MAYBE

Problem:
 diff(x,y) -> cond1(equal(x,y),x,y)
 cond1(true(),x,y) -> 0()
 cond1(false(),x,y) -> cond2(gt(x,y),x,y)
 cond2(true(),x,y) -> s(diff(x,s(y)))
 cond2(false(),x,y) -> s(diff(s(x),y))
 gt(0(),v) -> false()
 gt(s(u),0()) -> true()
 gt(s(u),s(v)) -> gt(u,v)
 equal(0(),0()) -> true()
 equal(s(x),0()) -> false()
 equal(0(),s(y)) -> false()
 equal(s(x),s(y)) -> equal(x,y)

Proof:
 DP Processor:
  DPs:
   diff#(x,y) -> equal#(x,y)
   diff#(x,y) -> cond1#(equal(x,y),x,y)
   cond1#(false(),x,y) -> gt#(x,y)
   cond1#(false(),x,y) -> cond2#(gt(x,y),x,y)
   cond2#(true(),x,y) -> diff#(x,s(y))
   cond2#(false(),x,y) -> diff#(s(x),y)
   gt#(s(u),s(v)) -> gt#(u,v)
   equal#(s(x),s(y)) -> equal#(x,y)
  TRS:
   diff(x,y) -> cond1(equal(x,y),x,y)
   cond1(true(),x,y) -> 0()
   cond1(false(),x,y) -> cond2(gt(x,y),x,y)
   cond2(true(),x,y) -> s(diff(x,s(y)))
   cond2(false(),x,y) -> s(diff(s(x),y))
   gt(0(),v) -> false()
   gt(s(u),0()) -> true()
   gt(s(u),s(v)) -> gt(u,v)
   equal(0(),0()) -> true()
   equal(s(x),0()) -> false()
   equal(0(),s(y)) -> false()
   equal(s(x),s(y)) -> equal(x,y)
  TDG Processor:
   DPs:
    diff#(x,y) -> equal#(x,y)
    diff#(x,y) -> cond1#(equal(x,y),x,y)
    cond1#(false(),x,y) -> gt#(x,y)
    cond1#(false(),x,y) -> cond2#(gt(x,y),x,y)
    cond2#(true(),x,y) -> diff#(x,s(y))
    cond2#(false(),x,y) -> diff#(s(x),y)
    gt#(s(u),s(v)) -> gt#(u,v)
    equal#(s(x),s(y)) -> equal#(x,y)
   TRS:
    diff(x,y) -> cond1(equal(x,y),x,y)
    cond1(true(),x,y) -> 0()
    cond1(false(),x,y) -> cond2(gt(x,y),x,y)
    cond2(true(),x,y) -> s(diff(x,s(y)))
    cond2(false(),x,y) -> s(diff(s(x),y))
    gt(0(),v) -> false()
    gt(s(u),0()) -> true()
    gt(s(u),s(v)) -> gt(u,v)
    equal(0(),0()) -> true()
    equal(s(x),0()) -> false()
    equal(0(),s(y)) -> false()
    equal(s(x),s(y)) -> equal(x,y)
   graph:
    cond2#(false(),x,y) -> diff#(s(x),y) ->
    diff#(x,y) -> cond1#(equal(x,y),x,y)
    cond2#(false(),x,y) -> diff#(s(x),y) ->
    diff#(x,y) -> equal#(x,y)
    cond2#(true(),x,y) -> diff#(x,s(y)) ->
    diff#(x,y) -> cond1#(equal(x,y),x,y)
    cond2#(true(),x,y) -> diff#(x,s(y)) -> diff#(x,y) -> equal#(x,y)
    gt#(s(u),s(v)) -> gt#(u,v) ->
    gt#(s(u),s(v)) -> gt#(u,v)
    cond1#(false(),x,y) -> cond2#(gt(x,y),x,y) ->
    cond2#(false(),x,y) -> diff#(s(x),y)
    cond1#(false(),x,y) -> cond2#(gt(x,y),x,y) ->
    cond2#(true(),x,y) -> diff#(x,s(y))
    cond1#(false(),x,y) -> gt#(x,y) -> gt#(s(u),s(v)) -> gt#(u,v)
    equal#(s(x),s(y)) -> equal#(x,y) ->
    equal#(s(x),s(y)) -> equal#(x,y)
    diff#(x,y) -> cond1#(equal(x,y),x,y) ->
    cond1#(false(),x,y) -> cond2#(gt(x,y),x,y)
    diff#(x,y) -> cond1#(equal(x,y),x,y) ->
    cond1#(false(),x,y) -> gt#(x,y)
    diff#(x,y) -> equal#(x,y) -> equal#(s(x),s(y)) -> equal#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 6
    #arcs: 12/64
    DPs:
     cond2#(false(),x,y) -> diff#(s(x),y)
     diff#(x,y) -> cond1#(equal(x,y),x,y)
     cond1#(false(),x,y) -> cond2#(gt(x,y),x,y)
     cond2#(true(),x,y) -> diff#(x,s(y))
    TRS:
     diff(x,y) -> cond1(equal(x,y),x,y)
     cond1(true(),x,y) -> 0()
     cond1(false(),x,y) -> cond2(gt(x,y),x,y)
     cond2(true(),x,y) -> s(diff(x,s(y)))
     cond2(false(),x,y) -> s(diff(s(x),y))
     gt(0(),v) -> false()
     gt(s(u),0()) -> true()
     gt(s(u),s(v)) -> gt(u,v)
     equal(0(),0()) -> true()
     equal(s(x),0()) -> false()
     equal(0(),s(y)) -> false()
     equal(s(x),s(y)) -> equal(x,y)
    Open
    
