Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270585

details

property	value
status	complete
benchmark	Applicative_05__Ex7Sorting.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n141.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	2.02559 seconds
cpu usage	2.02596
user time	1.94474
system time	0.08122
max virtual memory	157784.0
max residence set size	42212.0

stage attributes

key	value
starexec-result	YES

output

1.89/1.98	YES
1.97/2.02	We consider the system theBenchmark.
1.97/2.02	
1.97/2.02	  Alphabet:
1.97/2.02	
1.97/2.02	    0 : [] --> a 
1.97/2.02	    asort : [b] --> b 
1.97/2.02	    cons : [a * b] --> b 
1.97/2.02	    dsort : [b] --> b 
1.97/2.02	    insert : [a -> a -> a * a -> a -> a * b * a] --> b 
1.97/2.02	    max : [] --> a -> a -> a 
1.97/2.02	    min : [] --> a -> a -> a 
1.97/2.02	    nil : [] --> b 
1.97/2.02	    s : [a] --> a 
1.97/2.02	    sort : [a -> a -> a * a -> a -> a * b] --> b 
1.97/2.02	
1.97/2.02	  Rules:
1.97/2.02	
1.97/2.02	    sort(f, g, nil) => nil 
1.97/2.02	    sort(f, g, cons(x, y)) => insert(f, g, sort(f, g, y), x) 
1.97/2.02	    insert(f, g, nil, x) => cons(x, nil) 
1.97/2.02	    insert(f, g, cons(x, y), z) => cons(f x z, insert(f, g, y, g x z)) 
1.97/2.02	    max 0 x => x 
1.97/2.02	    max x 0 => x 
1.97/2.02	    max s(x) s(y) => max x y 
1.97/2.02	    min 0 x => 0 
1.97/2.02	    min x 0 => 0 
1.97/2.02	    min s(x) s(y) => min x y 
1.97/2.02	    asort(x) => sort(min, max, x) 
1.97/2.02	    dsort(x) => sort(max, min, x) 
1.97/2.02	
1.97/2.02	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
1.97/2.02	
1.97/2.02	We use rule removal, following [Kop12, Theorem 2.23].
1.97/2.02	
1.97/2.02	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
1.97/2.02	
1.97/2.02	  sort(F, G, nil) >? nil 
1.97/2.02	  sort(F, G, cons(X, Y)) >? insert(F, G, sort(F, G, Y), X) 
1.97/2.02	  insert(F, G, nil, X) >? cons(X, nil) 
1.97/2.02	  insert(F, G, cons(X, Y), Z) >? cons(F X Z, insert(F, G, Y, G X Z)) 
1.97/2.02	  max 0 X >? X 
1.97/2.02	  max X 0 >? X 
1.97/2.02	  max s(X) s(Y) >? max X Y 
1.97/2.02	  min 0 X >? 0 
1.97/2.02	  min X 0 >? 0 
1.97/2.02	  min s(X) s(Y) >? min X Y 
1.97/2.02	  asort(X) >? sort(min, max, X) 
1.97/2.02	  dsort(X) >? sort(max, min, X) 
1.97/2.02	
1.97/2.02	We use a recursive path ordering as defined in [Kop12, Chapter 5].
1.97/2.02	
1.97/2.02	Argument functions:
1.97/2.02	
1.97/2.02	  [[0]] = _|_ 
1.97/2.02	  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_2, x_3, x_4, x_1) 
1.97/2.02	  [[nil]] = _|_ 
1.97/2.02	
1.97/2.02	We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, asort, cons, dsort, max, min, s, sort}, and the following precedence: dsort > asort = max > min > s > sort > insert > @_{o -> o} > cons > @_{o -> o -> o}
1.97/2.02	
1.97/2.02	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
1.97/2.02	
1.97/2.02	  sort(F, G, _|_) >= _|_ 
1.97/2.02	  sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X) 
1.97/2.02	  insert(F, G, _|_, X) >= cons(X, _|_) 
1.97/2.02	  insert(F, G, cons(X, Y), Z) > cons(@_{o -> o}(@_{o -> o -> o}(F, X), Z), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z))) 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(max, _|_), X) > X 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(max, X), _|_) >= X 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(max, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(max, X), Y) 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(min, _|_), X) >= _|_ 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(min, X), _|_) >= _|_ 
1.97/2.02	  @_{o -> o}(@_{o -> o -> o}(min, s(X)), s(Y)) >= @_{o -> o}(@_{o -> o -> o}(min, X), Y) 
1.97/2.02	  asort(X) >= sort(min, max, X) 
1.97/2.02	  dsort(X) >= sort(max, min, X) 
1.97/2.02	
1.97/2.02	With these choices, we have:
1.97/2.02	
1.97/2.02	  1] sort(F, G, _|_) >= _|_  by (Bot) 
1.97/2.02	
1.97/2.02	  2] sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because [3], by (Star) 
1.97/2.02	  3] sort*(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because sort > insert, [4], [6], [8] and [14], by (Copy) 
1.97/2.02	  4] sort*(F, G, cons(X, Y)) >= F  because [5], by (Select) 
1.97/2.02	  5] F >= F  by (Meta) 
1.97/2.02	  6] sort*(F, G, cons(X, Y)) >= G  because [7], by (Select) 
1.97/2.02	  7] G >= G  by (Meta) 
1.97/2.02	  8] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [9], [10] and [11], by (Stat) 
1.97/2.02	  9] F >= F  by (Meta) 
1.97/2.02	  10] G >= G  by (Meta) 
1.97/2.02	  11] cons(X, Y) > Y  because [12], by definition 
1.97/2.02	  12] cons*(X, Y) >= Y  because [13], by (Select) 
1.97/2.02	  13] Y >= Y  by (Meta) 
1.97/2.02	  14] sort*(F, G, cons(X, Y)) >= X  because [15], by (Select) 
1.97/2.02	  15] cons(X, Y) >= X  because [16], by (Star) 
1.97/2.02	  16] cons*(X, Y) >= X  because [17], by (Select) 
1.97/2.02	  17] X >= X  by (Meta) 
1.97/2.02	
1.97/2.02	  18] insert(F, G, _|_, X) >= cons(X, _|_)  because [19], by (Star) 
1.97/2.02	  19] insert*(F, G, _|_, X) >= cons(X, _|_)  because insert > cons, [20] and [22], by (Copy) 
1.97/2.02	  20] insert*(F, G, _|_, X) >= X  because [21], by (Select) 
1.97/2.02	  21] X >= X  by (Meta)

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