YES
Welcome to CiME version 2.02 - Built on 14/04/2004 13:29:51

- : unit = ()
- : unit = ()
X : variable_set = <variable set>

F : signature = <signature>
R : HTRS = { U11(tt,V2) -> U12(isNat(V2)),
             U12(tt) -> tt,
             U21(tt) -> tt,
             U31(tt,V2) -> U32(isNat(V2)),
             U32(tt) -> tt,
             U41(tt,N) -> N,
             U51(tt,M,N) -> U52(isNat(N),M,N),
             U52(tt,M,N) -> s(plus(N,M)),
             U61(tt) -> 0,
             U71(tt,M,N) -> U72(isNat(N),M,N),
             U72(tt,M,N) -> plus(x(N,M),N),
             isNat(0) -> tt,
             isNat(plus(V1,V2)) -> U11(isNat(V1),V2),
             isNat(s(V1)) -> U21(isNat(V1)),
             isNat(x(V1,V2)) -> U31(isNat(V1),V2),
             plus(N,0) -> U41(isNat(N),N),
             plus(N,s(M)) -> U51(isNat(M),M,N),
             x(N,0) -> U61(isNat(N)),
             x(N,s(M)) -> U71(isNat(M),M,N) } (19 rules)
Termination now uses minimal decomposition
- : unit = ()
Entering the termination expert for modules. Verbose level = 0
Checking module:
{}

The dependency graph is
(0 nodes)
Checking each of the 0 strongly connected components :

Checking module:
{}

The dependency graph is
(0 nodes)
Checking each of the 0 strongly connected components :

Checking module:
{}

The dependency graph is
(0 nodes)
Checking each of the 0 strongly connected components :

Checking module:
{ U32(tt) -> tt } (1 rules)

The dependency graph is
(0 nodes)
Checking each of the 0 strongly connected components :

Checking module:
{ U21(tt) -> tt } (1 rules)

The dependency graph is
(0 nodes)
Checking each of the 0 strongly connected components :

Checking module:
{ U12(tt) -> tt } (1 rules)

The dependency graph is
(0 nodes)
Checking each of the 0 strongly connected components :

Checking module:
{ U41(tt,V_1) -> V_1 } (1 rules)

The dependency graph is
(0 nodes)
Checking each of the 0 strongly connected components :

Checking module:
{ U61(tt) -> 0 } (1 rules)

The dependency graph is
(0 nodes)
Checking each of the 0 strongly connected components :

Checking module:
{ x(V_1,0) -> U61(isNat(V_1)),
  x(V_1,s(V_0)) -> U71(isNat(V_0),V_0,V_1),
  isNat(x(V_2,V_3)) -> U31(isNat(V_2),V_3),
  isNat(s(V_2)) -> U21(isNat(V_2)),
  isNat(plus(V_2,V_3)) -> U11(isNat(V_2),V_3),
  isNat(0) -> tt,
  U11(tt,V_3) -> U12(isNat(V_3)),
  U31(tt,V_3) -> U32(isNat(V_3)),
  plus(V_1,0) -> U41(isNat(V_1),V_1),
  plus(V_1,s(V_0)) -> U51(isNat(V_0),V_0,V_1),
  U52(tt,V_0,V_1) -> s(plus(V_1,V_0)),
  U51(tt,V_0,V_1) -> U52(isNat(V_1),V_0,V_1),
  U72(tt,V_0,V_1) -> plus(x(V_1,V_0),V_1),
  U71(tt,V_0,V_1) -> U72(isNat(V_1),V_0,V_1) } (14 rules)

The dependency graph is
(1 nodes)
Checking each of the 3 strongly connected components :
Checking component 1
Trying simple graph criterion.
Trying to solve the following constraints:
(22 termination constraints)
Search parameters: linear polynomials, coefficient bound is 2.
Time out for these parameters.
Search parameters: simple polynomials, coefficient bound is 3.
Solution found for these constraints:
[s](X0) = X0 + 3; 
[tt] = 0; 
[0] = 0; 
[U61](X0) = 0; 
[U21](X0) = 0; 
[U12](X0) = 0; 
[U32](X0) = 0; 
[U41](X0,X1) = X1; 
[U71](X0,X1,X2) = X2*X1 + X2; 
[U72](X0,X1,X2) = X2*X1 + X2; 
[U51](X0,X1,X2) = X2 + X1 + 3; 
[U52](X0,X1,X2) = X2 + X1 + 3; 
[plus](X0,X1) = X1 + X0; 
[U31](X0,X1) = 0; 
[U11](X0,X1) = 0; 
[isNat](X0) = 0; 
[x](X0,X1) = X1*X0; 
['U71`](X0,X1,X2) = X1 + 2; 
['U72`](X0,X1,X2) = X1 + 1; 
['x`](X0,X1) = X1; 

