Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270589

details
property	value
status	complete
benchmark	Applicative_05__Ex9Maps.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n070.star.cs.uiowa.edu
space	Uncurried_Applicative_11
run statistics
property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	17.3859 seconds
cpu usage	17.3859
user time	17.0306
system time	0.355364
max virtual memory	325972.0
max residence set size	209440.0
stage attributes
key	value
starexec-result	YES
output
17.25/17.34	YES
17.35/17.37	We consider the system theBenchmark.
17.35/17.37	
17.35/17.37	  Alphabet:
17.35/17.37	
17.35/17.37	    cons : [d * e] --> e 
17.35/17.37	    g : [] --> b 
17.35/17.37	    map!fac62201 : [d -> d * e] --> e 
17.35/17.37	    map!fac62202 : [d -> a -> d * a * e] --> e 
17.35/17.37	    map!fac62203 : [b -> d -> c -> d * b * c * e] --> e 
17.35/17.37	
17.35/17.37	  Rules:
17.35/17.37	
17.35/17.37	    map!fac62201(f, cons(x, y)) => cons(f x, map!fac62201(f, y)) 
17.35/17.37	    map!fac62202(f, x, cons(y, z)) => cons(f y x, map!fac62202(f, x, z)) 
17.35/17.37	    map!fac62203(f, g, x, cons(y, z)) => cons(f g y x, map!fac62203(f, g, x, z)) 
17.35/17.37	
17.35/17.37	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
17.35/17.37	
17.35/17.37	We use rule removal, following [Kop12, Theorem 2.23].
17.35/17.37	
17.35/17.37	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
17.35/17.37	
17.35/17.37	  map!fac62201(F, cons(X, Y)) >? cons(F X, map!fac62201(F, Y)) 
17.35/17.37	  map!fac62202(F, X, cons(Y, Z)) >? cons(F Y X, map!fac62202(F, X, Z)) 
17.35/17.37	  map!fac62203(F, g, X, cons(Y, Z)) >? cons(F g Y X, map!fac62203(F, g, X, Z)) 
17.35/17.37	
17.35/17.37	We use a recursive path ordering as defined in [Kop12, Chapter 5].
17.35/17.37	
17.35/17.37	Argument functions:
17.35/17.37	
17.35/17.37	  [[g]] = _|_ 
17.35/17.37	
17.35/17.37	We choose Lex = {} and Mul = {@_{o -> o -> o -> o}, @_{o -> o -> o}, @_{o -> o}, cons, map!fac62201, map!fac62202, map!fac62203}, and the following precedence: map!fac62201 > map!fac62203 > @_{o -> o -> o} = map!fac62202 > @_{o -> o} > cons > @_{o -> o -> o -> o}
17.35/17.37	
17.35/17.37	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
17.35/17.37	
17.35/17.37	  map!fac62201(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map!fac62201(F, Y)) 
17.35/17.37	  map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z)) 
17.35/17.37	  map!fac62203(F, _|_, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z)) 
17.35/17.37	
17.35/17.37	With these choices, we have:
17.35/17.37	
17.35/17.37	  1] map!fac62201(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because [2], by definition 
17.35/17.37	  2] map!fac62201*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map!fac62201(F, Y))  because map!fac62201 > cons, [3] and [10], by (Copy) 
17.35/17.37	  3] map!fac62201*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map!fac62201 > @_{o -> o}, [4] and [6], by (Copy) 
17.35/17.37	  4] map!fac62201*(F, cons(X, Y)) >= F  because [5], by (Select) 
17.35/17.37	  5] F >= F  by (Meta) 
17.35/17.37	  6] map!fac62201*(F, cons(X, Y)) >= X  because [7], by (Select) 
17.35/17.37	  7] cons(X, Y) >= X  because [8], by (Star) 
17.35/17.37	  8] cons*(X, Y) >= X  because [9], by (Select) 
17.35/17.37	  9] X >= X  by (Meta) 
17.35/17.37	  10] map!fac62201*(F, cons(X, Y)) >= map!fac62201(F, Y)  because map!fac62201 in Mul, [11] and [12], by (Stat) 
17.35/17.37	  11] F >= F  by (Meta) 
