Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270459

details

property	value
status	complete
benchmark	h48.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n008.star.cs.uiowa.edu
space	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	4.22755 seconds
cpu usage	4.22773
user time	4.1274
system time	0.100332
max virtual memory	171732.0
max residence set size	56804.0

stage attributes

key	value
starexec-result	YES

output

4.18/4.20	YES
4.18/4.22	We consider the system theBenchmark.
4.18/4.22	
4.18/4.22	  Alphabet:
4.18/4.22	
4.18/4.22	    0 : [] --> nat 
4.18/4.22	    add : [nat * nat] --> nat 
4.18/4.22	    rec : [nat -> nat -> nat * nat * nat] --> nat 
4.18/4.22	    s : [nat] --> nat 
4.18/4.22	    succ : [] --> nat -> nat -> nat 
4.18/4.22	    xap : [nat -> nat -> nat * nat] --> nat -> nat 
4.18/4.22	    yap : [nat -> nat * nat] --> nat 
4.18/4.22	
4.18/4.22	  Rules:
4.18/4.22	
4.18/4.22	    rec(/\x./\y.yap(xap(f, x), y), z, 0) => z 
4.18/4.22	    rec(/\x./\y.yap(xap(f, x), y), z, s(u)) => yap(xap(f, u), rec(/\v./\w.yap(xap(f, v), w), z, u)) 
4.18/4.22	    succ x y => s(y) 
4.18/4.22	    add(x, y) => rec(/\z./\u.yap(xap(succ, z), u), x, y) 
4.18/4.22	    add(x, 0) => x 
4.18/4.22	    add(x, s(y)) => s(add(x, y)) 
4.18/4.22	    xap(f, x) => f x 
4.18/4.22	    yap(f, x) => f x 
4.18/4.22	
4.18/4.22	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
4.18/4.22	
4.18/4.22	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:
4.18/4.22	
4.18/4.22	  Alphabet:
4.18/4.22	
4.18/4.22	    0 : [] --> nat 
4.18/4.22	    add : [nat * nat] --> nat 
4.18/4.22	    rec : [nat -> nat -> nat * nat * nat] --> nat 
4.18/4.22	    s : [nat] --> nat 
4.18/4.22	    succ : [nat] --> nat -> nat 
4.18/4.22	    yap : [nat -> nat * nat] --> nat 
4.18/4.22	
4.18/4.22	  Rules:
4.18/4.22	
4.18/4.22	    rec(/\x./\y.yap(F(x), y), X, 0) => X 
4.18/4.22	    rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) 
4.18/4.22	    succ(X) Y => s(Y) 
4.18/4.22	    add(X, Y) => rec(/\x./\y.yap(succ(x), y), X, Y) 
4.18/4.22	    add(X, 0) => X 
4.18/4.22	    add(X, s(Y)) => s(add(X, Y)) 
4.18/4.22	    yap(F, X) => F X 
4.18/4.22	
4.18/4.22	We use rule removal, following [Kop12, Theorem 2.23].
4.18/4.22	
4.18/4.22	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
4.18/4.22	
4.18/4.22	  rec(/\x./\y.yap(F(x), y), X, 0) >? X 
4.18/4.22	  rec(/\x./\y.yap(F(x), y), X, s(Y)) >? yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) 
4.18/4.22	  succ(X) Y >? s(Y) 
4.18/4.22	  add(X, Y) >? rec(/\x./\y.yap(succ(x), y), X, Y) 
4.18/4.22	  add(X, 0) >? X 
4.18/4.22	  add(X, s(Y)) >? s(add(X, Y)) 
4.18/4.22	  yap(F, X) >? F X 
4.18/4.22	
4.18/4.22	We use a recursive path ordering as defined in [Kop12, Chapter 5].
4.18/4.22	
4.18/4.22	We choose Lex = {} and Mul = {0, @_{o -> o}, add, rec, s, succ, yap}, and the following precedence: 0 > add > succ > s > rec > yap > @_{o -> o}
4.18/4.22	
4.18/4.22	With these choices, we have:
4.18/4.22	
4.18/4.22	  1] rec(/\x./\y.yap(F(x), y), X, 0) > X  because [2], by definition 
4.18/4.22	  2] rec*(/\x./\y.yap(F(x), y), X, 0) >= X  because [3], by (Select) 
4.18/4.22	  3] X >= X  by (Meta) 
4.18/4.22	
4.18/4.22	  4] rec(/\x./\y.yap(F(x), y), X, s(Y)) > yap(F(Y), rec(/\x./\y.yap(F(x), y), X, Y))  because [5], by definition 
4.18/4.22	  5] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(Y), rec(/\x./\y.yap(F(x), y), X, Y))  because rec > yap, [6] and [13], by (Copy) 
4.18/4.22	  6] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= F(Y)  because [7], by (Select) 
4.18/4.22	  7] /\x.yap(F(rec*(/\y./\z.yap(F(y), z), X, s(Y))), x) >= F(Y)  because [8], by (Eta)[Kop13:2] 
4.18/4.22	  8] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(Y)  because [9], by (Meta) 
4.18/4.22	  9] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= Y  because [10], by (Select) 
4.18/4.22	  10] s(Y) >= Y  because [11], by (Star) 
4.18/4.22	  11] s*(Y) >= Y  because [12], by (Select) 
4.18/4.22	  12] Y >= Y  by (Meta) 
4.18/4.22	  13] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec in Mul, [14], [20] and [21], by (Stat) 
4.18/4.22	  14] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [15], by (Abs) 
4.18/4.22	  15] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [16], by (Abs) 
4.18/4.22	  16] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [17] and [19], by (Fun) 
4.18/4.22	  17] F(y) >= F(y)  because [18], by (Meta) 
4.18/4.22	  18] y >= y  by (Var) 
4.18/4.22	  19] x >= x  by (Var) 
4.18/4.22	  20] X >= X  by (Meta) 
4.18/4.22	  21] s(Y) > Y  because [22], by definition 
4.18/4.22	  22] s*(Y) >= Y  because [12], by (Select) 
4.18/4.22	
4.18/4.22	  23] @_{o -> o}(succ(X), Y) > s(Y)  because [24], by definition 
4.18/4.22	  24] @_{o -> o}*(succ(X), Y) >= s(Y)  because [25], by (Select) 
4.18/4.22	  25] succ(X) @_{o -> o}*(succ(X), Y) >= s(Y)  because [26] 
4.18/4.22	  26] succ*(X, @_{o -> o}*(succ(X), Y)) >= s(Y)  because succ > s and [27], by (Copy) 
4.18/4.22	  27] succ*(X, @_{o -> o}*(succ(X), Y)) >= Y  because [28], by (Select) 
4.18/4.22	  28] @_{o -> o}*(succ(X), Y) >= Y  because [29], by (Select) 
4.18/4.22	  29] Y >= Y  by (Meta) 
4.18/4.22	
4.18/4.22	  30] add(X, Y) >= rec(/\x./\y.yap(succ(x), y), X, Y)  because [31], by (Star) 
4.18/4.22	  31] add*(X, Y) >= rec(/\x./\y.yap(succ(x), y), X, Y)  because add > rec, [32], [40] and [42], by (Copy) 
4.18/4.22	  32] add*(X, Y) >= /\y./\z.yap(succ(y), z)  because [33], by (F-Abs)

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