Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270707

details

property	value
status	complete
benchmark	Applicative_first_order_05__29.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n150.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	0.970063 seconds
cpu usage	0.970964
user time	0.890676
system time	0.080288
max virtual memory	137536.0
max residence set size	22476.0

stage attributes

key	value
starexec-result	YES

output

0.85/0.88	YES
0.91/0.96	We consider the system theBenchmark.
0.91/0.96	
0.91/0.96	  Alphabet:
0.91/0.96	
0.91/0.96	    0 : [] --> a 
0.91/0.96	    ack : [a * a] --> a 
0.91/0.96	    cons : [c * d] --> d 
0.91/0.96	    false : [] --> b 
0.91/0.96	    filter : [c -> b * d] --> d 
0.91/0.96	    filter2 : [b * c -> b * c * d] --> d 
0.91/0.96	    map : [c -> c * d] --> d 
0.91/0.96	    nil : [] --> d 
0.91/0.96	    succ : [a] --> a 
0.91/0.96	    true : [] --> b 
0.91/0.96	
0.91/0.96	  Rules:
0.91/0.96	
0.91/0.96	    ack(0, x) => succ(x) 
0.91/0.96	    ack(succ(x), y) => ack(x, succ(0)) 
0.91/0.96	    ack(succ(x), succ(y)) => ack(x, ack(succ(x), y)) 
0.91/0.96	    map(f, nil) => nil 
0.91/0.96	    map(f, cons(x, y)) => cons(f x, map(f, y)) 
0.91/0.96	    filter(f, nil) => nil 
0.91/0.96	    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
0.91/0.96	    filter2(true, f, x, y) => cons(x, filter(f, y)) 
0.91/0.96	    filter2(false, f, x, y) => filter(f, y) 
0.91/0.96	
0.91/0.96	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.91/0.96	
0.91/0.96	We use rule removal, following [Kop12, Theorem 2.23].
0.91/0.96	
0.91/0.96	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.91/0.96	
0.91/0.96	  ack(0, X) >? succ(X) 
0.91/0.96	  ack(succ(X), Y) >? ack(X, succ(0)) 
0.91/0.96	  ack(succ(X), succ(Y)) >? ack(X, ack(succ(X), Y)) 
0.91/0.96	  map(F, nil) >? nil 
0.91/0.96	  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 
0.91/0.96	  filter(F, nil) >? nil 
0.91/0.96	  filter(F, cons(X, Y)) >? filter2(F X, F, X, Y) 
0.91/0.96	  filter2(true, F, X, Y) >? cons(X, filter(F, Y)) 
0.91/0.96	  filter2(false, F, X, Y) >? filter(F, Y) 
0.91/0.96	
0.91/0.96	We use a recursive path ordering as defined in [Kop12, Chapter 5].
0.91/0.96	
0.91/0.96	Argument functions:
0.91/0.96	
0.91/0.96	  [[0]] = _|_ 
0.91/0.96	  [[filter(x_1, x_2)]] = filter(x_2, x_1) 
0.91/0.96	  [[filter2(x_1, x_2, x_3, x_4)]] = filter2(x_4, x_2, x_1, x_3) 
0.91/0.96	  [[nil]] = _|_ 
0.91/0.96	
0.91/0.96	We choose Lex = {ack, filter, filter2} and Mul = {@_{o -> o}, cons, false, map, succ, true}, and the following precedence: ack > false > filter = filter2 > map > @_{o -> o} > cons > succ > true
0.91/0.96	
0.91/0.96	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
0.91/0.96	
0.91/0.96	  ack(_|_, X) >= succ(X) 
0.91/0.96	  ack(succ(X), Y) >= ack(X, succ(_|_)) 
0.91/0.96	  ack(succ(X), succ(Y)) >= ack(X, ack(succ(X), Y)) 
0.91/0.96	  map(F, _|_) >= _|_ 
0.91/0.96	  map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) 
0.91/0.96	  filter(F, _|_) >= _|_ 
0.91/0.96	  filter(F, cons(X, Y)) > filter2(@_{o -> o}(F, X), F, X, Y) 
0.91/0.96	  filter2(true, F, X, Y) >= cons(X, filter(F, Y)) 
0.91/0.96	  filter2(false, F, X, Y) >= filter(F, Y) 
0.91/0.96	
0.91/0.96	With these choices, we have:
0.91/0.96	
0.91/0.96	  1] ack(_|_, X) >= succ(X)  because [2], by (Star) 
0.91/0.96	  2] ack*(_|_, X) >= succ(X)  because ack > succ and [3], by (Copy) 
0.91/0.96	  3] ack*(_|_, X) >= X  because [4], by (Select) 
0.91/0.96	  4] X >= X  by (Meta) 
0.91/0.96	
0.91/0.96	  5] ack(succ(X), Y) >= ack(X, succ(_|_))  because [6], by (Star) 
0.91/0.96	  6] ack*(succ(X), Y) >= ack(X, succ(_|_))  because [7], [10] and [12], by (Stat) 
0.91/0.96	  7] succ(X) > X  because [8], by definition 
0.91/0.96	  8] succ*(X) >= X  because [9], by (Select) 
0.91/0.96	  9] X >= X  by (Meta) 
0.91/0.96	  10] ack*(succ(X), Y) >= X  because [11], by (Select) 
0.91/0.96	  11] succ(X) >= X  because [8], by (Star) 
0.91/0.96	  12] ack*(succ(X), Y) >= succ(_|_)  because ack > succ and [13], by (Copy) 
0.91/0.96	  13] ack*(succ(X), Y) >= _|_  by (Bot) 
0.91/0.96	
0.91/0.96	  14] ack(succ(X), succ(Y)) >= ack(X, ack(succ(X), Y))  because [15], by (Star) 
0.91/0.96	  15] ack*(succ(X), succ(Y)) >= ack(X, ack(succ(X), Y))  because [16], [19] and [21], by (Stat) 
0.91/0.96	  16] succ(X) > X  because [17], by definition 
0.91/0.96	  17] succ*(X) >= X  because [18], by (Select) 
0.91/0.96	  18] X >= X  by (Meta) 
0.91/0.96	  19] ack*(succ(X), succ(Y)) >= X  because [20], by (Select) 
0.91/0.96	  20] succ(X) >= X  because [17], by (Star) 
0.91/0.96	  21] ack*(succ(X), succ(Y)) >= ack(succ(X), Y)  because [22], [24], [27] and [28], by (Stat) 
0.91/0.96	  22] succ(X) >= succ(X)  because succ in Mul and [23], by (Fun) 
0.91/0.96	  23] X >= X  by (Meta) 
0.91/0.96	  24] succ(Y) > Y  because [25], by definition 
0.91/0.96	  25] succ*(Y) >= Y  because [26], by (Select) 
0.91/0.96	  26] Y >= Y  by (Meta) 
0.91/0.96	  27] ack*(succ(X), succ(Y)) >= succ(X)  because [22], by (Select) 
0.91/0.96	  28] ack*(succ(X), succ(Y)) >= Y  because [29], by (Select) 
0.91/0.96	  29] succ(Y) >= Y  because [25], by (Star)

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