17.35/17.37	  12] cons(X, Y) > Y  because [13], by definition 
17.35/17.37	  13] cons*(X, Y) >= Y  because [14], by (Select) 
17.35/17.37	  14] Y >= Y  by (Meta) 
17.35/17.37	
17.35/17.37	  15] map!fac62202(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because [16], by (Star) 
17.35/17.37	  16] map!fac62202*(F, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Y), X), map!fac62202(F, X, Z))  because map!fac62202 > cons, [17] and [25], by (Copy) 
17.35/17.37	  17] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), X)  because map!fac62202 > @_{o -> o}, [18] and [23], by (Copy) 
17.35/17.37	  18] map!fac62202*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, Y)  because map!fac62202 = @_{o -> o -> o}, map!fac62202 in Mul, [19] and [20], by (Stat) 
17.35/17.37	  19] F >= F  by (Meta) 
17.35/17.37	  20] cons(Y, Z) > Y  because [21], by definition 
17.35/17.37	  21] cons*(Y, Z) >= Y  because [22], by (Select) 
17.35/17.37	  22] Y >= Y  by (Meta) 
17.35/17.37	  23] map!fac62202*(F, X, cons(Y, Z)) >= X  because [24], by (Select) 
17.35/17.37	  24] X >= X  by (Meta) 
17.35/17.37	  25] map!fac62202*(F, X, cons(Y, Z)) >= map!fac62202(F, X, Z)  because map!fac62202 in Mul, [19], [26] and [27], by (Stat) 
17.35/17.37	  26] X >= X  by (Meta) 
17.35/17.37	  27] cons(Y, Z) > Z  because [28], by definition 
17.35/17.37	  28] cons*(Y, Z) >= Z  because [29], by (Select) 
17.35/17.37	  29] Z >= Z  by (Meta) 
17.35/17.37	
17.35/17.37	  30] map!fac62203(F, _|_, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z))  because [31], by (Star) 
17.35/17.37	  31] map!fac62203*(F, _|_, X, cons(Y, Z)) >= cons(@_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X), map!fac62203(F, _|_, X, Z))  because map!fac62203 > cons, [32] and [44], by (Copy) 
17.35/17.37	  32] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y), X)  because map!fac62203 > @_{o -> o}, [33] and [42], by (Copy) 
17.35/17.37	  33] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o -> o}(@_{o -> o -> o -> o}(F, _|_), Y)  because map!fac62203 > @_{o -> o -> o}, [34] and [38], by (Copy) 
17.35/17.37	  34] map!fac62203*(F, _|_, X, cons(Y, Z)) >= @_{o -> o -> o -> o}(F, _|_)  because map!fac62203 > @_{o -> o -> o -> o}, [35] and [37], by (Copy) 
17.35/17.37	  35] map!fac62203*(F, _|_, X, cons(Y, Z)) >= F  because [36], by (Select) 
17.35/17.37	  36] F >= F  by (Meta) 
17.35/17.37	  37] map!fac62203*(F, _|_, X, cons(Y, Z)) >= _|_  by (Bot) 
17.35/17.37	  38] map!fac62203*(F, _|_, X, cons(Y, Z)) >= Y  because [39], by (Select) 
17.35/17.37	  39] cons(Y, Z) >= Y  because [40], by (Star) 
17.35/17.37	  40] cons*(Y, Z) >= Y  because [41], by (Select) 
17.35/17.37	  41] Y >= Y  by (Meta) 
17.35/17.37	  42] map!fac62203*(F, _|_, X, cons(Y, Z)) >= X  because [43], by (Select) 
17.35/17.37	  43] X >= X  by (Meta) 
17.35/17.37	  44] map!fac62203*(F, _|_, X, cons(Y, Z)) >= map!fac62203(F, _|_, X, Z)  because map!fac62203 in Mul, [45], [46], [47] and [48], by (Stat) 
17.35/17.37	  45] F >= F  by (Meta) 
17.35/17.37	  46] _|_ >= _|_  by (Bot) 
17.35/17.37	  47] X >= X  by (Meta) 
17.35/17.37	  48] cons(Y, Z) > Z  because [49], by definition 
17.35/17.37	  49] cons*(Y, Z) >= Z  because [50], by (Select) 
17.35/17.37	  50] Z >= Z  by (Meta) 
17.35/17.37	
17.35/17.37	We can thus remove the following rules:
17.35/17.37	
17.35/17.37	  map!fac62201(F, cons(X, Y)) => cons(F X, map!fac62201(F, Y)) 
17.35/17.37
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