Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270469

details

property	value
status	complete
benchmark	h53.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n075.star.cs.uiowa.edu
space	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	11.491 seconds
cpu usage	11.4907
user time	11.291
system time	0.199734
max virtual memory	237328.0
max residence set size	120672.0

stage attributes

key	value
starexec-result	YES

output

11.40/11.44	YES
11.43/11.48	We consider the system theBenchmark.
11.43/11.48	
11.43/11.48	  Alphabet:
11.43/11.48	
11.43/11.48	    cons : [a * alist] --> alist 
11.43/11.48	    foldl : [a -> a -> a * a * alist] --> a 
11.43/11.48	    nil : [] --> alist 
11.43/11.48	    xap : [a -> a -> a * a] --> a -> a 
11.43/11.48	    yap : [a -> a * a] --> a 
11.43/11.48	
11.43/11.48	  Rules:
11.43/11.48	
11.43/11.48	    foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z 
11.43/11.48	    foldl(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => foldl(/\w./\x'.yap(xap(f, w), x'), yap(xap(f, z), u), v) 
11.43/11.48	    xap(f, x) => f x 
11.43/11.48	    yap(f, x) => f x 
11.43/11.48	
11.43/11.48	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
11.43/11.48	
11.43/11.48	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:
11.43/11.48	
11.43/11.48	  Alphabet:
11.43/11.48	
11.43/11.48	    cons : [a * alist] --> alist 
11.43/11.48	    foldl : [a -> a -> a * a * alist] --> a 
11.43/11.48	    nil : [] --> alist 
11.43/11.48	    yap : [a -> a * a] --> a 
11.43/11.48	
11.43/11.48	  Rules:
11.43/11.48	
11.43/11.48	    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
11.43/11.48	    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
11.43/11.48	    yap(F, X) => F X 
11.43/11.48	
11.43/11.48	We use rule removal, following [Kop12, Theorem 2.23].
11.43/11.48	
11.43/11.48	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
11.43/11.48	
11.43/11.48	  foldl(/\x./\y.yap(F(x), y), X, nil) >? X 
11.43/11.48	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
11.43/11.48	  yap(F, X) >? F X 
11.43/11.48	
11.43/11.48	We use a recursive path ordering as defined in [Kop12, Chapter 5].
11.43/11.48	
11.43/11.48	Argument functions:
11.43/11.48	
11.43/11.48	  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
11.43/11.48	
11.43/11.48	We choose Lex = {foldl} and Mul = {@_{o -> o}, cons, nil, yap}, and the following precedence: cons > nil > foldl > yap > @_{o -> o}
11.43/11.48	
11.43/11.48	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
11.43/11.48	
11.43/11.48	  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
11.43/11.48	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
11.43/11.48	  yap(F, X) > @_{o -> o}(F, X) 
11.43/11.48	
11.43/11.48	With these choices, we have:
11.43/11.48	
11.43/11.48	  1] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [2], by (Star) 
11.43/11.48	  2] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [3], by (Select) 
11.43/11.48	  3] X >= X  by (Meta) 
11.43/11.48	
11.43/11.48	  4] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [5], by (Star) 
11.43/11.48	  5] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [6], [9], [19] and [29], by (Stat) 
11.43/11.48	  6] cons(Y, Z) > Z  because [7], by definition 
11.43/11.48	  7] cons*(Y, Z) >= Z  because [8], by (Select) 
11.43/11.48	  8] Z >= Z  by (Meta) 
11.43/11.48	  9] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [10], by (F-Abs) 
11.43/11.48	  10] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x)  because [11], by (F-Abs) 
11.43/11.48	  11] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= yap(F(z), u)  because foldl > yap, [12] and [17], by (Copy) 
11.43/11.48	  12] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= F(z)  because [13], by (Select) 
11.43/11.48	  13] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z, u)), x) >= F(z)  because [14], by (Eta)[Kop13:2] 
11.43/11.48	  14] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)) >= F(z)  because [15], by (Meta) 
11.43/11.48	  15] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z  because [16], by (Select) 
11.43/11.48	  16] z >= z  by (Var) 
11.43/11.48	  17] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= u  because [18], by (Select) 
11.43/11.48	  18] u >= u  by (Var) 
11.43/11.48	  19] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [20] and [25], by (Copy) 
11.43/11.48	  20] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [21], by (Select) 
11.43/11.48	  21] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z))), x) >= F(X)  because [22], by (Eta)[Kop13:2] 
11.43/11.48	  22] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [23], by (Meta) 
11.43/11.48	  23] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [24], by (Select) 
11.43/11.48	  24] X >= X  by (Meta) 
11.43/11.48	  25] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [26], by (Select) 
11.43/11.48	  26] cons(Y, Z) >= Y  because [27], by (Star) 
11.43/11.48	  27] cons*(Y, Z) >= Y  because [28], by (Select) 
11.43/11.48	  28] Y >= Y  by (Meta) 
11.43/11.48	  29] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [30], by (Select) 
11.43/11.48	  30] cons(Y, Z) >= Z  because [7], by (Star) 
11.43/11.48	
11.43/11.48	  31] yap(F, X) > @_{o -> o}(F, X)  because [32], by definition 
11.43/11.48	  32] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [33] and [35], by (Copy) 
11.43/11.48	  33] yap*(F, X) >= F  because [34], by (Select) 
11.43/11.48	  34] F >= F  by (Meta) 
11.43/11.48	  35] yap*(F, X) >= X  because [36], by (Select) 
11.43/11.48	  36] X >= X  by (Meta) 
11.43/11.48	
11.43/11.48	We can thus remove the following rules:
11.43/11.48

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