/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem:
 .(.(x,y),z) -> .(x,.(y,z))
 a(f(x)) -> f(a(x))
 a(.(x,y)) -> .(a(x),y)
 a(b1(x)) -> b1(a(x))
 f(b(x)) -> b(f(x))
 .(b(x),y) -> b(.(x,y))
 b1(b(x)) -> b(b(x))
 a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
 a(f(0())) -> b1(.(f(0()),0()))
 f(.(0(),x)) -> b(.(0(),f(x)))
 f(0()) -> b(0())
 c(b(x)) -> c(a(x))
 a(b(x)) -> b(a(x))
 .(0(),x) -> x

Proof:
 DP Processor:
  DPs:
   .#(.(x,y),z) -> .#(y,z)
   .#(.(x,y),z) -> .#(x,.(y,z))
   a#(f(x)) -> a#(x)
   a#(f(x)) -> f#(a(x))
   a#(.(x,y)) -> a#(x)
   a#(.(x,y)) -> .#(a(x),y)
   a#(b1(x)) -> a#(x)
   a#(b1(x)) -> b1#(a(x))
   f#(b(x)) -> f#(x)
   .#(b(x),y) -> .#(x,y)
   a#(f(.(0(),x))) -> f#(x)
   a#(f(.(0(),x))) -> .#(0(),f(x))
   a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x)))
   a#(f(.(0(),x))) -> b1#(.(f(.(0(),x)),.(0(),f(x))))
   a#(f(0())) -> .#(f(0()),0())
   a#(f(0())) -> b1#(.(f(0()),0()))
   f#(.(0(),x)) -> f#(x)
   f#(.(0(),x)) -> .#(0(),f(x))
   c#(b(x)) -> a#(x)
   c#(b(x)) -> c#(a(x))
   a#(b(x)) -> a#(x)
  TRS:
   .(.(x,y),z) -> .(x,.(y,z))
   a(f(x)) -> f(a(x))
   a(.(x,y)) -> .(a(x),y)
   a(b1(x)) -> b1(a(x))
   f(b(x)) -> b(f(x))
   .(b(x),y) -> b(.(x,y))
   b1(b(x)) -> b(b(x))
   a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
   a(f(0())) -> b1(.(f(0()),0()))
   f(.(0(),x)) -> b(.(0(),f(x)))
   f(0()) -> b(0())
   c(b(x)) -> c(a(x))
   a(b(x)) -> b(a(x))
   .(0(),x) -> x
  EDG Processor:
   DPs:
    .#(.(x,y),z) -> .#(y,z)
    .#(.(x,y),z) -> .#(x,.(y,z))
    a#(f(x)) -> a#(x)
    a#(f(x)) -> f#(a(x))
    a#(.(x,y)) -> a#(x)
    a#(.(x,y)) -> .#(a(x),y)
    a#(b1(x)) -> a#(x)
    a#(b1(x)) -> b1#(a(x))
    f#(b(x)) -> f#(x)
    .#(b(x),y) -> .#(x,y)
    a#(f(.(0(),x))) -> f#(x)
    a#(f(.(0(),x))) -> .#(0(),f(x))
    a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x)))
    a#(f(.(0(),x))) -> b1#(.(f(.(0(),x)),.(0(),f(x))))
    a#(f(0())) -> .#(f(0()),0())
    a#(f(0())) -> b1#(.(f(0()),0()))
    f#(.(0(),x)) -> f#(x)
    f#(.(0(),x)) -> .#(0(),f(x))
    c#(b(x)) -> a#(x)
    c#(b(x)) -> c#(a(x))
    a#(b(x)) -> a#(x)
   TRS:
    .(.(x,y),z) -> .(x,.(y,z))
    a(f(x)) -> f(a(x))
    a(.(x,y)) -> .(a(x),y)
    a(b1(x)) -> b1(a(x))
    f(b(x)) -> b(f(x))
    .(b(x),y) -> b(.(x,y))
    b1(b(x)) -> b(b(x))
    a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
    a(f(0())) -> b1(.(f(0()),0()))
    f(.(0(),x)) -> b(.(0(),f(x)))
    f(0()) -> b(0())
    c(b(x)) -> c(a(x))
    a(b(x)) -> b(a(x))
    .(0(),x) -> x
   graph:
    c#(b(x)) -> c#(a(x)) -> c#(b(x)) -> a#(x)
    c#(b(x)) -> c#(a(x)) -> c#(b(x)) -> c#(a(x))
    c#(b(x)) -> a#(x) -> a#(f(x)) -> a#(x)
    c#(b(x)) -> a#(x) -> a#(f(x)) -> f#(a(x))
    c#(b(x)) -> a#(x) -> a#(.(x,y)) -> a#(x)
    c#(b(x)) -> a#(x) -> a#(.(x,y)) -> .#(a(x),y)
    c#(b(x)) -> a#(x) -> a#(b1(x)) -> a#(x)
    c#(b(x)) -> a#(x) -> a#(b1(x)) -> b1#(a(x))
    c#(b(x)) -> a#(x) -> a#(f(.(0(),x))) -> f#(x)
    c#(b(x)) -> a#(x) -> a#(f(.(0(),x))) -> .#(0(),f(x))
    c#(b(x)) -> a#(x) -> a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x)))