    DPs:
     gt#(s(u),s(v)) -> gt#(u,v)
    TRS:
     diff(x,y) -> cond1(equal(x,y),x,y)
     cond1(true(),x,y) -> 0()
     cond1(false(),x,y) -> cond2(gt(x,y),x,y)
     cond2(true(),x,y) -> s(diff(x,s(y)))
     cond2(false(),x,y) -> s(diff(s(x),y))
     gt(0(),v) -> false()
     gt(s(u),0()) -> true()
     gt(s(u),s(v)) -> gt(u,v)
     equal(0(),0()) -> true()
     equal(s(x),0()) -> false()
     equal(0(),s(y)) -> false()
     equal(s(x),s(y)) -> equal(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(gt#) = 1
     problem:
      DPs:
       
      TRS:
       diff(x,y) -> cond1(equal(x,y),x,y)
       cond1(true(),x,y) -> 0()
       cond1(false(),x,y) -> cond2(gt(x,y),x,y)
       cond2(true(),x,y) -> s(diff(x,s(y)))
       cond2(false(),x,y) -> s(diff(s(x),y))
       gt(0(),v) -> false()
       gt(s(u),0()) -> true()
       gt(s(u),s(v)) -> gt(u,v)
       equal(0(),0()) -> true()
       equal(s(x),0()) -> false()
       equal(0(),s(y)) -> false()
       equal(s(x),s(y)) -> equal(x,y)
     Qed
    
    DPs:
     equal#(s(x),s(y)) -> equal#(x,y)
    TRS:
     diff(x,y) -> cond1(equal(x,y),x,y)
     cond1(true(),x,y) -> 0()
     cond1(false(),x,y) -> cond2(gt(x,y),x,y)
     cond2(true(),x,y) -> s(diff(x,s(y)))
     cond2(false(),x,y) -> s(diff(s(x),y))
     gt(0(),v) -> false()
     gt(s(u),0()) -> true()
     gt(s(u),s(v)) -> gt(u,v)
     equal(0(),0()) -> true()
     equal(s(x),0()) -> false()
     equal(0(),s(y)) -> false()
     equal(s(x),s(y)) -> equal(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(equal#) = 1
     problem:
      DPs:
       
      TRS:
       diff(x,y) -> cond1(equal(x,y),x,y)
       cond1(true(),x,y) -> 0()
       cond1(false(),x,y) -> cond2(gt(x,y),x,y)
       cond2(true(),x,y) -> s(diff(x,s(y)))
       cond2(false(),x,y) -> s(diff(s(x),y))
       gt(0(),v) -> false()
       gt(s(u),0()) -> true()
       gt(s(u),s(v)) -> gt(u,v)
       equal(0(),0()) -> true()
       equal(s(x),0()) -> false()
       equal(0(),s(y)) -> false()
       equal(s(x),s(y)) -> equal(x,y)
     Qed