Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270281

details

property	value
status	complete
benchmark	from.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n035.star.cs.uiowa.edu
space	Mixed_HO_10

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	1.34477 seconds
cpu usage	2.30246
user time	2.15054
system time	0.151914
max virtual memory	7021216.0
max residence set size	113896.0

stage attributes

key	value
starexec-result	YES

output

2.26/1.32	YES
2.26/1.34	We consider the system theBenchmark.
2.26/1.34	
2.26/1.34	  Alphabet:
2.26/1.34	
2.26/1.34	    chain : [N -> N * list] --> list 
2.26/1.34	    cons : [N * list] --> list 
2.26/1.34	    false : [] --> B 
2.26/1.34	    from : [N * list] --> list 
2.26/1.34	    if : [B * list * list] --> list 
2.26/1.34	    incch : [list] --> list 
2.26/1.34	    lteq : [N * N] --> B 
2.26/1.34	    nil : [] --> list 
2.26/1.34	    o : [] --> N 
2.26/1.34	    s : [N] --> N 
2.26/1.34	    true : [] --> B 
2.26/1.34	
2.26/1.34	  Rules:
2.26/1.34	
2.26/1.34	    if(true, x, y) => x 
2.26/1.34	    if(false, x, y) => y 
2.26/1.34	    lteq(s(x), o) => false 
2.26/1.34	    lteq(o, x) => true 
2.26/1.34	    lteq(s(x), s(y)) => lteq(x, y) 
2.26/1.34	    from(x, nil) => nil 
2.26/1.34	    from(x, cons(y, z)) => if(lteq(x, y), cons(y, z), from(x, z)) 
2.26/1.34	    chain(f, nil) => nil 
2.26/1.34	    chain(f, cons(x, y)) => cons(f x, chain(f, from(f x, y))) 
2.26/1.34	    incch(x) => chain(/\y.s(y), x) 
2.26/1.34	
2.26/1.34	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
2.26/1.34	
2.26/1.34	We observe that the rules contain a first-order subset:
2.26/1.34	
2.26/1.34	  if(true, X, Y) => X 
2.26/1.34	  if(false, X, Y) => Y 
2.26/1.34	  lteq(s(X), o) => false 
2.26/1.34	  lteq(o, X) => true 
2.26/1.34	  lteq(s(X), s(Y)) => lteq(X, Y) 
2.26/1.34	  from(X, nil) => nil 
2.26/1.34	  from(X, cons(Y, Z)) => if(lteq(X, Y), cons(Y, Z), from(X, Z)) 
2.26/1.34	
2.26/1.34	Moreover, the system is orthogonal.  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 terminating when seen as a many-sorted first-order TRS.
2.26/1.34	
2.26/1.34	According to the external first-order termination prover, this system is indeed terminating:
2.26/1.34	
2.26/1.34	 || proof of resources/system.trs
2.26/1.34	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
2.26/1.34	 || 
2.26/1.34	 || 
2.26/1.34	 || Termination w.r.t. Q of the given QTRS could be proven:
2.26/1.34	 || 
2.26/1.34	 || (0) QTRS
2.26/1.34	 || (1) QTRSRRRProof [EQUIVALENT]
2.26/1.34	 || (2) QTRS
2.26/1.34	 || (3) RisEmptyProof [EQUIVALENT]
2.26/1.34	 || (4) YES
2.26/1.34	 || 
2.26/1.34	 || 
2.26/1.34	 || ----------------------------------------
2.26/1.34	 || 
2.26/1.34	 || (0)
2.26/1.34	 || Obligation:
2.26/1.34	 || Q restricted rewrite system:
2.26/1.34	 || The TRS R consists of the following rules:
2.26/1.34	 || 
2.26/1.34	 ||    if(true, %X, %Y) -> %X
2.26/1.34	 ||    if(false, %X, %Y) -> %Y
2.26/1.34	 ||    lteq(s(%X), o) -> false
2.26/1.34	 ||    lteq(o, %X) -> true
2.26/1.34	 ||    lteq(s(%X), s(%Y)) -> lteq(%X, %Y)
2.26/1.34	 ||    from(%X, nil) -> nil
2.26/1.34	 ||    from(%X, cons(%Y, %Z)) -> if(lteq(%X, %Y), cons(%Y, %Z), from(%X, %Z))
2.26/1.34	 || 
2.26/1.34	 || Q is empty.
2.26/1.34	 || 
2.26/1.34	 || ----------------------------------------
2.26/1.34	 || 
2.26/1.34	 || (1) QTRSRRRProof (EQUIVALENT)
2.26/1.34	 || Used ordering:
2.26/1.34	 || Quasi precedence:
2.26/1.34	 || from_2 > [if_3, cons_2]
2.26/1.34	 || from_2 > lteq_2 > [true, false, o]
2.26/1.34	 || from_2 > nil
2.26/1.34	 || 
2.26/1.34	 || 
2.26/1.34	 || Status:
2.26/1.34	 || if_3: multiset status
2.26/1.34	 || true: multiset status
2.26/1.34	 || false: multiset status
2.26/1.34	 || lteq_2: [2,1]
2.26/1.34	 || s_1: [1]
2.26/1.34	 || o: multiset status
2.26/1.34	 || from_2: multiset status
2.26/1.34	 || nil: multiset status
2.26/1.34	 || cons_2: multiset status
2.26/1.34	 || 
2.26/1.34	 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.26/1.34	 || 
2.26/1.34	 ||    if(true, %X, %Y) -> %X

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