Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270573

details

property	value
status	complete
benchmark	Applicative_05__Ex5Folding.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n117.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	0.42004 seconds
cpu usage	0.4204
user time	0.383612
system time	0.036788
max virtual memory	113176.0
max residence set size	13420.0

stage attributes

key	value
starexec-result	YES

output

0.00/0.40	YES
0.00/0.41	We consider the system theBenchmark.
0.00/0.41	
0.00/0.41	  Alphabet:
0.00/0.41	
0.00/0.41	    0 : [] --> c 
0.00/0.41	    add : [] --> a -> c -> c 
0.00/0.41	    cons : [a * b] --> b 
0.00/0.41	    fold : [a -> c -> c * c] --> b -> c 
0.00/0.41	    mul : [] --> a -> c -> c 
0.00/0.41	    nil : [] --> b 
0.00/0.41	    plus : [c * c] --> c 
0.00/0.41	    prod : [] --> b -> c 
0.00/0.41	    s : [c] --> c 
0.00/0.41	    sum : [] --> b -> c 
0.00/0.41	    times : [c * c] --> c 
0.00/0.41	
0.00/0.41	  Rules:
0.00/0.41	
0.00/0.41	    fold(f, x) nil => x 
0.00/0.41	    fold(f, x) cons(y, z) => f y (fold(f, x) z) 
0.00/0.41	    plus(0, x) => x 
0.00/0.41	    plus(s(x), y) => s(plus(x, y)) 
0.00/0.41	    times(0, x) => 0 
0.00/0.41	    times(s(x), y) => plus(times(x, y), y) 
0.00/0.41	    sum => fold(add, 0) 
0.00/0.41	    prod => fold(mul, s(0)) 
0.00/0.41	
0.00/0.41	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.00/0.41	
0.00/0.41	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.41	
0.00/0.41	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.41	
0.00/0.41	  fold(F, X) nil >? X 
0.00/0.41	  fold(F, X) cons(Y, Z) >? F Y (fold(F, X) Z) 
0.00/0.41	  plus(0, X) >? X 
0.00/0.41	  plus(s(X), Y) >? s(plus(X, Y)) 
0.00/0.41	  times(0, X) >? 0 
0.00/0.41	  times(s(X), Y) >? plus(times(X, Y), Y) 
0.00/0.41	  sum >? fold(add, 0) 
0.00/0.41	  prod >? fold(mul, s(0)) 
0.00/0.41	
0.00/0.41	We use a recursive path ordering as defined in [Kop12, Chapter 5].
0.00/0.41	
0.00/0.41	Argument functions:
0.00/0.41	
0.00/0.41	  [[0]] = _|_ 
0.00/0.41	  [[add]] = _|_ 
0.00/0.41	  [[mul]] = _|_ 
0.00/0.41	
0.00/0.41	We choose Lex = {} and Mul = {@_{o -> o -> o}, @_{o -> o}, cons, fold, nil, plus, prod, s, sum, times}, and the following precedence: nil > prod > sum > fold > @_{o -> o -> o} > @_{o -> o} > cons > times > plus > s
0.00/0.41	
0.00/0.41	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
0.00/0.41	
0.00/0.41	  @_{o -> o}(fold(F, X), nil) > X 
0.00/0.41	  @_{o -> o}(fold(F, X), cons(Y, Z)) > @_{o -> o}(@_{o -> o -> o}(F, Y), @_{o -> o}(fold(F, X), Z)) 
0.00/0.41	  plus(_|_, X) >= X 
0.00/0.41	  plus(s(X), Y) >= s(plus(X, Y)) 
0.00/0.41	  times(_|_, X) >= _|_ 
0.00/0.41	  times(s(X), Y) > plus(times(X, Y), Y) 
0.00/0.41	  sum >= fold(_|_, _|_) 