Checking component 2
Trying simple graph criterion.
Trying to solve the following constraints:
(22 termination constraints)
Search parameters: linear polynomials, coefficient bound is 2.
Time out for these parameters.
Search parameters: simple polynomials, coefficient bound is 3.
Solution found for these constraints:
[s](X0) = X0 + 3; 
[tt] = 0; 
[0] = 0; 
[U61](X0) = 0; 
[U21](X0) = 0; 
[U12](X0) = 0; 
[U32](X0) = 0; 
[U41](X0,X1) = X1; 
[U71](X0,X1,X2) = X2*X1 + X2; 
[U72](X0,X1,X2) = X2*X1 + X2; 
[U51](X0,X1,X2) = X2 + X1 + 3; 
[U52](X0,X1,X2) = X2 + X1 + 3; 
[plus](X0,X1) = X1 + X0; 
[U31](X0,X1) = 0; 
[U11](X0,X1) = 0; 
[isNat](X0) = 0; 
[x](X0,X1) = X1*X0; 
['U51`](X0,X1,X2) = X1 + 2; 
['U52`](X0,X1,X2) = X1 + 1; 
['plus`](X0,X1) = X1; 

Checking component 3
Trying simple graph criterion.
Trying to solve the following constraints:
(26 termination constraints)
Search parameters: linear polynomials, coefficient bound is 2.
No solution found for these parameters.
Search parameters: simple polynomials, coefficient bound is 3.
Solution found for these constraints:
[s](X0) = X0 + 1; 
[tt] = 0; 
[0] = 0; 
[U61](X0) = 0; 
[U21](X0) = 0; 
[U12](X0) = 0; 
[U32](X0) = 0; 
[U41](X0,X1) = X1; 
[U71](X0,X1,X2) = X2*X1 + 2*X2 + X1 + 2; 
[U72](X0,X1,X2) = X2*X1 + 2*X2 + X1 + 2; 
[U51](X0,X1,X2) = X2 + X1 + 2; 
[U52](X0,X1,X2) = X2 + X1 + 2; 
[plus](X0,X1) = X1 + X0 + 1; 
[U31](X0,X1) = 0; 
[U11](X0,X1) = 0; 
[isNat](X0) = 0; 
[x](X0,X1) = X1*X0 + X1 + X0 + 1; 
['U31`](X0,X1) = 2*X1 + 1; 
['U11`](X0,X1) = 2*X1 + 1; 
['isNat`](X0) = 2*X0; 


Modular termination proof found.
- : unit = ()
The module
  {}
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has no strongly connected components.

The module
  {}
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has no strongly connected components.

The module
  {}
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has no strongly connected components.

The module
  { U32(tt) -> tt } (1 rules)
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has no strongly connected components.

The module
  { U21(tt) -> tt } (1 rules)
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has no strongly connected components.

The module
  { U12(tt) -> tt } (1 rules)
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has no strongly connected components.

The module
  { U41(tt,V_1) -> V_1 } (1 rules)
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has no strongly connected components.

The module
  { U61(tt) -> 0 } (1 rules)
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has no strongly connected components.

