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()
 ge(x,0()) -> true()
 ge(0(),s(x)) -> false()
 ge(s(x),s(y)) -> ge(x,y)
 gt(0(),y) -> false()
 gt(s(x),0()) -> true()
 gt(s(x),s(y)) -> gt(x,y)
 div(x,y) -> if1(ge(x,y),x,y)
 if1(true(),x,y) -> if2(gt(y,0()),x,y)
 if1(false(),x,y) -> 0()
 if2(true(),x,y) -> s(div(minus(x,y),y))
 if2(false(),x,y) -> 0()

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)
   ge#(s(x),s(y)) -> ge#(x,y)
   gt#(s(x),s(y)) -> gt#(x,y)
   div#(x,y) -> ge#(x,y)
   div#(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)
   if2#(true(),x,y) -> minus#(x,y)
   if2#(true(),x,y) -> div#(minus(x,y),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()
   ge(x,0()) -> true()
   ge(0(),s(x)) -> false()
   ge(s(x),s(y)) -> ge(x,y)
   gt(0(),y) -> false()
   gt(s(x),0()) -> true()
   gt(s(x),s(y)) -> gt(x,y)
   div(x,y) -> if1(ge(x,y),x,y)
   if1(true(),x,y) -> if2(gt(y,0()),x,y)
   if1(false(),x,y) -> 0()
   if2(true(),x,y) -> s(div(minus(x,y),y))
   if2(false(),x,y) -> 0()
  EDG 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)
    ge#(s(x),s(y)) -> ge#(x,y)
    gt#(s(x),s(y)) -> gt#(x,y)
    div#(x,y) -> ge#(x,y)
    div#(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)
    if2#(true(),x,y) -> minus#(x,y)
    if2#(true(),x,y) -> div#(minus(x,y),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()
    ge(x,0()) -> true()
    ge(0(),s(x)) -> false()
    ge(s(x),s(y)) -> ge(x,y)
    gt(0(),y) -> false()
    gt(s(x),0()) -> true()
    gt(s(x),s(y)) -> gt(x,y)
    div(x,y) -> if1(ge(x,y),x,y)
    if1(true(),x,y) -> if2(gt(y,0()),x,y)
    if1(false(),x,y) -> 0()
    if2(true(),x,y) -> s(div(minus(x,y),y))
    if2(false(),x,y) -> 0()
   graph:
    if2#(true(),x,y) -> div#(minus(x,y),y) ->
    div#(x,y) -> ge#(x,y)
    if2#(true(),x,y) -> div#(minus(x,y),y) ->
    div#(x,y) -> if1#(ge(x,y),x,y)
    if2#(true(),x,y) -> minus#(x,y) -> minus#(s(x),y) -> gt#(s(x),y)
    if2#(true(),x,y) -> minus#(x,y) ->
    minus#(s(x),y) -> if#(gt(s(x),y),x,y)
    if1#(true(),x,y) -> if2#(gt(y,0()),x,y) ->
    if2#(true(),x,y) -> minus#(x,y)
    if1#(true(),x,y) -> if2#(gt(y,0()),x,y) ->
    if2#(true(),x,y) -> div#(minus(x,y),y)
    div#(x,y) -> if1#(ge(x,y),x,y) -> if1#(true(),x,y) -> gt#(y,0())
    div#(x,y) -> if1#(ge(x,y),x,y) -> if1#(true(),x,y) -> if2#(gt(y,0()),x,y)
    div#(x,y) -> ge#(x,y) -> ge#(s(x),s(y)) -> ge#(x,y)
    ge#(s(x),s(y)) -> ge#(x,y) -> ge#(s(x),s(y)) -> ge#(x,y)
    if#(true(),x,y) -> minus#(x,y) -> minus#(s(x),y) -> gt#(s(x),y)
    if#(true(),x,y) -> minus#(x,y) ->
    minus#(s(x),y) -> if#(gt(s(x),y),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: 7
    #arcs: 15/121
    DPs:
     if2#(true(),x,y) -> div#(minus(x,y),y)
     div#(x,y) -> if1#(ge(x,y),x,y)
     if1#(true(),x,y) -> if2#(gt(y,0()),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()
     ge(x,0()) -> true()
     ge(0(),s(x)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     div(x,y) -> if1(ge(x,y),x,y)
     if1(true(),x,y) -> if2(gt(y,0()),x,y)
     if1(false(),x,y) -> 0()
     if2(true(),x,y) -> s(div(minus(x,y),y))
     if2(false(),x,y) -> 0()
    Open
    
