Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270319

details

property	value
status	complete
benchmark	qsort.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n113.star.cs.uiowa.edu
space	Mixed_HO_10

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	2.16807 seconds
cpu usage	3.01813
user time	2.80856
system time	0.209566
max virtual memory	6759440.0
max residence set size	110644.0

stage attributes

key	value
starexec-result	YES

output

2.85/2.09	YES
2.93/2.16	We consider the system theBenchmark.
2.93/2.16	
2.93/2.16	  Alphabet:
2.93/2.16	
2.93/2.16	    0 : [] --> nat 
2.93/2.16	    app : [list * list] --> list 
2.93/2.16	    cons : [nat * list] --> list 
2.93/2.16	    false : [] --> bool 
2.93/2.16	    filter : [nat -> bool * list] --> list 
2.93/2.16	    gr : [nat * nat] --> bool 
2.93/2.16	    if : [bool * list * list] --> list 
2.93/2.16	    le : [nat * nat] --> bool 
2.93/2.16	    nil : [] --> list 
2.93/2.16	    qsort : [list] --> list 
2.93/2.16	    s : [nat] --> nat 
2.93/2.16	    true : [] --> bool 
2.93/2.16	
2.93/2.16	  Rules:
2.93/2.16	
2.93/2.16	    if(true, x, y) => x 
2.93/2.16	    if(false, x, y) => y 
2.93/2.16	    app(nil, x) => x 
2.93/2.16	    app(cons(x, y), z) => cons(x, app(y, z)) 
2.93/2.16	    le(0, x) => true 
2.93/2.16	    le(s(x), 0) => false 
2.93/2.16	    le(s(x), s(y)) => le(x, y) 
2.93/2.16	    gr(0, x) => false 
2.93/2.16	    gr(s(x), 0) => true 
2.93/2.16	    gr(s(x), s(y)) => gr(x, y) 
2.93/2.16	    filter(f, nil) => nil 
2.93/2.16	    filter(f, cons(x, y)) => if(f x, cons(x, filter(f, y)), filter(f, y)) 
2.93/2.16	    qsort(nil) => nil 
2.93/2.16	    qsort(cons(x, y)) => app(qsort(filter(/\z.le(z, x), y)), app(cons(x, nil), qsort(filter(/\u.gr(u, x), y)))) 
2.93/2.16	
2.93/2.16	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
2.93/2.16	
2.93/2.16	We observe that the rules contain a first-order subset:
2.93/2.16	
2.93/2.16	  if(true, X, Y) => X 
2.93/2.16	  if(false, X, Y) => Y 
2.93/2.16	  app(nil, X) => X 
2.93/2.16	  app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
2.93/2.16	  le(0, X) => true 
2.93/2.16	  le(s(X), 0) => false 
2.93/2.16	  le(s(X), s(Y)) => le(X, Y) 
2.93/2.16	  gr(0, X) => false 
2.93/2.16	  gr(s(X), 0) => true 
2.93/2.16	  gr(s(X), s(Y)) => gr(X, Y) 
2.93/2.16	
2.93/2.16	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.
2.93/2.16	
2.93/2.16	According to the external first-order termination prover, this system is indeed terminating:
2.93/2.16	
2.93/2.16	 || proof of resources/system.trs
2.93/2.16	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
2.93/2.16	 || 
2.93/2.16	 || 
2.93/2.16	 || Termination w.r.t. Q of the given QTRS could be proven:
2.93/2.16	 || 
2.93/2.16	 || (0) QTRS
2.93/2.16	 || (1) QTRSRRRProof [EQUIVALENT]
2.93/2.16	 || (2) QTRS
2.93/2.16	 || (3) RisEmptyProof [EQUIVALENT]
2.93/2.16	 || (4) YES
2.93/2.16	 || 
2.93/2.16	 || 
2.93/2.16	 || ----------------------------------------
2.93/2.16	 || 
2.93/2.16	 || (0)
2.93/2.16	 || Obligation:
2.93/2.16	 || Q restricted rewrite system:
2.93/2.16	 || The TRS R consists of the following rules:
2.93/2.16	 || 
2.93/2.16	 ||    if(true, %X, %Y) -> %X
2.93/2.16	 ||    if(false, %X, %Y) -> %Y
2.93/2.16	 ||    app(nil, %X) -> %X
2.93/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
2.93/2.16	 ||    le(0, %X) -> true
2.93/2.16	 ||    le(s(%X), 0) -> false
2.93/2.16	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
2.93/2.16	 ||    gr(0, %X) -> false
2.93/2.16	 ||    gr(s(%X), 0) -> true
2.93/2.16	 ||    gr(s(%X), s(%Y)) -> gr(%X, %Y)
2.93/2.16	 || 
2.93/2.16	 || Q is empty.
2.93/2.16	 || 
2.93/2.16	 || ----------------------------------------
2.93/2.16	 || 
2.93/2.16	 || (1) QTRSRRRProof (EQUIVALENT)
2.93/2.16	 || Used ordering:
2.93/2.16	 || Knuth-Bendix order [KBO] with precedence:s_1 > gr_2 > if_3 > 0 > le_2 > app_2 > cons_2 > nil > false > true
2.93/2.16	 || 
2.93/2.16	 || and weight map:
2.93/2.16	 || 
2.93/2.16	 ||    true=2
2.93/2.16	 ||    false=2
2.93/2.16	 ||    nil=1
2.93/2.16	 ||    0=1
2.93/2.16	 ||    s_1=0

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