    c#(b(x)) -> a#(x) -> a#(f(.(0(),x))) -> b1#(.(f(.(0(),x)),.(0(),f(x))))
    c#(b(x)) -> a#(x) -> a#(f(0())) -> .#(f(0()),0())
    c#(b(x)) -> a#(x) -> a#(f(0())) -> b1#(.(f(0()),0()))
    c#(b(x)) -> a#(x) -> a#(b(x)) -> a#(x)
    f#(b(x)) -> f#(x) -> f#(b(x)) -> f#(x)
    f#(b(x)) -> f#(x) -> f#(.(0(),x)) -> f#(x)
    f#(b(x)) -> f#(x) -> f#(.(0(),x)) -> .#(0(),f(x))
    f#(.(0(),x)) -> f#(x) -> f#(b(x)) -> f#(x)
    f#(.(0(),x)) -> f#(x) -> f#(.(0(),x)) -> f#(x)
    f#(.(0(),x)) -> f#(x) -> f#(.(0(),x)) -> .#(0(),f(x))
    a#(b(x)) -> a#(x) -> a#(f(x)) -> a#(x)
    a#(b(x)) -> a#(x) -> a#(f(x)) -> f#(a(x))
    a#(b(x)) -> a#(x) -> a#(.(x,y)) -> a#(x)
    a#(b(x)) -> a#(x) -> a#(.(x,y)) -> .#(a(x),y)
    a#(b(x)) -> a#(x) -> a#(b1(x)) -> a#(x)
    a#(b(x)) -> a#(x) -> a#(b1(x)) -> b1#(a(x))
    a#(b(x)) -> a#(x) -> a#(f(.(0(),x))) -> f#(x)
    a#(b(x)) -> a#(x) -> a#(f(.(0(),x))) -> .#(0(),f(x))
    a#(b(x)) -> a#(x) -> a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x)))
    a#(b(x)) -> a#(x) -> a#(f(.(0(),x))) -> b1#(.(f(.(0(),x)),.(0(),f(x))))
    a#(b(x)) -> a#(x) -> a#(f(0())) -> .#(f(0()),0())
    a#(b(x)) -> a#(x) -> a#(f(0())) -> b1#(.(f(0()),0()))
    a#(b(x)) -> a#(x) -> a#(b(x)) -> a#(x)
    a#(b1(x)) -> a#(x) -> a#(f(x)) -> a#(x)
    a#(b1(x)) -> a#(x) -> a#(f(x)) -> f#(a(x))
    a#(b1(x)) -> a#(x) -> a#(.(x,y)) -> a#(x)
    a#(b1(x)) -> a#(x) -> a#(.(x,y)) -> .#(a(x),y)
    a#(b1(x)) -> a#(x) -> a#(b1(x)) -> a#(x)
    a#(b1(x)) -> a#(x) -> a#(b1(x)) -> b1#(a(x))
    a#(b1(x)) -> a#(x) -> a#(f(.(0(),x))) -> f#(x)
    a#(b1(x)) -> a#(x) -> a#(f(.(0(),x))) -> .#(0(),f(x))
    a#(b1(x)) -> a#(x) -> a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x)))
    a#(b1(x)) -> a#(x) -> a#(f(.(0(),x))) -> b1#(.(f(.(0(),x)),.(0(),f(x))))
    a#(b1(x)) -> a#(x) -> a#(f(0())) -> .#(f(0()),0())
    a#(b1(x)) -> a#(x) -> a#(f(0())) -> b1#(.(f(0()),0()))
    a#(b1(x)) -> a#(x) -> a#(b(x)) -> a#(x)
    a#(f(0())) -> .#(f(0()),0()) -> .#(.(x,y),z) -> .#(y,z)
    a#(f(0())) -> .#(f(0()),0()) -> .#(.(x,y),z) -> .#(x,.(y,z))
    a#(f(0())) -> .#(f(0()),0()) -> .#(b(x),y) -> .#(x,y)
    a#(f(.(0(),x))) -> f#(x) -> f#(b(x)) -> f#(x)
    a#(f(.(0(),x))) -> f#(x) -> f#(.(0(),x)) -> f#(x)
    a#(f(.(0(),x))) -> f#(x) ->
    f#(.(0(),x)) -> .#(0(),f(x))
    a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x))) ->
    .#(.(x,y),z) -> .#(y,z)
    a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x))) ->
    .#(.(x,y),z) -> .#(x,.(y,z))
    a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x))) -> .#(b(x),y) -> .#(x,y)
    a#(f(x)) -> f#(a(x)) -> f#(b(x)) -> f#(x)
    a#(f(x)) -> f#(a(x)) -> f#(.(0(),x)) -> f#(x)
    a#(f(x)) -> f#(a(x)) -> f#(.(0(),x)) -> .#(0(),f(x))
    a#(f(x)) -> a#(x) -> a#(f(x)) -> a#(x)
    a#(f(x)) -> a#(x) -> a#(f(x)) -> f#(a(x))
    a#(f(x)) -> a#(x) -> a#(.(x,y)) -> a#(x)
    a#(f(x)) -> a#(x) -> a#(.(x,y)) -> .#(a(x),y)
    a#(f(x)) -> a#(x) -> a#(b1(x)) -> a#(x)
    a#(f(x)) -> a#(x) -> a#(b1(x)) -> b1#(a(x))
    a#(f(x)) -> a#(x) -> a#(f(.(0(),x))) -> f#(x)
    a#(f(x)) -> a#(x) -> a#(f(.(0(),x))) -> .#(0(),f(x))
