MAYBE

Problem:
 minus(s(x),y) -> if(gt(s(x),y),x,y)
 if(true(),x,y) -> s(minus(x,y))
 if(false(),x,y) -> 0()
 gcd(x,y) -> if1(ge(x,y),x,y)
 if1(true(),x,y) -> if2(gt(y,0()),x,y)
 if1(false(),x,y) -> gcd(y,x)
 if2(true(),x,y) -> gcd(minus(x,y),y)
 if2(false(),x,y) -> x
 gt(0(),y) -> false()
 gt(s(x),0()) -> true()
 gt(s(x),s(y)) -> gt(x,y)
 ge(x,0()) -> true()
 ge(0(),s(x)) -> false()
 ge(s(x),s(y)) -> ge(x,y)

Proof:
 DP Processor:
  DPs:
   minus#(s(x),y) -> gt#(s(x),y)
   minus#(s(x),y) -> if#(gt(s(x),y),x,y)
   if#(true(),x,y) -> minus#(x,y)
   gcd#(x,y) -> ge#(x,y)
   gcd#(x,y) -> if1#(ge(x,y),x,y)
   if1#(true(),x,y) -> gt#(y,0())
   if1#(true(),x,y) -> if2#(gt(y,0()),x,y)
   if1#(false(),x,y) -> gcd#(y,x)
   if2#(true(),x,y) -> minus#(x,y)
   if2#(true(),x,y) -> gcd#(minus(x,y),y)
   gt#(s(x),s(y)) -> gt#(x,y)
   ge#(s(x),s(y)) -> ge#(x,y)
  TRS:
   minus(s(x),y) -> if(gt(s(x),y),x,y)
   if(true(),x,y) -> s(minus(x,y))
   if(false(),x,y) -> 0()
   gcd(x,y) -> if1(ge(x,y),x,y)
   if1(true(),x,y) -> if2(gt(y,0()),x,y)
   if1(false(),x,y) -> gcd(y,x)
   if2(true(),x,y) -> gcd(minus(x,y),y)
   if2(false(),x,y) -> x
   gt(0(),y) -> false()
   gt(s(x),0()) -> true()
   gt(s(x),s(y)) -> gt(x,y)
   ge(x,0()) -> true()
   ge(0(),s(x)) -> false()
   ge(s(x),s(y)) -> ge(x,y)
  TDG Processor:
   DPs:
    minus#(s(x),y) -> gt#(s(x),y)
    minus#(s(x),y) -> if#(gt(s(x),y),x,y)
    if#(true(),x,y) -> minus#(x,y)
    gcd#(x,y) -> ge#(x,y)
    gcd#(x,y) -> if1#(ge(x,y),x,y)
    if1#(true(),x,y) -> gt#(y,0())
    if1#(true(),x,y) -> if2#(gt(y,0()),x,y)
    if1#(false(),x,y) -> gcd#(y,x)
    if2#(true(),x,y) -> minus#(x,y)
    if2#(true(),x,y) -> gcd#(minus(x,y),y)
    gt#(s(x),s(y)) -> gt#(x,y)
    ge#(s(x),s(y)) -> ge#(x,y)
   TRS:
    minus(s(x),y) -> if(gt(s(x),y),x,y)
    if(true(),x,y) -> s(minus(x,y))
    if(false(),x,y) -> 0()
    gcd(x,y) -> if1(ge(x,y),x,y)
    if1(true(),x,y) -> if2(gt(y,0()),x,y)
    if1(false(),x,y) -> gcd(y,x)
    if2(true(),x,y) -> gcd(minus(x,y),y)
    if2(false(),x,y) -> x
    gt(0(),y) -> false()
    gt(s(x),0()) -> true()
    gt(s(x),s(y)) -> gt(x,y)
    ge(x,0()) -> true()
    ge(0(),s(x)) -> false()
    ge(s(x),s(y)) -> ge(x,y)
   graph:
    if2#(true(),x,y) -> gcd#(minus(x,y),y) ->
    gcd#(x,y) -> if1#(ge(x,y),x,y)
    if2#(true(),x,y) -> gcd#(minus(x,y),y) -> gcd#(x,y) -> ge#(x,y)
    if2#(true(),x,y) -> minus#(x,y) ->
    minus#(s(x),y) -> if#(gt(s(x),y),x,y)
    if2#(true(),x,y) -> minus#(x,y) -> minus#(s(x),y) -> gt#(s(x),y)
    if1#(false(),x,y) -> gcd#(y,x) -> gcd#(x,y) -> if1#(ge(x,y),x,y)
    if1#(false(),x,y) -> gcd#(y,x) -> gcd#(x,y) -> ge#(x,y)
    if1#(true(),x,y) -> if2#(gt(y,0()),x,y) ->
    if2#(true(),x,y) -> gcd#(minus(x,y),y)
    if1#(true(),x,y) -> if2#(gt(y,0()),x,y) ->
    if2#(true(),x,y) -> minus#(x,y)
    if1#(true(),x,y) -> gt#(y,0()) -> gt#(s(x),s(y)) -> gt#(x,y)
    ge#(s(x),s(y)) -> ge#(x,y) -> ge#(s(x),s(y)) -> ge#(x,y)
    gcd#(x,y) -> if1#(ge(x,y),x,y) -> if1#(false(),x,y) -> gcd#(y,x)
    gcd#(x,y) -> if1#(ge(x,y),x,y) ->
    if1#(true(),x,y) -> if2#(gt(y,0()),x,y)
    gcd#(x,y) -> if1#(ge(x,y),x,y) -> if1#(true(),x,y) -> gt#(y,0())
    gcd#(x,y) -> ge#(x,y) -> ge#(s(x),s(y)) -> ge#(x,y)
    if#(true(),x,y) -> minus#(x,y) ->
    minus#(s(x),y) -> if#(gt(s(x),y),x,y)
    if#(true(),x,y) -> minus#(x,y) -> minus#(s(x),y) -> gt#(s(x),y)
    gt#(s(x),s(y)) -> gt#(x,y) -> gt#(s(x),s(y)) -> gt#(x,y)
    minus#(s(x),y) -> if#(gt(s(x),y),x,y) ->
    if#(true(),x,y) -> minus#(x,y)
    minus#(s(x),y) -> gt#(s(x),y) -> gt#(s(x),s(y)) -> gt#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 8
    #arcs: 19/144
    DPs:
     if2#(true(),x,y) -> gcd#(minus(x,y),y)
     gcd#(x,y) -> if1#(ge(x,y),x,y)
     if1#(true(),x,y) -> if2#(gt(y,0()),x,y)
     if1#(false(),x,y) -> gcd#(y,x)
    TRS:
     minus(s(x),y) -> if(gt(s(x),y),x,y)
     if(true(),x,y) -> s(minus(x,y))
     if(false(),x,y) -> 0()
     gcd(x,y) -> if1(ge(x,y),x,y)
     if1(true(),x,y) -> if2(gt(y,0()),x,y)
     if1(false(),x,y) -> gcd(y,x)
     if2(true(),x,y) -> gcd(minus(x,y),y)
     if2(false(),x,y) -> x
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     ge(x,0()) -> true()
     ge(0(),s(x)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
    Open
    
