Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270479

details

property	value
status	complete
benchmark	h58.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n042.star.cs.uiowa.edu
space	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	14.7488 seconds
cpu usage	14.7493
user time	14.4689
system time	0.280439
max virtual memory	438048.0
max residence set size	154228.0

stage attributes

key	value
starexec-result	YES

output

14.67/14.69	YES
14.71/14.73	We consider the system theBenchmark.
14.71/14.73	
14.71/14.73	  Alphabet:
14.71/14.73	
14.71/14.73	    0 : [] --> nat 
14.71/14.73	    cons : [nat * list] --> list 
14.71/14.73	    foldr : [nat -> nat -> nat * nat * list] --> nat 
14.71/14.73	    length : [list] --> nat 
14.71/14.73	    nil : [] --> list 
14.71/14.73	    s : [nat] --> nat 
14.71/14.73	    succ : [] --> nat -> nat -> nat 
14.71/14.73	    xap : [nat -> nat -> nat * nat] --> nat -> nat 
14.71/14.73	    yap : [nat -> nat * nat] --> nat 
14.71/14.73	
14.71/14.73	  Rules:
14.71/14.73	
14.71/14.73	    foldr(/\x./\y.yap(xap(f, x), y), z, nil) => z 
14.71/14.73	    foldr(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => yap(xap(f, u), foldr(/\w./\x'.yap(xap(f, w), x'), z, v)) 
14.71/14.73	    succ x y => s(y) 
14.71/14.73	    length(x) => foldr(/\y./\z.yap(xap(succ, y), z), 0, x) 
14.71/14.73	    xap(f, x) => f x 
14.71/14.73	    yap(f, x) => f x 
14.71/14.73	
14.71/14.73	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
14.71/14.73	
14.71/14.73	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:
14.71/14.73	
14.71/14.73	  Alphabet:
14.71/14.73	
14.71/14.73	    0 : [] --> nat 
14.71/14.73	    cons : [nat * list] --> list 
14.71/14.73	    foldr : [nat -> nat -> nat * nat * list] --> nat 
14.71/14.73	    length : [list] --> nat 
14.71/14.73	    nil : [] --> list 
14.71/14.73	    s : [nat] --> nat 
14.71/14.73	    succ : [nat] --> nat -> nat 
14.71/14.73	    yap : [nat -> nat * nat] --> nat 
14.71/14.73	
14.71/14.73	  Rules:
14.71/14.73	
14.71/14.73	    foldr(/\x./\y.yap(F(x), y), X, nil) => X 
14.71/14.73	    foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 
14.71/14.73	    succ(X) Y => s(Y) 
14.71/14.73	    length(X) => foldr(/\x./\y.yap(succ(x), y), 0, X) 
14.71/14.73	    yap(F, X) => F X 
14.71/14.73	
14.71/14.73	We use rule removal, following [Kop12, Theorem 2.23].
14.71/14.73	
14.71/14.73	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
14.71/14.73	
14.71/14.73	  foldr(/\x./\y.yap(F(x), y), X, nil) >? X 
14.71/14.73	  foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap(F(Y), foldr(/\z./\u.yap(F(z), u), X, Z)) 
14.71/14.73	  succ(X) Y >? s(Y) 
14.71/14.73	  length(X) >? foldr(/\x./\y.yap(succ(x), y), 0, X) 
14.71/14.73	  yap(F, X) >? F X 
14.71/14.73	
14.71/14.73	We use a recursive path ordering as defined in [Kop12, Chapter 5].
14.71/14.73	
14.71/14.73	Argument functions:
14.71/14.73	
14.71/14.73	  [[0]] = _|_ 
14.71/14.73	  [[s(x_1)]] = x_1 
14.71/14.73	
14.71/14.73	We choose Lex = {} and Mul = {@_{o -> o}, cons, foldr, length, nil, succ, yap}, and the following precedence: cons > length > foldr > nil > succ > yap > @_{o -> o}
14.71/14.73	
14.71/14.73	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
14.71/14.73	
14.71/14.73	  foldr(/\x./\y.yap(F(x), y), X, nil) >= X 
14.71/14.73	  foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z)) 
14.71/14.73	  @_{o -> o}(succ(X), Y) > Y 
14.71/14.73	  length(X) >= foldr(/\x./\y.yap(succ(x), y), _|_, X) 
14.71/14.73	  yap(F, X) > @_{o -> o}(F, X) 
14.71/14.73	
14.71/14.73	With these choices, we have:
14.71/14.73	
14.71/14.73	  1] foldr(/\x./\y.yap(F(x), y), X, nil) >= X  because [2], by (Star) 
14.71/14.73	  2] foldr*(/\x./\y.yap(F(x), y), X, nil) >= X  because [3], by (Select) 
14.71/14.73	  3] X >= X  by (Meta) 
14.71/14.73	
14.71/14.73	  4] foldr(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because [5], by (Star) 
14.71/14.73	  5] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(Y), foldr(/\x./\y.yap(F(x), y), X, Z))  because foldr > yap, [6] and [13], by (Copy) 
14.71/14.73	  6] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(Y)  because [7], by (Select) 
14.71/14.73	  7] /\x.yap(F(foldr*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(Y)  because [8], by (Eta)[Kop13:2] 
14.71/14.73	  8] F(foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(Y)  because [9], by (Meta) 
14.71/14.73	  9] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [10], by (Select) 
14.71/14.73	  10] cons(Y, Z) >= Y  because [11], by (Star) 
14.71/14.73	  11] cons*(Y, Z) >= Y  because [12], by (Select) 
14.71/14.73	  12] Y >= Y  by (Meta) 
14.71/14.73	  13] foldr*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldr(/\x./\y.yap(F(x), y), X, Z)  because foldr in Mul, [14], [20] and [21], by (Stat) 
14.71/14.73	  14] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [15], by (Abs) 
14.71/14.73	  15] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [16], by (Abs) 
14.71/14.73	  16] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [17] and [19], by (Fun) 
14.71/14.73	  17] F(y) >= F(y)  because [18], by (Meta) 
14.71/14.73	  18] y >= y  by (Var) 
14.71/14.73	  19] x >= x  by (Var) 
14.71/14.73	  20] X >= X  by (Meta) 
14.71/14.73	  21] cons(Y, Z) > Z  because [22], by definition 
14.71/14.73	  22] cons*(Y, Z) >= Z  because [23], by (Select) 
14.71/14.73	  23] Z >= Z  by (Meta)

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actions all output return to Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10