/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  a!6220!6220head : [o] --> o 
  a!6220!6220incr : [o] --> o 
  a!6220!6220nats : [] --> o 
  a!6220!6220odds : [] --> o 
  a!6220!6220pairs : [] --> o 
  a!6220!6220tail : [o] --> o 
  cons : [o * o] --> o 
  head : [o] --> o 
  incr : [o] --> o 
  mark : [o] --> o 
  nats : [] --> o 
  nil : [] --> o 
  odds : [] --> o 
  pairs : [] --> o 
  s : [o] --> o 
  tail : [o] --> o 

  a!6220!6220nats => cons(0, incr(nats)) 
  a!6220!6220pairs => cons(0, incr(odds)) 
  a!6220!6220odds => a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) => cons(s(mark(X)), incr(Y)) 
  a!6220!6220head(cons(X, Y)) => mark(X) 
  a!6220!6220tail(cons(X, Y)) => mark(Y) 
  mark(nats) => a!6220!6220nats 
  mark(pairs) => a!6220!6220pairs 
  mark(odds) => a!6220!6220odds 
  mark(incr(X)) => a!6220!6220incr(mark(X)) 
  mark(head(X)) => a!6220!6220head(mark(X)) 
  mark(tail(X)) => a!6220!6220tail(mark(X)) 
  mark(0) => 0 
  mark(s(X)) => s(mark(X)) 
  mark(nil) => nil 
  mark(cons(X, Y)) => cons(mark(X), Y) 
  a!6220!6220nats => nats 
  a!6220!6220pairs => pairs 
  a!6220!6220odds => odds 
  a!6220!6220incr(X) => incr(X) 
  a!6220!6220head(X) => head(X) 
  a!6220!6220tail(X) => tail(X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  a!6220!6220nats >? cons(0, incr(nats)) 
  a!6220!6220pairs >? cons(0, incr(odds)) 
  a!6220!6220odds >? a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >? cons(s(mark(X)), incr(Y)) 
  a!6220!6220head(cons(X, Y)) >? mark(X) 
  a!6220!6220tail(cons(X, Y)) >? mark(Y) 
  mark(nats) >? a!6220!6220nats 
  mark(pairs) >? a!6220!6220pairs 
  mark(odds) >? a!6220!6220odds 
  mark(incr(X)) >? a!6220!6220incr(mark(X)) 
  mark(head(X)) >? a!6220!6220head(mark(X)) 
  mark(tail(X)) >? a!6220!6220tail(mark(X)) 
  mark(0) >? 0 
  mark(s(X)) >? s(mark(X)) 
  mark(nil) >? nil 
  mark(cons(X, Y)) >? cons(mark(X), Y) 
  a!6220!6220nats >? nats 
  a!6220!6220pairs >? pairs 
  a!6220!6220odds >? odds 
  a!6220!6220incr(X) >? incr(X) 
  a!6220!6220head(X) >? head(X) 
  a!6220!6220tail(X) >? tail(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220head = \y0.y0 
  a!6220!6220incr = \y0.2y0 
  a!6220!6220nats = 0 
  a!6220!6220odds = 0 
  a!6220!6220pairs = 0 
  a!6220!6220tail = \y0.2 + y0 
  cons = \y0y1.2y0 + 2y1 
  head = \y0.y0 
  incr = \y0.2y0 
  mark = \y0.y0 
  nats = 0 
  nil = 0 
  odds = 0 
  pairs = 0 
  s = \y0.y0 
  tail = \y0.2 + y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220nats]] = 0 >= 0 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = 4x0 + 4x1 >= 2x0 + 4x1 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[a!6220!6220head(cons(_x0, _x1))]] = 2x0 + 2x1 >= x0 = [[mark(_x0)]] 
  [[a!6220!6220tail(cons(_x0, _x1))]] = 2 + 2x0 + 2x1 > x1 = [[mark(_x1)]] 
  [[mark(nats)]] = 0 >= 0 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 0 >= 0 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(head(_x0))]] = x0 >= x0 = [[a!6220!6220head(mark(_x0))]] 
  [[mark(tail(_x0))]] = 2 + x0 >= 2 + x0 = [[a!6220!6220tail(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[mark(nil)]] = 0 >= 0 = [[nil]] 
  [[mark(cons(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220nats]] = 0 >= 0 = [[nats]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = 2x0 >= 2x0 = [[incr(_x0)]] 
  [[a!6220!6220head(_x0)]] = x0 >= x0 = [[head(_x0)]] 
  [[a!6220!6220tail(_x0)]] = 2 + x0 >= 2 + x0 = [[tail(_x0)]] 

