Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270393

details

property	value
status	complete
benchmark	h15.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	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	12.3852 seconds
cpu usage	12.3861
user time	12.1451
system time	0.241035
max virtual memory	244072.0
max residence set size	127676.0

stage attributes

key	value
starexec-result	YES

output

12.23/12.32	YES
12.33/12.37	We consider the system theBenchmark.
12.33/12.37	
12.33/12.37	  Alphabet:
12.33/12.37	
12.33/12.37	    0 : [] --> nat 
12.33/12.37	    cons : [nat * natlist] --> natlist 
12.33/12.37	    foldl : [nat -> nat -> nat * nat * natlist] --> nat 
12.33/12.37	    nil : [] --> natlist 
12.33/12.37	    plus : [] --> nat -> nat -> nat 
12.33/12.37	    sum : [natlist] --> nat 
12.33/12.37	    xap : [nat -> nat -> nat * nat] --> nat -> nat 
12.33/12.37	    yap : [nat -> nat * nat] --> nat 
12.33/12.37	
12.33/12.37	  Rules:
12.33/12.37	
12.33/12.37	    foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z 
12.33/12.37	    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) 
12.33/12.37	    sum(x) => foldl(/\y./\z.yap(xap(plus, y), z), 0, x) 
12.33/12.37	    xap(f, x) => f x 
12.33/12.37	    yap(f, x) => f x 
12.33/12.37	
12.33/12.37	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
12.33/12.37	
12.33/12.37	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:
12.33/12.37	
12.33/12.37	  Alphabet:
12.33/12.37	
12.33/12.37	    0 : [] --> nat 
12.33/12.37	    cons : [nat * natlist] --> natlist 
12.33/12.37	    foldl : [nat -> nat -> nat * nat * natlist] --> nat 
12.33/12.37	    nil : [] --> natlist 
12.33/12.37	    plus : [nat] --> nat -> nat 
12.33/12.37	    sum : [natlist] --> nat 
12.33/12.37	    yap : [nat -> nat * nat] --> nat 
12.33/12.37	
12.33/12.37	  Rules:
12.33/12.37	
12.33/12.37	    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
12.33/12.37	    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
12.33/12.37	    sum(X) => foldl(/\x./\y.yap(plus(x), y), 0, X) 
12.33/12.37	    yap(F, X) => F X 
12.33/12.37	
12.33/12.37	We use rule removal, following [Kop12, Theorem 2.23].
12.33/12.37	
12.33/12.37	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
12.33/12.37	
12.33/12.37	  foldl(/\x./\y.yap(F(x), y), X, nil) >? X 
12.33/12.37	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
12.33/12.37	  sum(X) >? foldl(/\x./\y.yap(plus(x), y), 0, X) 
12.33/12.37	  yap(F, X) >? F X 
12.33/12.37	
12.33/12.37	We use a recursive path ordering as defined in [Kop12, Chapter 5].
12.33/12.37	
12.33/12.37	Argument functions:
12.33/12.37	
12.33/12.37	  [[0]] = _|_ 
12.33/12.37	  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
12.33/12.37	
12.33/12.37	We choose Lex = {foldl} and Mul = {@_{o -> o}, cons, nil, plus, sum, yap}, and the following precedence: cons > nil > sum > plus > yap > @_{o -> o} > foldl
12.33/12.37	
12.33/12.37	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
12.33/12.37	
12.33/12.37	  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
12.33/12.37	  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
12.33/12.37	  sum(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X) 
12.33/12.37	  yap(F, X) > @_{o -> o}(F, X) 
12.33/12.37	
12.33/12.37	With these choices, we have:
12.33/12.37	
12.33/12.37	  1] foldl(/\x./\y.yap(F(x), y), X, nil) >= X  because [2], by (Star) 
12.33/12.37	  2] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X  because [3], by (Select) 
12.33/12.37	  3] X >= X  by (Meta) 
12.33/12.37	
12.33/12.37	  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) 
12.33/12.37	  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], [17] and [26], by (Stat) 
12.33/12.37	  6] cons(Y, Z) > Z  because [7], by definition 
12.33/12.37	  7] cons*(Y, Z) >= Z  because [8], by (Select) 
12.33/12.37	  8] Z >= Z  by (Meta) 
12.33/12.37	  9] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [10], by (F-Abs) 
12.33/12.37	  10] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x)  because [11], by (Select) 
12.33/12.37	  11] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z)), x) >= /\x.yap(F(z), x)  because [12], by (Abs) 
12.33/12.37	  12] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z)), u) >= yap(F(z), u)  because yap in Mul, [13] and [16], by (Fun) 
12.33/12.37	  13] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z)) >= F(z)  because [14], by (Meta) 
12.33/12.37	  14] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= z  because [15], by (Select) 
12.33/12.37	  15] z >= z  by (Var) 
12.33/12.37	  16] u >= u  by (Var) 
12.33/12.37	  17] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because [18], by (Select) 
12.33/12.37	  18] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\v./\w.yap(F(v), w), X, cons(Y, Z))) >= yap(F(X), Y)  because yap in Mul, [19] and [22], by (Fun) 
12.33/12.37	  19] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [20], by (Meta) 
12.33/12.37	  20] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [21], by (Select) 
12.33/12.37	  21] X >= X  by (Meta) 
12.33/12.37	  22] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [23], by (Select) 
12.33/12.37	  23] cons(Y, Z) >= Y  because [24], by (Star) 
12.33/12.37	  24] cons*(Y, Z) >= Y  because [25], by (Select) 
12.33/12.37	  25] Y >= Y  by (Meta) 
12.33/12.37	  26] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [27], by (Select) 
12.33/12.37	  27] cons(Y, Z) >= Z  because [7], by (Star) 
12.33/12.37	
12.33/12.37	  28] sum(X) >= foldl(/\x./\y.yap(plus(x), y), _|_, X)  because [29], by (Star)

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