    DPs:
     minus#(s(x),y) -> if#(gt(s(x),y),x,y)
     if#(true(),x,y) -> minus#(x,y)
    TRS:
     minus(s(x),y) -> if(gt(s(x),y),x,y)
     if(true(),x,y) -> s(minus(x,y))
     if(false(),x,y) -> 0()
     gcd(x,y) -> if1(ge(x,y),x,y)
     if1(true(),x,y) -> if2(gt(y,0()),x,y)
     if1(false(),x,y) -> gcd(y,x)
     if2(true(),x,y) -> gcd(minus(x,y),y)
     if2(false(),x,y) -> x
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     ge(x,0()) -> true()
     ge(0(),s(x)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
      pi(if#) = 1
     problem:
      DPs:
       if#(true(),x,y) -> minus#(x,y)
      TRS:
       minus(s(x),y) -> if(gt(s(x),y),x,y)
       if(true(),x,y) -> s(minus(x,y))
       if(false(),x,y) -> 0()
       gcd(x,y) -> if1(ge(x,y),x,y)
       if1(true(),x,y) -> if2(gt(y,0()),x,y)
       if1(false(),x,y) -> gcd(y,x)
       if2(true(),x,y) -> gcd(minus(x,y),y)
       if2(false(),x,y) -> x
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       ge(x,0()) -> true()
       ge(0(),s(x)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    DPs:
     gt#(s(x),s(y)) -> gt#(x,y)
    TRS:
     minus(s(x),y) -> if(gt(s(x),y),x,y)
     if(true(),x,y) -> s(minus(x,y))
     if(false(),x,y) -> 0()
     gcd(x,y) -> if1(ge(x,y),x,y)
     if1(true(),x,y) -> if2(gt(y,0()),x,y)
     if1(false(),x,y) -> gcd(y,x)
     if2(true(),x,y) -> gcd(minus(x,y),y)
     if2(false(),x,y) -> x
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     ge(x,0()) -> true()
     ge(0(),s(x)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(gt#) = 1
     problem:
      DPs:
       
      TRS:
       minus(s(x),y) -> if(gt(s(x),y),x,y)
       if(true(),x,y) -> s(minus(x,y))
       if(false(),x,y) -> 0()
       gcd(x,y) -> if1(ge(x,y),x,y)
       if1(true(),x,y) -> if2(gt(y,0()),x,y)
       if1(false(),x,y) -> gcd(y,x)
       if2(true(),x,y) -> gcd(minus(x,y),y)
       if2(false(),x,y) -> x
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       ge(x,0()) -> true()
       ge(0(),s(x)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
     Qed
    
    DPs:
     ge#(s(x),s(y)) -> ge#(x,y)
    TRS:
     minus(s(x),y) -> if(gt(s(x),y),x,y)
     if(true(),x,y) -> s(minus(x,y))
     if(false(),x,y) -> 0()
     gcd(x,y) -> if1(ge(x,y),x,y)
     if1(true(),x,y) -> if2(gt(y,0()),x,y)
     if1(false(),x,y) -> gcd(y,x)
     if2(true(),x,y) -> gcd(minus(x,y),y)
     if2(false(),x,y) -> x
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     ge(x,0()) -> true()
     ge(0(),s(x)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(ge#) = 1
     problem:
      DPs:
       
      TRS:
       minus(s(x),y) -> if(gt(s(x),y),x,y)
       if(true(),x,y) -> s(minus(x,y))
       if(false(),x,y) -> 0()
       gcd(x,y) -> if1(ge(x,y),x,y)
       if1(true(),x,y) -> if2(gt(y,0()),x,y)
       if1(false(),x,y) -> gcd(y,x)
       if2(true(),x,y) -> gcd(minus(x,y),y)
       if2(false(),x,y) -> x
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       ge(x,0()) -> true()
       ge(0(),s(x)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
     Qed