Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270351

details

property	value
status	complete
benchmark	restriction.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n129.star.cs.uiowa.edu
space	Hamana_17

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	10.0353 seconds
cpu usage	9.9879
user time	9.36367
system time	0.624225
max virtual memory	765196.0
max residence set size	426636.0

stage attributes

key	value
starexec-result	YES

output

9.85/9.98	YES
9.85/9.99	We consider the system theBenchmark.
9.85/9.99	
9.85/9.99	  Alphabet:
9.85/9.99	
9.85/9.99	    New : [] --> (N -> A) -> A 
9.85/9.99	
9.85/9.99	  Rules:
9.85/9.99	
9.85/9.99	    New (/\x.y) => y 
9.85/9.99	    New (/\x.New (/\y.f x y)) => New (/\z.New (/\u.f u z)) 
9.85/9.99	
9.85/9.99	Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.
9.85/9.99	
9.85/9.99	We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:
9.85/9.99	
9.85/9.99	  Alphabet:
9.85/9.99	
9.85/9.99	    New : [N -> A] --> A 
9.85/9.99	    ~AP1 : [N -> N -> A * N] --> N -> A 
9.85/9.99	
9.85/9.99	  Rules:
9.85/9.99	
9.85/9.99	    New(/\x.X) => X 
9.85/9.99	    New(/\x.New(/\y.~AP1(F, x) y)) => New(/\z.New(/\u.~AP1(F, u) z)) 
9.85/9.99	    ~AP1(F, X) => F X 
9.85/9.99	
9.85/9.99	We use rule removal, following [Kop12, Theorem 2.23].
9.85/9.99	
9.85/9.99	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
9.85/9.99	
9.85/9.99	  New(/\x.X) >? X 
9.85/9.99	  New(/\x.New(/\y.~AP1(F, x) y)) >? New(/\z.New(/\u.~AP1(F, u) z)) 
9.85/9.99	  ~AP1(F, X) >? F X 
9.85/9.99	
9.85/9.99	We orient these requirements with a polynomial interpretation in the natural numbers.
9.85/9.99	
9.85/9.99	The following interpretation satisfies the requirements:
9.85/9.99	
9.85/9.99	  New = \G0.3 + G0(0) 
9.85/9.99	  ~AP1 = \G0y1y2.3 + y1 + G0(y1,y2) 
9.85/9.99	
9.85/9.99	Using this interpretation, the requirements translate to:
9.85/9.99	
9.85/9.99	  [[New(/\x._x0)]] = 3 + x0 > x0 = [[_x0]] 
9.85/9.99	  [[New(/\x.New(/\y.~AP1(_F0, x) y))]] = 9 + F0(0,0) >= 9 + F0(0,0) = [[New(/\x.New(/\y.~AP1(_F0, y) x))]] 
9.85/9.99	  [[~AP1(_F0, _x1)]] = \y0.3 + x1 + F0(x1,y0) > \y0.x1 + F0(x1,y0) = [[_F0 _x1]] 
9.85/9.99	
9.85/9.99	We can thus remove the following rules:
9.85/9.99	
9.85/9.99	  New(/\x.X) => X 
9.85/9.99	  ~AP1(F, X) => F X 
9.85/9.99	
9.85/9.99	We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.
9.85/9.99	
9.85/9.99	We thus obtain the following dependency pair problem (P_0, R_0, minimal, all):
9.85/9.99	
9.85/9.99	  Dependency Pairs P_0:
9.85/9.99	
9.85/9.99	    0] New#(/\x.New(/\y.~AP1(F, x, y))) =#> New#(/\z.New(/\u.~AP1(F, u, z)))   
9.85/9.99	    1] New#(/\x.New(/\y.~AP1(F, x, y))) =#> New#(/\z.~AP1(F, z, u))   
9.85/9.99	
9.85/9.99	  Rules R_0:
9.85/9.99	
9.85/9.99	    New(/\x.New(/\y.~AP1(F, x, y))) => New(/\z.New(/\u.~AP1(F, u, z))) 
9.85/9.99	
9.85/9.99	Thus, the original system is terminating if (P_0, R_0, minimal, all) is finite.
9.85/9.99	
9.85/9.99	We consider the dependency pair problem (P_0, R_0, minimal, all).
9.85/9.99	
9.85/9.99	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
9.85/9.99	
9.85/9.99	    * 0 : 0, 1 
9.85/9.99	    * 1 :  
9.85/9.99	
9.85/9.99	This graph has the following strongly connected components:
9.85/9.99	
9.85/9.99	  P_1:
9.85/9.99	
9.85/9.99	    New#(/\x.New(/\y.~AP1(F, x, y))) =#> New#(/\z.New(/\u.~AP1(F, u, z)))   
9.85/9.99	
9.85/9.99	By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).
9.85/9.99	
9.85/9.99	Thus, the original system is terminating if (P_1, R_0, minimal, all) is finite.
9.85/9.99	
9.85/9.99	We consider the dependency pair problem (P_1, R_0, minimal, all).
9.85/9.99	
9.85/9.99	We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:
9.85/9.99	
9.85/9.99	  New#(/\x.New(/\y.~AP1(F, x, y))) >? New#(/\z.New(/\u.~AP1(F, u, z))) 
9.85/9.99	  New(/\x.New(/\y.~AP1(F, x, y))) >= New(/\z.New(/\u.~AP1(F, u, z))) 
9.85/9.99	
9.85/9.99	Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)
9.85/9.99	
9.85/9.99	We use a recursive path ordering as defined in [Kop12, Chapter 5].
9.85/9.99	
9.85/9.99	Argument functions:
9.85/9.99	
9.85/9.99	  [[~AP1(x_1, x_2, x_3)]] = ~AP1(x_1, x_2) 
9.85/9.99

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