Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270503

details

property	value
status	complete
benchmark	09ex.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n037.star.cs.uiowa.edu
space	Blanqui_15

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	0.0181689 seconds
cpu usage	0.018147
user time	0.01093
system time	0.007217
max virtual memory	113176.0
max residence set size	1876.0

stage attributes

key	value
starexec-result	YES

output

0.00/0.01	YES
0.00/0.01	We consider the system theBenchmark.
0.00/0.01	
0.00/0.01	  Alphabet:
0.00/0.01	
0.00/0.01	    c : [] --> ((C -> L) -> L) -> C 
0.00/0.01	    d : [] --> C 
0.00/0.01	    ex : [] --> C -> L 
0.00/0.01	    nil : [] --> L 
0.00/0.01	
0.00/0.01	  Rules:
0.00/0.01	
0.00/0.01	    ex d => nil 
0.00/0.01	    ex (c (/\f.g f)) => g ex 
0.00/0.01	
0.00/0.01	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.
0.00/0.01	
0.00/0.01	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:
0.00/0.01	
0.00/0.01	  Alphabet:
0.00/0.01	
0.00/0.01	    c : [(C -> L) -> L] --> C 
0.00/0.01	    d : [] --> C 
0.00/0.01	    ex : [] --> C -> L 
0.00/0.01	    nil : [] --> L 
0.00/0.01	    ~AP1 : [(C -> L) -> L * C -> L] --> L 
0.00/0.01	
0.00/0.01	  Rules:
0.00/0.01	
0.00/0.01	    ex d => nil 
0.00/0.01	    ex c(/\f.~AP1(F, f)) => ~AP1(F, ex) 
0.00/0.01	    ~AP1(F, G) => F G 
0.00/0.01	
0.00/0.01	Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:
0.00/0.01	
0.00/0.01	  Alphabet:
0.00/0.01	
0.00/0.01	    c : [(C -> L) -> L] --> C 
0.00/0.01	    d : [] --> C 
0.00/0.01	    ex : [] --> C -> L 
0.00/0.01	    nil : [] --> L 
0.00/0.01	
0.00/0.01	  Rules:
0.00/0.01	
0.00/0.01	    ex d => nil 
0.00/0.01	    ex c(/\f.X(f)) => X(ex) 
0.00/0.01	
0.00/0.01	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.01	
0.00/0.01	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.01	
0.00/0.01	  ex d >? nil 
0.00/0.01	  ex c(/\f.X(f)) >? X(ex) 
0.00/0.01	
0.00/0.01	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.01	
0.00/0.01	The following interpretation satisfies the requirements:
0.00/0.01	
0.00/0.01	  c = \G0.3 + G0(\y1.0) 
0.00/0.01	  d = 3 
0.00/0.01	  ex = \y0.0 
0.00/0.01	  nil = 0 
0.00/0.01	
0.00/0.01	Using this interpretation, the requirements translate to:
0.00/0.01	
0.00/0.01	  [[ex d]] = 3 > 0 = [[nil]] 
0.00/0.01	  [[ex c(/\f._x0(f))]] = 3 + F0(\y0.0) > F0(\y0.0) = [[_x0(ex)]] 
0.00/0.01	
0.00/0.01	We can thus remove the following rules:
0.00/0.01	
0.00/0.01	  ex d => nil 
0.00/0.01	  ex c(/\f.X(f)) => X(ex) 
0.00/0.01	
0.00/0.01	All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.
0.00/0.01	
0.00/0.01	
0.00/0.01	+++ Citations +++
0.00/0.01	
0.00/0.01	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
0.00/0.01	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.01	EOF

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