    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()
     ge(x,0()) -> true()
     ge(0(),s(x)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     div(x,y) -> if1(ge(x,y),x,y)
     if1(true(),x,y) -> if2(gt(y,0()),x,y)
     if1(false(),x,y) -> 0()
     if2(true(),x,y) -> s(div(minus(x,y),y))
     if2(false(),x,y) -> 0()
    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()
       ge(x,0()) -> true()
       ge(0(),s(x)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       div(x,y) -> if1(ge(x,y),x,y)
       if1(true(),x,y) -> if2(gt(y,0()),x,y)
       if1(false(),x,y) -> 0()
       if2(true(),x,y) -> s(div(minus(x,y),y))
       if2(false(),x,y) -> 0()
     Qed
    
    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()
     ge(x,0()) -> true()
     ge(0(),s(x)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     div(x,y) -> if1(ge(x,y),x,y)
     if1(true(),x,y) -> if2(gt(y,0()),x,y)
     if1(false(),x,y) -> 0()
     if2(true(),x,y) -> s(div(minus(x,y),y))
     if2(false(),x,y) -> 0()
    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()
       ge(x,0()) -> true()
       ge(0(),s(x)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       div(x,y) -> if1(ge(x,y),x,y)
       if1(true(),x,y) -> if2(gt(y,0()),x,y)
       if1(false(),x,y) -> 0()
       if2(true(),x,y) -> s(div(minus(x,y),y))
       if2(false(),x,y) -> 0()
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [if#](x0, x1, x2) = x0 + 1,
       
       [minus#](x0, x1) = 0,
       
       [if2](x0, x1, x2) = 0,
       
       [if1](x0, x1, x2) = 0,
       
       [div](x0, x1) = 0,
       
       [ge](x0, x1) = 1,
       
       [0] = 0,
       
       [false] = 1,
       
       [true] = 1,
       
       [if](x0, x1, x2) = 0,
       
       [gt](x0, x1) = 1,
       
       [minus](x0, x1) = 0,
       
       [s](x0) = 0
      orientation:
       if#(true(),x,y) = 2 >= 0 = minus#(x,y)
       
       minus(s(x),y) = 0 >= 0 = if(gt(s(x),y),x,y)
       
       if(true(),x,y) = 0 >= 0 = s(minus(x,y))
       
       if(false(),x,y) = 0 >= 0 = 0()
       
       ge(x,0()) = 1 >= 1 = true()
       
       ge(0(),s(x)) = 1 >= 1 = false()
       
       ge(s(x),s(y)) = 1 >= 1 = ge(x,y)
       
       gt(0(),y) = 1 >= 1 = false()
       
       gt(s(x),0()) = 1 >= 1 = true()
       
       gt(s(x),s(y)) = 1 >= 1 = gt(x,y)
       
       div(x,y) = 0 >= 0 = if1(ge(x,y),x,y)
       
       if1(true(),x,y) = 0 >= 0 = if2(gt(y,0()),x,y)
       
       if1(false(),x,y) = 0 >= 0 = 0()
       
       if2(true(),x,y) = 0 >= 0 = s(div(minus(x,y),y))
       
       if2(false(),x,y) = 0 >= 0 = 0()
      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()
        ge(x,0()) -> true()
        ge(0(),s(x)) -> false()
        ge(s(x),s(y)) -> ge(x,y)
        gt(0(),y) -> false()
        gt(s(x),0()) -> true()
        gt(s(x),s(y)) -> gt(x,y)
        div(x,y) -> if1(ge(x,y),x,y)
        if1(true(),x,y) -> if2(gt(y,0()),x,y)
        if1(false(),x,y) -> 0()
        if2(true(),x,y) -> s(div(minus(x,y),y))
        if2(false(),x,y) -> 0()
      Qed
    
    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()
     ge(x,0()) -> true()
     ge(0(),s(x)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
     gt(0(),y) -> false()
     gt(s(x),0()) -> true()
     gt(s(x),s(y)) -> gt(x,y)
     div(x,y) -> if1(ge(x,y),x,y)
     if1(true(),x,y) -> if2(gt(y,0()),x,y)
     if1(false(),x,y) -> 0()
     if2(true(),x,y) -> s(div(minus(x,y),y))
     if2(false(),x,y) -> 0()
    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()
       ge(x,0()) -> true()
       ge(0(),s(x)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
       gt(0(),y) -> false()
       gt(s(x),0()) -> true()
       gt(s(x),s(y)) -> gt(x,y)
       div(x,y) -> if1(ge(x,y),x,y)
       if1(true(),x,y) -> if2(gt(y,0()),x,y)
       if1(false(),x,y) -> 0()
       if2(true(),x,y) -> s(div(minus(x,y),y))
       if2(false(),x,y) -> 0()
     Qed