We can thus remove the following rules:

  a!6220!6220tail(cons(X, Y)) => mark(Y) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  a!6220!6220nats >? cons(0, incr(nats)) 
  a!6220!6220pairs >? cons(0, incr(odds)) 
  a!6220!6220odds >? a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >? cons(s(mark(X)), incr(Y)) 
  a!6220!6220head(cons(X, Y)) >? mark(X) 
  mark(nats) >? a!6220!6220nats 
  mark(pairs) >? a!6220!6220pairs 
  mark(odds) >? a!6220!6220odds 
  mark(incr(X)) >? a!6220!6220incr(mark(X)) 
  mark(head(X)) >? a!6220!6220head(mark(X)) 
  mark(tail(X)) >? a!6220!6220tail(mark(X)) 
  mark(0) >? 0 
  mark(s(X)) >? s(mark(X)) 
  mark(nil) >? nil 
  mark(cons(X, Y)) >? cons(mark(X), Y) 
  a!6220!6220nats >? nats 
  a!6220!6220pairs >? pairs 
  a!6220!6220odds >? odds 
  a!6220!6220incr(X) >? incr(X) 
  a!6220!6220head(X) >? head(X) 
  a!6220!6220tail(X) >? tail(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220head = \y0.1 + y0 
  a!6220!6220incr = \y0.y0 
  a!6220!6220nats = 0 
  a!6220!6220odds = 0 
  a!6220!6220pairs = 0 
  a!6220!6220tail = \y0.y0 
  cons = \y0y1.2y0 + 2y1 
  head = \y0.1 + y0 
  incr = \y0.y0 
  mark = \y0.y0 
  nats = 0 
  nil = 0 
  odds = 0 
  pairs = 0 
  s = \y0.y0 
  tail = \y0.y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220nats]] = 0 >= 0 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[a!6220!6220head(cons(_x0, _x1))]] = 1 + 2x0 + 2x1 > x0 = [[mark(_x0)]] 
  [[mark(nats)]] = 0 >= 0 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 0 >= 0 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = x0 >= x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(head(_x0))]] = 1 + x0 >= 1 + x0 = [[a!6220!6220head(mark(_x0))]] 
  [[mark(tail(_x0))]] = x0 >= x0 = [[a!6220!6220tail(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[mark(nil)]] = 0 >= 0 = [[nil]] 
  [[mark(cons(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220nats]] = 0 >= 0 = [[nats]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = x0 >= x0 = [[incr(_x0)]] 
  [[a!6220!6220head(_x0)]] = 1 + x0 >= 1 + x0 = [[head(_x0)]] 
  [[a!6220!6220tail(_x0)]] = x0 >= x0 = [[tail(_x0)]] 

We can thus remove the following rules:

  a!6220!6220head(cons(X, Y)) => mark(X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  a!6220!6220nats >? cons(0, incr(nats)) 
  a!6220!6220pairs >? cons(0, incr(odds)) 
  a!6220!6220odds >? a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >? cons(s(mark(X)), incr(Y)) 
  mark(nats) >? a!6220!6220nats 
  mark(pairs) >? a!6220!6220pairs 
  mark(odds) >? a!6220!6220odds 
  mark(incr(X)) >? a!6220!6220incr(mark(X)) 
  mark(head(X)) >? a!6220!6220head(mark(X)) 
  mark(tail(X)) >? a!6220!6220tail(mark(X)) 
  mark(0) >? 0 
  mark(s(X)) >? s(mark(X)) 
  mark(nil) >? nil 
  mark(cons(X, Y)) >? cons(mark(X), Y) 
  a!6220!6220nats >? nats 
  a!6220!6220pairs >? pairs 
  a!6220!6220odds >? odds 
  a!6220!6220incr(X) >? incr(X) 
  a!6220!6220head(X) >? head(X) 
  a!6220!6220tail(X) >? tail(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220head = \y0.y0 
  a!6220!6220incr = \y0.2y0 
  a!6220!6220nats = 0 
  a!6220!6220odds = 0 
  a!6220!6220pairs = 0 
  a!6220!6220tail = \y0.y0 
  cons = \y0y1.2y0 + 2y1 
  head = \y0.y0 
  incr = \y0.2y0 
  mark = \y0.2y0 
  nats = 0 
  nil = 2 
  odds = 0 
  pairs = 0 
  s = \y0.y0 
  tail = \y0.y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220nats]] = 0 >= 0 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = 4x0 + 4x1 >= 4x0 + 4x1 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[mark(nats)]] = 0 >= 0 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 0 >= 0 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(head(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220head(mark(_x0))]] 
  [[mark(tail(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220tail(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] 
  [[mark(nil)]] = 4 > 2 = [[nil]] 
  [[mark(cons(_x0, _x1))]] = 4x0 + 4x1 >= 2x1 + 4x0 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220nats]] = 0 >= 0 = [[nats]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = 2x0 >= 2x0 = [[incr(_x0)]] 
  [[a!6220!6220head(_x0)]] = x0 >= x0 = [[head(_x0)]] 
  [[a!6220!6220tail(_x0)]] = x0 >= x0 = [[tail(_x0)]] 