    a#(f(x)) -> a#(x) -> a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x)))
    a#(f(x)) -> a#(x) -> a#(f(.(0(),x))) -> b1#(.(f(.(0(),x)),.(0(),f(x))))
    a#(f(x)) -> a#(x) -> a#(f(0())) -> .#(f(0()),0())
    a#(f(x)) -> a#(x) -> a#(f(0())) -> b1#(.(f(0()),0()))
    a#(f(x)) -> a#(x) -> a#(b(x)) -> a#(x)
    a#(.(x,y)) -> a#(x) -> a#(f(x)) -> a#(x)
    a#(.(x,y)) -> a#(x) -> a#(f(x)) -> f#(a(x))
    a#(.(x,y)) -> a#(x) -> a#(.(x,y)) -> a#(x)
    a#(.(x,y)) -> a#(x) -> a#(.(x,y)) -> .#(a(x),y)
    a#(.(x,y)) -> a#(x) -> a#(b1(x)) -> a#(x)
    a#(.(x,y)) -> a#(x) -> a#(b1(x)) -> b1#(a(x))
    a#(.(x,y)) -> a#(x) -> a#(f(.(0(),x))) -> f#(x)
    a#(.(x,y)) -> a#(x) -> a#(f(.(0(),x))) -> .#(0(),f(x))
    a#(.(x,y)) -> a#(x) -> a#(f(.(0(),x))) -> .#(f(.(0(),x)),.(0(),f(x)))
    a#(.(x,y)) -> a#(x) -> a#(f(.(0(),x))) -> b1#(.(f(.(0(),x)),.(0(),f(x))))
    a#(.(x,y)) -> a#(x) -> a#(f(0())) -> .#(f(0()),0())
    a#(.(x,y)) -> a#(x) -> a#(f(0())) -> b1#(.(f(0()),0()))
    a#(.(x,y)) -> a#(x) -> a#(b(x)) -> a#(x)
    a#(.(x,y)) -> .#(a(x),y) -> .#(.(x,y),z) -> .#(y,z)
    a#(.(x,y)) -> .#(a(x),y) -> .#(.(x,y),z) -> .#(x,.(y,z))
    a#(.(x,y)) -> .#(a(x),y) -> .#(b(x),y) -> .#(x,y)
    .#(b(x),y) -> .#(x,y) -> .#(.(x,y),z) -> .#(y,z)
    .#(b(x),y) -> .#(x,y) -> .#(.(x,y),z) -> .#(x,.(y,z))
    .#(b(x),y) -> .#(x,y) -> .#(b(x),y) -> .#(x,y)
    .#(.(x,y),z) -> .#(y,z) -> .#(.(x,y),z) -> .#(y,z)
    .#(.(x,y),z) -> .#(y,z) -> .#(.(x,y),z) -> .#(x,.(y,z))
    .#(.(x,y),z) -> .#(y,z) -> .#(b(x),y) -> .#(x,y)
    .#(.(x,y),z) -> .#(x,.(y,z)) -> .#(.(x,y),z) -> .#(y,z)
    .#(.(x,y),z) -> .#(x,.(y,z)) -> .#(.(x,y),z) -> .#(x,.(y,z))
    .#(.(x,y),z) -> .#(x,.(y,z)) -> .#(b(x),y) -> .#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 10
    #arcs: 97/441
    DPs:
     c#(b(x)) -> c#(a(x))
    TRS:
     .(.(x,y),z) -> .(x,.(y,z))
     a(f(x)) -> f(a(x))
     a(.(x,y)) -> .(a(x),y)
     a(b1(x)) -> b1(a(x))
     f(b(x)) -> b(f(x))
     .(b(x),y) -> b(.(x,y))
     b1(b(x)) -> b(b(x))
     a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
     a(f(0())) -> b1(.(f(0()),0()))
     f(.(0(),x)) -> b(.(0(),f(x)))
     f(0()) -> b(0())
     c(b(x)) -> c(a(x))
     a(b(x)) -> b(a(x))
     .(0(),x) -> x
    Higher Ordinal Interpretation Processor:
     degree: 2
     reverse arguments: true
     interpretation:
      c(x52) =  (+) x52 (+) 1
      b(x53) =  (+) x53
      a(x54) =  (+) x54
      .(x55,x56) = x55 + x56
      f(x57) = omega^( (+) x57 (+) 1)
      b1(x58) =  (+) x58
      0() =  (+) 1
      c(x59) =  (+) x59 (+) 1
     problem:
      DPs:
       c#(b(x)) -> c#(a(x))
      TRS:
       .(.(x,y),z) -> .(x,.(y,z))
       a(f(x)) -> f(a(x))
       a(.(x,y)) -> .(a(x),y)
       a(b1(x)) -> b1(a(x))
       f(b(x)) -> b(f(x))
       .(b(x),y) -> b(.(x,y))
       b1(b(x)) -> b(b(x))
       a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
       a(f(0())) -> b1(.(f(0()),0()))
       c(b(x)) -> c(a(x))
       a(b(x)) -> b(a(x))
       f12(x56) -> .(x56,x55)
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [f12](x0) = 2x0,
       
