Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270343

details

property	value
status	complete
benchmark	lambda_prod.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n077.star.cs.uiowa.edu
space	Hamana_17

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	1.33302 seconds
cpu usage	1.48346
user time	1.38009
system time	0.103372
max virtual memory	7015168.0
max residence set size	108488.0

stage attributes

key	value
starexec-result	YES

output

1.44/1.32	YES
1.44/1.33	We consider the system theBenchmark.
1.44/1.33	
1.44/1.33	  Alphabet:
1.44/1.33	
1.44/1.33	    app : [] --> arrab -> a -> b 
1.44/1.33	    lam : [] --> (a -> b) -> arrab 
1.44/1.33	    pair : [] --> a -> b -> prodab 
1.44/1.33	    pia : [] --> prodab -> a 
1.44/1.33	    pib : [] --> prodab -> b 
1.44/1.33	
1.44/1.33	  Rules:
1.44/1.33	
1.44/1.33	    app (lam (/\x.f x)) y => f y 
1.44/1.33	    lam (/\x.app y x) => y 
1.44/1.33	    pia (pair x y) => x 
1.44/1.33	    pib (pair x y) => y 
1.44/1.33	    pair (pia x) (pib x) => x 
1.44/1.33	
1.44/1.33	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.
1.44/1.33	
1.44/1.33	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:
1.44/1.33	
1.44/1.33	  Alphabet:
1.44/1.33	
1.44/1.33	    app : [arrab * a] --> b 
1.44/1.33	    lam : [a -> b] --> arrab 
1.44/1.33	    pair : [a * b] --> prodab 
1.44/1.33	    pia : [prodab] --> a 
1.44/1.33	    pib : [prodab] --> b 
1.44/1.33	    ~AP1 : [a -> b * a] --> b 
1.44/1.33	
1.44/1.33	  Rules:
1.44/1.33	
1.44/1.33	    app(lam(/\x.~AP1(F, x)), X) => ~AP1(F, X) 
1.44/1.33	    lam(/\x.app(X, x)) => X 
1.44/1.33	    pia(pair(X, Y)) => X 
1.44/1.33	    pib(pair(X, Y)) => Y 
1.44/1.33	    pair(pia(X), pib(X)) => X 
1.44/1.33	    app(lam(/\x.app(X, x)), Y) => app(X, Y) 
1.44/1.33	    ~AP1(F, X) => F X 
1.44/1.33	
1.44/1.33	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.  Additionally, we can remove some (now-)redundant rules.  This gives:
1.44/1.33	
1.44/1.33	  Alphabet:
1.44/1.33	
1.44/1.33	    app : [arrab * a] --> b 
1.44/1.33	    lam : [a -> b] --> arrab 
1.44/1.33	    pair : [a * b] --> prodab 
1.44/1.33	    pia : [prodab] --> a 
1.44/1.33	    pib : [prodab] --> b 
1.44/1.33	
1.44/1.33	  Rules:
1.44/1.33	
1.44/1.33	    app(lam(/\x.X(x)), Y) => X(Y) 
1.44/1.33	    lam(/\x.app(X, x)) => X 
1.44/1.33	    pia(pair(X, Y)) => X 
1.44/1.33	    pib(pair(X, Y)) => Y 
1.44/1.33	    pair(pia(X), pib(X)) => X 
1.44/1.33	
1.44/1.33	We observe that the rules contain a first-order subset:
1.44/1.33	
1.44/1.33	  pia(pair(X, Y)) => X 
1.44/1.33	  pib(pair(X, Y)) => Y 
1.44/1.33	  pair(pia(X), pib(X)) => X 
1.44/1.33	
1.44/1.33	Moreover, the system is finitely branching.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS.
1.44/1.33	
1.44/1.33	According to the external first-order termination prover, this system is indeed Ce-terminating:
1.44/1.33	
1.44/1.33	 || proof of resources/system.trs
1.44/1.33	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
1.44/1.33	 || 
1.44/1.33	 || 
1.44/1.33	 || Termination w.r.t. Q of the given QTRS could be proven:
1.44/1.33	 || 
1.44/1.33	 || (0) QTRS
1.44/1.33	 || (1) QTRSRRRProof [EQUIVALENT]
1.44/1.33	 || (2) QTRS
1.44/1.33	 || (3) RisEmptyProof [EQUIVALENT]
1.44/1.33	 || (4) YES
1.44/1.33	 || 
1.44/1.33	 || 
1.44/1.33	 || ----------------------------------------
1.44/1.33	 || 
1.44/1.33	 || (0)
1.44/1.33	 || Obligation:
1.44/1.33	 || Q restricted rewrite system:
1.44/1.33	 || The TRS R consists of the following rules:
1.44/1.33	 || 
1.44/1.33	 ||    pia(pair(%X, %Y)) -> %X
1.44/1.33	 ||    pib(pair(%X, %Y)) -> %Y
1.44/1.33	 ||    pair(pia(%X), pib(%X)) -> %X
1.44/1.33	 ||    ~PAIR(%X, %Y) -> %X
1.44/1.33	 ||    ~PAIR(%X, %Y) -> %Y
1.44/1.33	 || 
1.44/1.33	 || Q is empty.
1.44/1.33	 || 
1.44/1.33	 || ----------------------------------------
1.44/1.33	 ||

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