YES(O(1),O(1))
187.09/91.97	YES(O(1),O(1))
187.09/91.97	
187.09/91.97	We are left with following problem, upon which TcT provides the
187.09/91.97	certificate YES(O(1),O(1)).
187.09/91.97	
187.09/91.97	Strict Trs:
187.09/91.97	  { f(x, y, z) -> g(<=(x, y), x, y, z)
187.09/91.97	  , g(true(), x, y, z) -> z
187.09/91.97	  , g(false(), x, y, z) ->
187.09/91.97	    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
187.09/91.97	  , p(0()) -> 0()
187.09/91.97	  , p(s(x)) -> x }
187.09/91.97	Obligation:
187.09/91.97	  innermost runtime complexity
187.09/91.97	Answer:
187.09/91.97	  YES(O(1),O(1))
187.09/91.97	
187.09/91.97	We add the following dependency tuples:
187.09/91.97	
187.09/91.97	Strict DPs:
187.09/91.97	  { f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
187.09/91.97	  , g^#(true(), x, y, z) -> c_2()
187.09/91.97	  , g^#(false(), x, y, z) ->
187.09/91.97	    c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
187.09/91.97	        f^#(p(x), y, z),
187.09/91.97	        p^#(x),
187.09/91.97	        f^#(p(y), z, x),
187.09/91.97	        p^#(y),
187.09/91.97	        f^#(p(z), x, y),
187.09/91.97	        p^#(z))
187.09/91.97	  , p^#(0()) -> c_4()
187.09/91.97	  , p^#(s(x)) -> c_5() }
187.09/91.97	
187.09/91.97	and mark the set of starting terms.
187.09/91.97	
187.09/91.97	We are left with following problem, upon which TcT provides the
187.09/91.97	certificate YES(O(1),O(1)).
187.09/91.97	
187.09/91.97	Strict DPs:
187.09/91.97	  { f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
187.09/91.97	  , g^#(true(), x, y, z) -> c_2()
187.09/91.97	  , g^#(false(), x, y, z) ->
187.09/91.97	    c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
187.09/91.97	        f^#(p(x), y, z),
187.09/91.97	        p^#(x),
187.09/91.97	        f^#(p(y), z, x),
187.09/91.97	        p^#(y),
187.09/91.97	        f^#(p(z), x, y),
187.09/91.97	        p^#(z))
187.09/91.97	  , p^#(0()) -> c_4()
187.09/91.97	  , p^#(s(x)) -> c_5() }
187.09/91.97	Weak Trs:
187.09/91.97	  { f(x, y, z) -> g(<=(x, y), x, y, z)
187.09/91.97	  , g(true(), x, y, z) -> z
187.09/91.97	  , g(false(), x, y, z) ->
187.09/91.97	    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
187.09/91.97	  , p(0()) -> 0()
187.09/91.97	  , p(s(x)) -> x }
187.09/91.97	Obligation:
187.09/91.97	  innermost runtime complexity
187.09/91.97	Answer:
187.09/91.97	  YES(O(1),O(1))
187.09/91.97	
187.09/91.97	We estimate the number of application of {1,2,4,5} by applications
187.09/91.97	of Pre({1,2,4,5}) = {3}. Here rules are labeled as follows:
187.09/91.97	
187.09/91.97	  DPs:
187.09/91.97	    { 1: f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
187.09/91.97	    , 2: g^#(true(), x, y, z) -> c_2()
187.09/91.97	    , 3: g^#(false(), x, y, z) ->
187.09/91.97	         c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
187.09/91.97	             f^#(p(x), y, z),
187.09/91.97	             p^#(x),
187.09/91.97	             f^#(p(y), z, x),
187.09/91.97	             p^#(y),
187.09/91.97	             f^#(p(z), x, y),
187.09/91.97	             p^#(z))
187.09/91.97	    , 4: p^#(0()) -> c_4()
187.09/91.97	    , 5: p^#(s(x)) -> c_5() }
187.09/91.97	
187.09/91.97	We are left with following problem, upon which TcT provides the
187.09/91.97	certificate YES(O(1),O(1)).
187.09/91.97	
187.09/91.97	Strict DPs:
187.09/91.97	  { g^#(false(), x, y, z) ->
187.09/91.97	    c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
187.09/91.97	        f^#(p(x), y, z),
187.09/91.97	        p^#(x),
187.09/91.97	        f^#(p(y), z, x),
187.09/91.97	        p^#(y),
187.09/91.97	        f^#(p(z), x, y),
187.09/91.97	        p^#(z)) }
187.09/91.97	Weak DPs:
187.09/91.97	  { f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
187.09/91.97	  , g^#(true(), x, y, z) -> c_2()
187.09/91.97	  , p^#(0()) -> c_4()
187.09/91.97	  , p^#(s(x)) -> c_5() }
187.09/91.97	Weak Trs:
187.09/91.97	  { f(x, y, z) -> g(<=(x, y), x, y, z)
187.09/91.97	  , g(true(), x, y, z) -> z
187.09/91.97	  , g(false(), x, y, z) ->
187.09/91.97	    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
187.09/91.97	  , p(0()) -> 0()
187.09/91.97	  , p(s(x)) -> x }
187.09/91.97	Obligation:
187.09/91.97	  innermost runtime complexity
187.09/91.97	Answer:
187.09/91.97	  YES(O(1),O(1))
187.09/91.97	
187.09/91.97	We estimate the number of application of {1} by applications of
187.09/91.97	Pre({1}) = {}. Here rules are labeled as follows:
187.09/91.97	
187.09/91.97	  DPs:
187.09/91.97	    { 1: g^#(false(), x, y, z) ->
187.09/91.97	         c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
187.09/91.97	             f^#(p(x), y, z),
187.09/91.97	             p^#(x),
187.09/91.97	             f^#(p(y), z, x),
187.09/91.97	             p^#(y),
187.09/91.97	             f^#(p(z), x, y),
187.09/91.97	             p^#(z))
187.09/91.97	    , 2: f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
187.09/91.97	    , 3: g^#(true(), x, y, z) -> c_2()
187.09/91.97	    , 4: p^#(0()) -> c_4()
187.09/91.97	    , 5: p^#(s(x)) -> c_5() }
187.09/91.97	
187.09/91.97	We are left with following problem, upon which TcT provides the
187.09/91.97	certificate YES(O(1),O(1)).
187.09/91.97	
187.09/91.97	Weak DPs:
187.09/91.97	  { f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
187.09/91.97	  , g^#(true(), x, y, z) -> c_2()
187.09/91.97	  , g^#(false(), x, y, z) ->
187.09/91.97	    c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
187.09/91.97	        f^#(p(x), y, z),
187.09/91.97	        p^#(x),
187.09/91.97	        f^#(p(y), z, x),
187.09/91.97	        p^#(y),
187.09/91.97	        f^#(p(z), x, y),
187.09/91.97	        p^#(z))
187.09/91.97	  , p^#(0()) -> c_4()
187.09/91.97	  , p^#(s(x)) -> c_5() }
187.09/91.97	Weak Trs:
187.09/91.97	  { f(x, y, z) -> g(<=(x, y), x, y, z)
187.09/91.97	  , g(true(), x, y, z) -> z
187.09/91.97	  , g(false(), x, y, z) ->
187.09/91.97	    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
187.09/91.97	  , p(0()) -> 0()
187.09/91.97	  , p(s(x)) -> x }
187.09/91.97	Obligation:
187.09/91.97	  innermost runtime complexity
187.09/91.97	Answer:
187.09/91.97	  YES(O(1),O(1))
187.09/91.97	
187.09/91.97	The following weak DPs constitute a sub-graph of the DG that is
187.09/91.97	closed under successors. The DPs are removed.
187.09/91.97	
187.09/91.97	{ f^#(x, y, z) -> c_1(g^#(<=(x, y), x, y, z))
187.09/91.97	, g^#(true(), x, y, z) -> c_2()
187.09/91.97	, g^#(false(), x, y, z) ->
187.09/91.97	  c_3(f^#(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)),
187.09/91.97	      f^#(p(x), y, z),
187.09/91.97	      p^#(x),
187.09/91.97	      f^#(p(y), z, x),
187.09/91.97	      p^#(y),
187.09/91.97	      f^#(p(z), x, y),
187.09/91.97	      p^#(z))
187.09/91.97	, p^#(0()) -> c_4()
187.09/91.97	, p^#(s(x)) -> c_5() }
187.09/91.97	
187.09/91.97	We are left with following problem, upon which TcT provides the
187.09/91.97	certificate YES(O(1),O(1)).
187.09/91.97	
187.09/91.97	Weak Trs:
187.09/91.97	  { f(x, y, z) -> g(<=(x, y), x, y, z)
187.09/91.97	  , g(true(), x, y, z) -> z
187.09/91.97	  , g(false(), x, y, z) ->
187.09/91.97	    f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
187.09/91.97	  , p(0()) -> 0()
187.09/91.97	  , p(s(x)) -> x }
187.09/91.97	Obligation:
187.09/91.97	  innermost runtime complexity
187.09/91.97	Answer:
187.09/91.97	  YES(O(1),O(1))
187.09/91.97	
187.09/91.97	No rule is usable, rules are removed from the input problem.
187.09/91.97	
187.09/91.97	We are left with following problem, upon which TcT provides the
187.09/91.97	certificate YES(O(1),O(1)).
187.09/91.97	
187.09/91.97	Rules: Empty
187.09/91.97	Obligation:
187.09/91.97	  innermost runtime complexity
187.09/91.97	Answer:
187.09/91.97	  YES(O(1),O(1))
187.09/91.97	
187.09/91.97	Empty rules are trivially bounded
187.09/91.97	
187.09/91.97	Hurray, we answered YES(O(1),O(1))
187.36/92.05	EOF