       [c#](x0) = 2x0 + 1,
       
       [c](x0) = 1,
       
       [0] = 0,
       
       [b](x0) = x0 + 1,
       
       [b1](x0) = 2x0,
       
       [a](x0) = x0,
       
       [f](x0) = x0,
       
       [.](x0, x1) = x0
      orientation:
       c#(b(x)) = 2x + 3 >= 2x + 1 = c#(a(x))
       
       .(.(x,y),z) = x >= x = .(x,.(y,z))
       
       a(f(x)) = x >= x = f(a(x))
       
       a(.(x,y)) = x >= x = .(a(x),y)
       
       a(b1(x)) = 2x >= 2x = b1(a(x))
       
       f(b(x)) = x + 1 >= x + 1 = b(f(x))
       
       .(b(x),y) = x + 1 >= x + 1 = b(.(x,y))
       
       b1(b(x)) = 2x + 2 >= x + 2 = b(b(x))
       
       a(f(.(0(),x))) = 0 >= 0 = b1(.(f(.(0(),x)),.(0(),f(x))))
       
       a(f(0())) = 0 >= 0 = b1(.(f(0()),0()))
       
       c(b(x)) = 1 >= 1 = c(a(x))
       
       a(b(x)) = x + 1 >= x + 1 = b(a(x))
       
       f12(x56) = 2x56 >= x56 = .(x56,x55)
      problem:
       DPs:
        
       TRS:
        .(.(x,y),z) -> .(x,.(y,z))
        a(f(x)) -> f(a(x))
        a(.(x,y)) -> .(a(x),y)
        a(b1(x)) -> b1(a(x))
        f(b(x)) -> b(f(x))
        .(b(x),y) -> b(.(x,y))
        b1(b(x)) -> b(b(x))
        a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
        a(f(0())) -> b1(.(f(0()),0()))
        c(b(x)) -> c(a(x))
        a(b(x)) -> b(a(x))
        f12(x56) -> .(x56,x55)
      Qed
    
