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,N) -> N,
             U41(tt,M,N) -> U42(isNat(N),M,N),
             U42(tt,M,N) -> s(plus(N,M)),
             isNat(0) -> tt,
             isNat(plus(V1,V2)) -> U11(isNat(V1),V2),
             isNat(s(V1)) -> U21(isNat(V1)),
             plus(N,0) -> U31(isNat(N),N),
             plus(N,s(M)) -> U41(isNat(M),M,N) } (11 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:
{ 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:
{ U31(tt,V_1) -> V_1 } (1 rules)

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

Checking module:
{ plus(V_1,0) -> U31(isNat(V_1),V_1),
  plus(V_1,s(V_0)) -> U41(isNat(V_0),V_0,V_1),
  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)),
  U42(tt,V_0,V_1) -> s(plus(V_1,V_0)),
  U41(tt,V_0,V_1) -> U42(isNat(V_1),V_0,V_1) } (8 rules)

The dependency graph is
(1 nodes)
Checking each of the 2 strongly connected components :
Checking component 1
Trying simple graph criterion.
Trying to solve the following constraints:
(14 termination constraints)
Search parameters: linear polynomials, coefficient bound is 2.
Solution found for these constraints:
[tt] = 0; 
[U31](X0,X1) = X1; 
[s](X0) = X0 + 2; 
[0] = 0; 
[U21](X0) = 0; 
[U12](X0) = 0; 
[U41](X0,X1,X2) = X2 + X1 + 2; 
[U42](X0,X1,X2) = X2 + X1 + 2; 
[U11](X0,X1) = 0; 
[isNat](X0) = 0; 
[plus](X0,X1) = X1 + X0; 
['U41`](X0,X1,X2) = 2*X1 + 2; 
['U42`](X0,X1,X2) = 2*X1 + 1; 
['plus`](X0,X1) = 2*X1; 

Checking component 2
Trying simple graph criterion.
Trying to solve the following constraints:
(15 termination constraints)
Search parameters: linear polynomials, coefficient bound is 2.
Solution found for these constraints:
[tt] = 0; 
[U31](X0,X1) = X1; 
[s](X0) = X0 + 1; 
[0] = 0; 
[U21](X0) = 0; 
[U12](X0) = 0; 
[U41](X0,X1,X2) = X2 + X1 + 2; 
[U42](X0,X1,X2) = X2 + X1 + 2; 
[U11](X0,X1) = 0; 
[isNat](X0) = 0; 
[plus](X0,X1) = X1 + X0 + 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
  { 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
  { U31(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
  { plus(V_1,0) -> U31(isNat(V_1),V_1),
    plus(V_1,s(V_0)) -> U41(isNat(V_0),V_0,V_1),
    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)),
    U42(tt,V_0,V_1) -> s(plus(V_1,V_0)),
    U41(tt,V_0,V_1) -> U42(isNat(V_1),V_0,V_1) } (8 rules)
  is CE-terminating by criterion with marks except on AC-symbols, and with graph;
  the dependency graph has 2 strongly connected components:
    component 1 is
      3: 'U11`(tt,V_3) ; 'isNat`(V_3)
      4: 'isNat`(plus(V_2,V_3)) ; 'isNat`(V_2)
      5: 'isNat`(plus(V_2,V_3)) ; 'U11`(isNat(V_2),V_3)
      6: 'isNat`(s(V_2)) ; 'isNat`(V_2) 
      3 -> 4, 3 -> 5, 3 -> 6, 4 -> 4, 4 -> 5, 4 -> 6, 5 -> 3, 6 -> 4, 6 -> 5,
      6 -> 6, 
      On this component, the interpretation
      [tt] = 0; 
      [U31](X0,X1) = X1; 
      [s](X0) = X0 + 1; 
      [0] = 0; 
      [U21](X0) = 0; 
      [U12](X0) = 0; 
      [U41](X0,X1,X2) = X2 + X1 + 2; 
      [U42](X0,X1,X2) = X2 + X1 + 2; 
      [U11](X0,X1) = 0; 
      [isNat](X0) = 0; 
      [plus](X0,X1) = X1 + X0 + 1; 
      ['U11`](X0,X1) = 2*X1 + 1; 
      ['isNat`](X0) = 2*X0; 
       strictly decreases on every cycle
    component 2 is
      1: 'U41`(tt,V_0,V_1) ; 'U42`(isNat(V_1),V_0,V_1)
      2: 'U42`(tt,V_0,V_1) ; 'plus`(V_1,V_0)
      8: 'plus`(V_1,s(V_0)) ; 'U41`(isNat(V_0),V_0,V_1) 
      1 -> 2, 2 -> 8, 8 -> 1, 
      On this component, the interpretation
      [tt] = 0; 
      [U31](X0,X1) = X1; 
      [s](X0) = X0 + 2; 
      [0] = 0; 
      [U21](X0) = 0; 
      [U12](X0) = 0; 
      [U41](X0,X1,X2) = X2 + X1 + 2; 
      [U42](X0,X1,X2) = X2 + X1 + 2; 
      [U11](X0,X1) = 0; 
      [isNat](X0) = 0; 
      [plus](X0,X1) = X1 + X0; 
      ['U41`](X0,X1,X2) = 2*X1 + 2; 
      ['U42`](X0,X1,X2) = 2*X1 + 1; 
      ['plus`](X0,X1) = 2*X1; 
       strictly decreases on every cycle 

- : unit = ()

Quitting.
Standard error: