27/20.35	  @_{o -> o}(@_{o -> o -> o}(plus3(X), Y), Z) > plus(X, plus(Y, Z)) 
20.27/20.35	  mult(X, Y) >= rec(X, _|_, plus3(Y)) 
20.27/20.35	  yap(F, X) >= @_{o -> o}(F, X) 
20.27/20.35	
20.27/20.35	With these choices, we have:
20.27/20.35	
20.27/20.35	  1] rec(_|_, X, /\x./\y.yap(F(x), y)) > X  because [2], by definition 
20.27/20.35	  2] rec*(_|_, X, /\x./\y.yap(F(x), y)) >= X  because [3], by (Select) 
20.27/20.35	  3] X >= X  by (Meta) 
20.27/20.35	
20.27/20.35	  4] rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [5], by (Star) 
20.27/20.35	  5] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because [6], by (Select) 
20.27/20.35	  6] yap(F(rec*(s(X), Y, /\x./\y.yap(F(x), y))), rec*(s(X), Y, /\z./\u.yap(F(z), u))) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y)))  because yap in Mul, [7] and [12], by (Fun) 
20.27/20.35	  7] F(rec*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X)  because [8], by (Meta) 
20.27/20.35	  8] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= X  because [9], by (Select) 
20.27/20.35	  9] s(X) >= X  because [10], by (Star) 
20.27/20.35	  10] s*(X) >= X  because [11], by (Select) 
20.27/20.35	  11] X >= X  by (Meta) 
20.27/20.35	  12] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y))  because rec in Mul, [13], [15] and [16], by (Stat) 
20.27/20.35	  13] s(X) > X  because [14], by definition 
20.27/20.35	  14] s*(X) >= X  because [11], by (Select) 
20.27/20.35	  15] Y >= Y  by (Meta) 
20.27/20.35	  16] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [17], by (Abs)

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