Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270471

details

property	value
status	complete
benchmark	h54.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n045.star.cs.uiowa.edu
space	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	9.34197 seconds
cpu usage	9.34203
user time	9.1758
system time	0.166231
max virtual memory	215368.0
max residence set size	99440.0

stage attributes

key	value
starexec-result	YES

output

9.22/9.29	YES
9.29/9.33	We consider the system theBenchmark.
9.29/9.33	
9.29/9.33	  Alphabet:
9.29/9.33	
9.29/9.33	    0 : [] --> nat 
9.29/9.33	    cons : [nat * list] --> list 
9.29/9.33	    foldl : [nat -> nat -> nat * nat * list] --> nat 
9.29/9.33	    nil : [] --> list 
9.29/9.33	    plusc : [] --> nat -> nat -> nat 
9.29/9.33	    s : [nat] --> nat 
9.29/9.33	    sum : [list] --> nat 
9.29/9.33	    xap : [nat -> nat -> nat * nat] --> nat -> nat 
9.29/9.33	    yap : [nat -> nat * nat] --> nat 
9.29/9.33	
9.29/9.33	  Rules:
9.29/9.33	
9.29/9.33	    foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z 
9.29/9.33	    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) 
9.29/9.33	    plusc x 0 => x 
9.29/9.33	    plusc x s(y) => s(plusc x y) 
9.29/9.33	    sum(x) => foldl(/\y./\z.yap(xap(plusc, y), z), 0, x) 
9.29/9.33	    xap(f, x) => f x 
9.29/9.33	    yap(f, x) => f x 
9.29/9.33	
9.29/9.33	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
9.29/9.33	
9.29/9.33	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:
9.29/9.33	
9.29/9.33	  Alphabet:
9.29/9.33	
9.29/9.33	    0 : [] --> nat 
9.29/9.33	    cons : [nat * list] --> list 
9.29/9.33	    foldl : [nat -> nat -> nat * nat * list] --> nat 
9.29/9.33	    nil : [] --> list 
9.29/9.33	    plusc : [nat] --> nat -> nat 
9.29/9.33	    s : [nat] --> nat 
9.29/9.33	    sum : [list] --> nat 
9.29/9.33	    yap : [nat -> nat * nat] --> nat 
9.29/9.33	
9.29/9.33	  Rules:
9.29/9.33	
9.29/9.33	    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
9.29/9.33	    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
9.29/9.33	    plusc(X) 0 => X 
9.29/9.33	    plusc(X) s(Y) => s(plusc(X) Y) 
9.29/9.33	    sum(X) => foldl(/\x./\y.yap(plusc(x), y), 0, X) 
9.29/9.33	    yap(F, X) => F X 
9.29/9.33	
9.29/9.33	We use rule removal, following [Kop12, Theorem 2.23].
9.29/9.33	
9.29/9.33	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, nil) >? X 
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
9.29/9.33	  plusc(X) 0 >? X 
9.29/9.33	  plusc(X) s(Y) >? s(plusc(X) Y) 
9.29/9.33	  sum(X) >? foldl(/\x./\y.yap(plusc(x), y), 0, X) 
9.29/9.33	  yap(F, X) >? F X 
9.29/9.33	
9.29/9.33	We use a recursive path ordering as defined in [Kop12, Chapter 5].
9.29/9.33	
9.29/9.33	Argument functions:
9.29/9.33	
9.29/9.33	  [[0]] = _|_ 
9.29/9.33	  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
9.29/9.33	  [[s(x_1)]] = x_1 
9.29/9.33	
9.29/9.33	We choose Lex = {foldl} and Mul = {@_{o -> o}, cons, nil, plusc, sum, yap}, and the following precedence: nil > sum > plusc > cons > yap > @_{o -> o} > foldl
9.29/9.33	
9.29/9.33	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
9.29/9.33	
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
9.29/9.33	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
9.29/9.33	  @_{o -> o}(plusc(X), _|_) >= X 
9.29/9.33	  @_{o -> o}(plusc(X), Y) >= @_{o -> o}(plusc(X), Y) 
9.29/9.33	  sum(X) >= foldl(/\x./\y.yap(plusc(x), y), _|_, X) 
9.29/9.33	  yap(F, X) > @_{o -> o}(F, X) 
9.29/9.33	
9.29/9.33	With these choices, we have:
9.29/9.33	
9.29/9.33	  1] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [2], by (Star) 
9.29/9.33	  2] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [3], by (Select) 
9.29/9.33	  3] X >= X  by (Meta) 
9.29/9.33	
9.29/9.33	  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) 
9.29/9.33	  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], [16] and [25], by (Stat) 
9.29/9.33	  6] cons(Y, Z) > Z  because [7], by definition 
9.29/9.33	  7] cons*(Y, Z) >= Z  because [8], by (Select) 
9.29/9.33	  8] Z >= Z  by (Meta) 
9.29/9.33	  9] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [10], by (Select) 
9.29/9.33	  10] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [11], by (Abs) 
9.29/9.33	  11] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [12], by (Abs) 
9.29/9.33	  12] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [13] and [15], by (Fun) 
9.29/9.33	  13] F(y) >= F(y)  because [14], by (Meta) 
9.29/9.33	  14] y >= y  by (Var) 
9.29/9.33	  15] x >= x  by (Var) 
9.29/9.33	  16] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y)  because [17], by (Select) 
9.29/9.33	  17] yap(F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= yap(F(X), Y)  because yap in Mul, [18] and [21], by (Fun) 
9.29/9.33	  18] F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [19], by (Meta)

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