0.00/0.41	  prod >= fold(_|_, s(_|_)) 
0.00/0.41	
0.00/0.41	With these choices, we have:
0.00/0.41	
0.00/0.41	  1] @_{o -> o}(fold(F, X), nil) > X  because [2], by definition 
0.00/0.41	  2] @_{o -> o}*(fold(F, X), nil) >= X  because [3], by (Select) 
0.00/0.41	  3] fold(F, X) @_{o -> o}*(fold(F, X), nil) >= X  because [4] 
0.00/0.41	  4] fold*(F, X, @_{o -> o}*(fold(F, X), nil)) >= X  because [5], by (Select) 
0.00/0.41	  5] X >= X  by (Meta) 
0.00/0.41	
0.00/0.41	  6] @_{o -> o}(fold(F, X), cons(Y, Z)) > @_{o -> o}(@_{o -> o -> o}(F, Y), @_{o -> o}(fold(F, X), Z))  because [7], by definition 
0.00/0.41	  7] @_{o -> o}*(fold(F, X), cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), @_{o -> o}(fold(F, X), Z))  because [8], by (Select) 
0.00/0.41	  8] fold(F, X) @_{o -> o}*(fold(F, X), cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, Y), @_{o -> o}(fold(F, X), Z))  because [9] 
0.00/0.41	  9] fold*(F, X, @_{o -> o}*(fold(F, X), cons(Y, Z))) >= @_{o -> o}(@_{o -> o -> o}(F, Y), @_{o -> o}(fold(F, X), Z))  because fold > @_{o -> o}, [10] and [18], by (Copy) 
0.00/0.41	  10] fold*(F, X, @_{o -> o}*(fold(F, X), cons(Y, Z))) >= @_{o -> o -> o}(F, Y)  because fold > @_{o -> o -> o}, [11] and [13], by (Copy) 
0.00/0.41	  11] fold*(F, X, @_{o -> o}*(fold(F, X), cons(Y, Z))) >= F  because [12], by (Select) 
0.00/0.41	  12] F >= F  by (Meta) 
0.00/0.41	  13] fold*(F, X, @_{o -> o}*(fold(F, X), cons(Y, Z))) >= Y  because [14], by (Select) 
0.00/0.41	  14] @_{o -> o}*(fold(F, X), cons(Y, Z)) >= Y  because [15], by (Select) 
0.00/0.41	  15] cons(Y, Z) >= Y  because [16], by (Star) 
0.00/0.41	  16] cons*(Y, Z) >= Y  because [17], by (Select) 
0.00/0.41	  17] Y >= Y  by (Meta) 
0.00/0.41	  18] fold*(F, X, @_{o -> o}*(fold(F, X), cons(Y, Z))) >= @_{o -> o}(fold(F, X), Z)  because [19], by (Select) 
0.00/0.41	  19] @_{o -> o}*(fold(F, X), cons(Y, Z)) >= @_{o -> o}(fold(F, X), Z)  because @_{o -> o} in Mul, [20] and [23], by (Stat) 
0.00/0.41	  20] fold(F, X) >= fold(F, X)  because fold in Mul, [21] and [22], by (Fun) 
0.00/0.41	  21] F >= F  by (Meta) 
0.00/0.41	  22] X >= X  by (Meta) 
0.00/0.41	  23] cons(Y, Z) > Z  because [24], by definition 
0.00/0.41	  24] cons*(Y, Z) >= Z  because [25], by (Select) 
0.00/0.41	  25] Z >= Z  by (Meta) 
0.00/0.41	
0.00/0.41	  26] plus(_|_, X) >= X  because [27], by (Star) 
0.00/0.41	  27] plus*(_|_, X) >= X  because [28], by (Select) 
0.00/0.41	  28] X >= X  by (Meta) 
0.00/0.41	
0.00/0.41	  29] plus(s(X), Y) >= s(plus(X, Y))  because [30], by (Star) 
0.00/0.41	  30] plus*(s(X), Y) >= s(plus(X, Y))  because plus > s and [31], by (Copy) 
0.00/0.41	  31] plus*(s(X), Y) >= plus(X, Y)  because plus in Mul, [32] and [35], by (Stat)

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