Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270603

details

property	value
status	complete
benchmark	Applicative_05__TreeFlatten.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n118.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	0.168582 seconds
cpu usage	0.14737
user time	0.126812
system time	0.020558
max virtual memory	113176.0
max residence set size	3460.0

stage attributes

key	value
starexec-result	YES

output

0.00/0.16	YES
0.00/0.16	We consider the system theBenchmark.
0.00/0.16	
0.00/0.16	  Alphabet:
0.00/0.16	
0.00/0.16	    append : [a * a] --> a 
0.00/0.16	    concat : [a] --> a 
0.00/0.16	    cons : [a * a] --> a 
0.00/0.16	    flatten : [] --> a -> a 
0.00/0.16	    map : [a -> a * a] --> a 
0.00/0.16	    nil : [] --> a 
0.00/0.16	    node : [a * a] --> a 
0.00/0.16	
0.00/0.16	  Rules:
0.00/0.16	
0.00/0.16	    map(f, nil) => nil 
0.00/0.16	    map(f, cons(x, y)) => cons(f x, map(f, y)) 
0.00/0.16	    flatten node(x, y) => cons(x, concat(map(flatten, y))) 
0.00/0.16	    concat(nil) => nil 
0.00/0.16	    concat(cons(x, y)) => append(x, concat(y)) 
0.00/0.16	    append(nil, x) => x 
0.00/0.16	    append(cons(x, y), z) => cons(x, append(y, z)) 
0.00/0.16	
0.00/0.16	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.00/0.16	
0.00/0.16	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.16	
0.00/0.16	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.16	
0.00/0.16	  map(F, nil) >? nil 
0.00/0.16	  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 
0.00/0.16	  flatten node(X, Y) >? cons(X, concat(map(flatten, Y))) 
0.00/0.16	  concat(nil) >? nil 
0.00/0.16	  concat(cons(X, Y)) >? append(X, concat(Y)) 
0.00/0.16	  append(nil, X) >? X 
0.00/0.16	  append(cons(X, Y), Z) >? cons(X, append(Y, Z)) 
0.00/0.16	
0.00/0.16	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.16	
0.00/0.16	The following interpretation satisfies the requirements:
0.00/0.16	
0.00/0.16	  append = \y0y1.y0 + y1 
0.00/0.16	  concat = \y0.y0 
0.00/0.16	  cons = \y0y1.1 + y0 + y1 
0.00/0.16	  flatten = \y0.0 
0.00/0.16	  map = \G0y1.2 + y1 + G0(y1) + 2y1G0(y1) 
0.00/0.16	  nil = 0 
0.00/0.16	  node = \y0y1.3 + 3y0 + 3y1 
0.00/0.16	
0.00/0.16	Using this interpretation, the requirements translate to:
0.00/0.16	
0.00/0.16	  [[map(_F0, nil)]] = 2 + F0(0) > 0 = [[nil]] 
0.00/0.16	  [[map(_F0, cons(_x1, _x2))]] = 3 + x1 + x2 + 2x1F0(1 + x1 + x2) + 2x2F0(1 + x1 + x2) + 3F0(1 + x1 + x2) >= 3 + x1 + x2 + F0(x1) + F0(x2) + 2x2F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 
0.00/0.16	  [[flatten node(_x0, _x1)]] = 3 + 3x0 + 3x1 >= 3 + x0 + x1 = [[cons(_x0, concat(map(flatten, _x1)))]] 
0.00/0.16	  [[concat(nil)]] = 0 >= 0 = [[nil]] 
0.00/0.16	  [[concat(cons(_x0, _x1))]] = 1 + x0 + x1 > x0 + x1 = [[append(_x0, concat(_x1))]] 
0.00/0.16	  [[append(nil, _x0)]] = x0 >= x0 = [[_x0]] 
0.00/0.16	  [[append(cons(_x0, _x1), _x2)]] = 1 + x0 + x1 + x2 >= 1 + x0 + x1 + x2 = [[cons(_x0, append(_x1, _x2))]] 
0.00/0.16	
0.00/0.16	We can thus remove the following rules:
0.00/0.16	
0.00/0.16	  map(F, nil) => nil 
0.00/0.16	  concat(cons(X, Y)) => append(X, concat(Y)) 
0.00/0.16	
0.00/0.16	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.16	
0.00/0.16	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.16	
0.00/0.16	  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 
0.00/0.16	  flatten node(X, Y) >? cons(X, concat(map(flatten, Y))) 
0.00/0.16	  concat(nil) >? nil 
0.00/0.16	  append(nil, X) >? X 
0.00/0.16	  append(cons(X, Y), Z) >? cons(X, append(Y, Z)) 
0.00/0.16	
0.00/0.16	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.16	
0.00/0.16	The following interpretation satisfies the requirements:
0.00/0.16	
0.00/0.16	  append = \y0y1.3 + y1 + 3y0 
0.00/0.16	  concat = \y0.y0 
0.00/0.16	  cons = \y0y1.3 + y0 + y1 
0.00/0.16	  flatten = \y0.0 
0.00/0.16	  map = \G0y1.2y1 + G0(0) + 3y1G0(y1) 
0.00/0.16	  nil = 0 
0.00/0.16	  node = \y0y1.3 + 3y0 + 3y1 
0.00/0.16	
0.00/0.16	Using this interpretation, the requirements translate to:
0.00/0.16	
0.00/0.16	  [[map(_F0, cons(_x1, _x2))]] = 6 + 2x1 + 2x2 + F0(0) + 3x1F0(3 + x1 + x2) + 3x2F0(3 + x1 + x2) + 9F0(3 + x1 + x2) > 3 + x1 + 2x2 + F0(0) + F0(x1) + 3x2F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 
0.00/0.16	  [[flatten node(_x0, _x1)]] = 3 + 3x0 + 3x1 >= 3 + x0 + 2x1 = [[cons(_x0, concat(map(flatten, _x1)))]] 
0.00/0.16	  [[concat(nil)]] = 0 >= 0 = [[nil]] 
0.00/0.16	  [[append(nil, _x0)]] = 3 + x0 > x0 = [[_x0]] 
0.00/0.16	  [[append(cons(_x0, _x1), _x2)]] = 12 + x2 + 3x0 + 3x1 > 6 + x0 + x2 + 3x1 = [[cons(_x0, append(_x1, _x2))]] 
0.00/0.16	
0.00/0.16	We can thus remove the following rules:
0.00/0.16	
0.00/0.16	  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
0.00/0.16	  append(nil, X) => X 
0.00/0.16	  append(cons(X, Y), Z) => cons(X, append(Y, Z)) 
0.00/0.16

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