The module
  { x(V_1,0) -> U61(isNat(V_1)),
    x(V_1,s(V_0)) -> U71(isNat(V_0),V_0,V_1),
    isNat(x(V_2,V_3)) -> U31(isNat(V_2),V_3),
    isNat(s(V_2)) -> U21(isNat(V_2)),
    isNat(plus(V_2,V_3)) -> U11(isNat(V_2),V_3),
    isNat(0) -> tt,
    U11(tt,V_3) -> U12(isNat(V_3)),
    U31(tt,V_3) -> U32(isNat(V_3)),
    plus(V_1,0) -> U41(isNat(V_1),V_1),
    plus(V_1,s(V_0)) -> U51(isNat(V_0),V_0,V_1),
    U52(tt,V_0,V_1) -> s(plus(V_1,V_0)),
    U51(tt,V_0,V_1) -> U52(isNat(V_1),V_0,V_1),
    U72(tt,V_0,V_1) -> plus(x(V_1,V_0),V_1),
    U71(tt,V_0,V_1) -> U72(isNat(V_1),V_0,V_1) } (14 rules)
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has 3 strongly connected components:
    component 1 is
      10: 'U31`(tt,V_3) ; 'isNat`(V_3) 11: 'U11`(tt,V_3) ; 'isNat`(V_3)
      12: 'isNat`(plus(V_2,V_3)) ; 'isNat`(V_2)
      13: 'isNat`(plus(V_2,V_3)) ; 'U11`(isNat(V_2),V_3)
      14: 'isNat`(s(V_2)) ; 'isNat`(V_2)
      15: 'isNat`(x(V_2,V_3)) ; 'isNat`(V_2)
      16: 'isNat`(x(V_2,V_3)) ; 'U31`(isNat(V_2),V_3) 
      10 -> 12, 10 -> 13, 10 -> 14, 10 -> 15, 10 -> 16, 11 -> 12, 11 -> 13,
      11 -> 14, 11 -> 15, 11 -> 16, 12 -> 12, 12 -> 13, 12 -> 14, 12 -> 15,
      12 -> 16, 13 -> 11, 14 -> 12, 14 -> 13, 14 -> 14, 14 -> 15, 14 -> 16,
      15 -> 12, 15 -> 13, 15 -> 14, 15 -> 15, 15 -> 16, 16 -> 10, 
      On this component, the interpretation
      [s](X0) = X0 + 1; 
      [tt] = 0; 
      [0] = 0; 
      [U61](X0) = 0; 
      [U21](X0) = 0; 
      [U12](X0) = 0; 
      [U32](X0) = 0; 
      [U41](X0,X1) = X1; 
      [U71](X0,X1,X2) = X2*X1 + 2*X2 + X1 + 2; 
      [U72](X0,X1,X2) = X2*X1 + 2*X2 + X1 + 2; 
      [U51](X0,X1,X2) = X2 + X1 + 2; 
      [U52](X0,X1,X2) = X2 + X1 + 2; 
      [plus](X0,X1) = X1 + X0 + 1; 
      [U31](X0,X1) = 0; 
      [U11](X0,X1) = 0; 
      [isNat](X0) = 0; 
      [x](X0,X1) = X1*X0 + X1 + X0 + 1; 
      ['U31`](X0,X1) = 2*X1 + 1; 
      ['U11`](X0,X1) = 2*X1 + 1; 
      ['isNat`](X0) = 2*X0; 
       strictly decreases on every cycle
    component 2 is
      5: 'U51`(tt,V_0,V_1) ; 'U52`(isNat(V_1),V_0,V_1)
      6: 'U52`(tt,V_0,V_1) ; 'plus`(V_1,V_0)
      8: 'plus`(V_1,s(V_0)) ; 'U51`(isNat(V_0),V_0,V_1) 
      5 -> 6, 6 -> 8, 8 -> 5, 
      On this component, the interpretation
      [s](X0) = X0 + 3; 
      [tt] = 0; 
      [0] = 0; 
      [U61](X0) = 0; 
      [U21](X0) = 0; 
      [U12](X0) = 0; 
      [U32](X0) = 0; 
      [U41](X0,X1) = X1; 
      [U71](X0,X1,X2) = X2*X1 + X2; 
      [U72](X0,X1,X2) = X2*X1 + X2; 
      [U51](X0,X1,X2) = X2 + X1 + 3; 
      [U52](X0,X1,X2) = X2 + X1 + 3; 
      [plus](X0,X1) = X1 + X0; 
      [U31](X0,X1) = 0; 
      [U11](X0,X1) = 0; 
      [isNat](X0) = 0; 
      [x](X0,X1) = X1*X0; 
      ['U51`](X0,X1,X2) = X1 + 2; 
      ['U52`](X0,X1,X2) = X1 + 1; 
      ['plus`](X0,X1) = X1; 
       strictly decreases on every cycle
    component 3 is
      1: 'U71`(tt,V_0,V_1) ; 'U72`(isNat(V_1),V_0,V_1)
      2: 'U72`(tt,V_0,V_1) ; 'x`(V_1,V_0)
      18: 'x`(V_1,s(V_0)) ; 'U71`(isNat(V_0),V_0,V_1) 
      1 -> 2, 2 -> 18, 18 -> 1, 
      On this component, the interpretation
      [s](X0) = X0 + 3; 
      [tt] = 0; 
      [0] = 0; 
      [U61](X0) = 0; 
      [U21](X0) = 0; 
      [U12](X0) = 0; 
      [U32](X0) = 0; 
      [U41](X0,X1) = X1; 
      [U71](X0,X1,X2) = X2*X1 + X2; 
      [U72](X0,X1,X2) = X2*X1 + X2; 
      [U51](X0,X1,X2) = X2 + X1 + 3; 
      [U52](X0,X1,X2) = X2 + X1 + 3; 
      [plus](X0,X1) = X1 + X0; 
      [U31](X0,X1) = 0; 
      [U11](X0,X1) = 0; 
      [isNat](X0) = 0; 
      [x](X0,X1) = X1*X0; 
      ['U71`](X0,X1,X2) = X1 + 2; 
      ['U72`](X0,X1,X2) = X1 + 1; 
      ['x`](X0,X1) = X1; 
       strictly decreases on every cycle 

- : unit = ()

Quitting.
Standard error: