details

property	value
status	complete
benchmark	quicksort.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n173.star.cs.uiowa.edu
space	AProVE_06

run statistics

property	value
solver	muterm 6.0.3
configuration	default
runtime (wallclock)	0.805816 seconds
cpu usage	0.756874
user time	0.628109
system time	0.128765
max virtual memory	714220.0
max residence set size	10932.0

stage attributes

key	value
starexec-result	YES

output

YES Problem 1: (VAR v_NonEmpty:S m:S n:S x:S y:S) (RULES app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S)) app(nil,y:S) -> y:S head(add(n:S,x:S)) -> n:S high(n:S,add(m:S,x:S)) -> if_high(le(m:S,n:S),n:S,add(m:S,x:S)) high(n:S,nil) -> nil if_high(ffalse,n:S,add(m:S,x:S)) -> add(m:S,high(n:S,x:S)) if_high(ttrue,n:S,add(m:S,x:S)) -> high(n:S,x:S) if_low(ffalse,n:S,add(m:S,x:S)) -> low(n:S,x:S) if_low(ttrue,n:S,add(m:S,x:S)) -> add(m:S,low(n:S,x:S)) if_qs(ffalse,x:S,n:S,y:S) -> app(quicksort(x:S),add(n:S,quicksort(y:S))) if_qs(ttrue,x:S,n:S,y:S) -> nil isempty(add(n:S,x:S)) -> ffalse isempty(nil) -> ttrue le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) low(n:S,add(m:S,x:S)) -> if_low(le(m:S,n:S),n:S,add(m:S,x:S)) low(n:S,nil) -> nil quicksort(x:S) -> if_qs(isempty(x:S),low(head(x:S),tail(x:S)),head(x:S),high(head(x:S),tail(x:S))) tail(add(n:S,x:S)) -> x:S ) Problem 1: Innermost Equivalent Processor: -> Rules: app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S)) app(nil,y:S) -> y:S head(add(n:S,x:S)) -> n:S high(n:S,add(m:S,x:S)) -> if_high(le(m:S,n:S),n:S,add(m:S,x:S)) high(n:S,nil) -> nil if_high(ffalse,n:S,add(m:S,x:S)) -> add(m:S,high(n:S,x:S)) if_high(ttrue,n:S,add(m:S,x:S)) -> high(n:S,x:S) if_low(ffalse,n:S,add(m:S,x:S)) -> low(n:S,x:S) if_low(ttrue,n:S,add(m:S,x:S)) -> add(m:S,low(n:S,x:S)) if_qs(ffalse,x:S,n:S,y:S) -> app(quicksort(x:S),add(n:S,quicksort(y:S))) if_qs(ttrue,x:S,n:S,y:S) -> nil isempty(add(n:S,x:S)) -> ffalse isempty(nil) -> ttrue le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) low(n:S,add(m:S,x:S)) -> if_low(le(m:S,n:S),n:S,add(m:S,x:S)) low(n:S,nil) -> nil quicksort(x:S) -> if_qs(isempty(x:S),low(head(x:S),tail(x:S)),head(x:S),high(head(x:S),tail(x:S))) tail(add(n:S,x:S)) -> x:S -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination. Problem 1: Dependency Pairs Processor: -> Pairs: APP(add(n:S,x:S),y:S) -> APP(x:S,y:S) HIGH(n:S,add(m:S,x:S)) -> IF_HIGH(le(m:S,n:S),n:S,add(m:S,x:S)) HIGH(n:S,add(m:S,x:S)) -> LE(m:S,n:S) IF_HIGH(ffalse,n:S,add(m:S,x:S)) -> HIGH(n:S,x:S) IF_HIGH(ttrue,n:S,add(m:S,x:S)) -> HIGH(n:S,x:S) IF_LOW(ffalse,n:S,add(m:S,x:S)) -> LOW(n:S,x:S) IF_LOW(ttrue,n:S,add(m:S,x:S)) -> LOW(n:S,x:S) IF_QS(ffalse,x:S,n:S,y:S) -> APP(quicksort(x:S),add(n:S,quicksort(y:S))) IF_QS(ffalse,x:S,n:S,y:S) -> QUICKSORT(x:S) IF_QS(ffalse,x:S,n:S,y:S) -> QUICKSORT(y:S) LE(s(x:S),s(y:S)) -> LE(x:S,y:S) LOW(n:S,add(m:S,x:S)) -> IF_LOW(le(m:S,n:S),n:S,add(m:S,x:S)) LOW(n:S,add(m:S,x:S)) -> LE(m:S,n:S) QUICKSORT(x:S) -> HEAD(x:S) QUICKSORT(x:S) -> HIGH(head(x:S),tail(x:S)) QUICKSORT(x:S) -> IF_QS(isempty(x:S),low(head(x:S),tail(x:S)),head(x:S),high(head(x:S),tail(x:S))) QUICKSORT(x:S) -> ISEMPTY(x:S) QUICKSORT(x:S) -> LOW(head(x:S),tail(x:S)) QUICKSORT(x:S) -> TAIL(x:S) -> Rules: app(add(n:S,x:S),y:S) -> add(n:S,app(x:S,y:S)) app(nil,y:S) -> y:S head(add(n:S,x:S)) -> n:S high(n:S,add(m:S,x:S)) -> if_high(le(m:S,n:S),n:S,add(m:S,x:S)) high(n:S,nil) -> nil if_high(ffalse,n:S,add(m:S,x:S)) -> add(m:S,high(n:S,x:S)) if_high(ttrue,n:S,add(m:S,x:S)) -> high(n:S,x:S) if_low(ffalse,n:S,add(m:S,x:S)) -> low(n:S,x:S) if_low(ttrue,n:S,add(m:S,x:S)) -> add(m:S,low(n:S,x:S)) if_qs(ffalse,x:S,n:S,y:S) -> app(quicksort(x:S),add(n:S,quicksort(y:S))) if_qs(ttrue,x:S,n:S,y:S) -> nil isempty(add(n:S,x:S)) -> ffalse isempty(nil) -> ttrue le(0,y:S) -> ttrue le(s(x:S),0) -> ffalse le(s(x:S),s(y:S)) -> le(x:S,y:S) low(n:S,add(m:S,x:S)) -> if_low(le(m:S,n:S),n:S,add(m:S,x:S)) low(n:S,nil) -> nil quicksort(x:S) -> if_qs(isempty(x:S),low(head(x:S),tail(x:S)),head(x:S),high(head(x:S),tail(x:S))) tail(add(n:S,x:S)) -> x:S popout

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actions

all output return to TRS Standard