    DPs:
     a#(b(x)) -> a#(x)
     a#(b1(x)) -> a#(x)
     a#(.(x,y)) -> a#(x)
     a#(f(x)) -> a#(x)
    TRS:
     .(.(x,y),z) -> .(x,.(y,z))
     a(f(x)) -> f(a(x))
     a(.(x,y)) -> .(a(x),y)
     a(b1(x)) -> b1(a(x))
     f(b(x)) -> b(f(x))
     .(b(x),y) -> b(.(x,y))
     b1(b(x)) -> b(b(x))
     a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
     a(f(0())) -> b1(.(f(0()),0()))
     f(.(0(),x)) -> b(.(0(),f(x)))
     f(0()) -> b(0())
     c(b(x)) -> c(a(x))
     a(b(x)) -> b(a(x))
     .(0(),x) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(a#) = 0
     problem:
      DPs:
       
      TRS:
       .(.(x,y),z) -> .(x,.(y,z))
       a(f(x)) -> f(a(x))
       a(.(x,y)) -> .(a(x),y)
       a(b1(x)) -> b1(a(x))
       f(b(x)) -> b(f(x))
       .(b(x),y) -> b(.(x,y))
       b1(b(x)) -> b(b(x))
       a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
       a(f(0())) -> b1(.(f(0()),0()))
       f(.(0(),x)) -> b(.(0(),f(x)))
       f(0()) -> b(0())
       c(b(x)) -> c(a(x))
       a(b(x)) -> b(a(x))
       .(0(),x) -> x
     Qed
    
    DPs:
     f#(.(0(),x)) -> f#(x)
     f#(b(x)) -> f#(x)
    TRS:
     .(.(x,y),z) -> .(x,.(y,z))
     a(f(x)) -> f(a(x))
     a(.(x,y)) -> .(a(x),y)
     a(b1(x)) -> b1(a(x))
     f(b(x)) -> b(f(x))
     .(b(x),y) -> b(.(x,y))
     b1(b(x)) -> b(b(x))
     a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
     a(f(0())) -> b1(.(f(0()),0()))
     f(.(0(),x)) -> b(.(0(),f(x)))
     f(0()) -> b(0())
     c(b(x)) -> c(a(x))
     a(b(x)) -> b(a(x))
     .(0(),x) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(f#) = 0
     problem:
      DPs:
       
      TRS:
       .(.(x,y),z) -> .(x,.(y,z))
       a(f(x)) -> f(a(x))
       a(.(x,y)) -> .(a(x),y)
       a(b1(x)) -> b1(a(x))
       f(b(x)) -> b(f(x))
       .(b(x),y) -> b(.(x,y))
       b1(b(x)) -> b(b(x))
       a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
       a(f(0())) -> b1(.(f(0()),0()))
       f(.(0(),x)) -> b(.(0(),f(x)))
       f(0()) -> b(0())
       c(b(x)) -> c(a(x))
       a(b(x)) -> b(a(x))
       .(0(),x) -> x
     Qed
    
    DPs:
     .#(b(x),y) -> .#(x,y)
     .#(.(x,y),z) -> .#(x,.(y,z))
     .#(.(x,y),z) -> .#(y,z)
    TRS:
     .(.(x,y),z) -> .(x,.(y,z))
     a(f(x)) -> f(a(x))
     a(.(x,y)) -> .(a(x),y)
     a(b1(x)) -> b1(a(x))
     f(b(x)) -> b(f(x))
     .(b(x),y) -> b(.(x,y))
     b1(b(x)) -> b(b(x))
     a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
     a(f(0())) -> b1(.(f(0()),0()))
     f(.(0(),x)) -> b(.(0(),f(x)))
     f(0()) -> b(0())
     c(b(x)) -> c(a(x))
     a(b(x)) -> b(a(x))
     .(0(),x) -> x
    Subterm Criterion Processor:
     simple projection:
      pi(.#) = 0
     problem:
      DPs:
       
      TRS:
       .(.(x,y),z) -> .(x,.(y,z))
       a(f(x)) -> f(a(x))
       a(.(x,y)) -> .(a(x),y)
       a(b1(x)) -> b1(a(x))
       f(b(x)) -> b(f(x))
       .(b(x),y) -> b(.(x,y))
       b1(b(x)) -> b(b(x))
       a(f(.(0(),x))) -> b1(.(f(.(0(),x)),.(0(),f(x))))
       a(f(0())) -> b1(.(f(0()),0()))
       f(.(0(),x)) -> b(.(0(),f(x)))
       f(0()) -> b(0())
       c(b(x)) -> c(a(x))
       a(b(x)) -> b(a(x))
       .(0(),x) -> x
     Qed