We can thus remove the following rules:

  mark(nil) => nil 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  a!6220!6220nats >? cons(0, incr(nats)) 
  a!6220!6220pairs >? cons(0, incr(odds)) 
  a!6220!6220odds >? a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >? cons(s(mark(X)), incr(Y)) 
  mark(nats) >? a!6220!6220nats 
  mark(pairs) >? a!6220!6220pairs 
  mark(odds) >? a!6220!6220odds 
  mark(incr(X)) >? a!6220!6220incr(mark(X)) 
  mark(head(X)) >? a!6220!6220head(mark(X)) 
  mark(tail(X)) >? a!6220!6220tail(mark(X)) 
  mark(0) >? 0 
  mark(s(X)) >? s(mark(X)) 
  mark(cons(X, Y)) >? cons(mark(X), Y) 
  a!6220!6220nats >? nats 
  a!6220!6220pairs >? pairs 
  a!6220!6220odds >? odds 
  a!6220!6220incr(X) >? incr(X) 
  a!6220!6220head(X) >? head(X) 
  a!6220!6220tail(X) >? tail(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220head = \y0.2 + y0 
  a!6220!6220incr = \y0.2y0 
  a!6220!6220nats = 0 
  a!6220!6220odds = 0 
  a!6220!6220pairs = 0 
  a!6220!6220tail = \y0.y0 
  cons = \y0y1.y1 + 2y0 
  head = \y0.1 + y0 
  incr = \y0.2y0 
  mark = \y0.2y0 
  nats = 0 
  odds = 0 
  pairs = 0 
  s = \y0.y0 
  tail = \y0.y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220nats]] = 0 >= 0 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = 2x1 + 4x0 >= 2x1 + 4x0 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[mark(nats)]] = 0 >= 0 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 0 >= 0 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(head(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[a!6220!6220head(mark(_x0))]] 
  [[mark(tail(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220tail(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] 
  [[mark(cons(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 4x0 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220nats]] = 0 >= 0 = [[nats]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = 2x0 >= 2x0 = [[incr(_x0)]] 
  [[a!6220!6220head(_x0)]] = 2 + x0 > 1 + x0 = [[head(_x0)]] 
  [[a!6220!6220tail(_x0)]] = x0 >= x0 = [[tail(_x0)]] 

We can thus remove the following rules:

  a!6220!6220head(X) => head(X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  a!6220!6220nats >? cons(0, incr(nats)) 
  a!6220!6220pairs >? cons(0, incr(odds)) 
  a!6220!6220odds >? a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >? cons(s(mark(X)), incr(Y)) 
  mark(nats) >? a!6220!6220nats 
  mark(pairs) >? a!6220!6220pairs 
  mark(odds) >? a!6220!6220odds 
  mark(incr(X)) >? a!6220!6220incr(mark(X)) 
  mark(head(X)) >? a!6220!6220head(mark(X)) 
  mark(tail(X)) >? a!6220!6220tail(mark(X)) 
  mark(0) >? 0 
  mark(s(X)) >? s(mark(X)) 
  mark(cons(X, Y)) >? cons(mark(X), Y) 
  a!6220!6220nats >? nats 
  a!6220!6220pairs >? pairs 
  a!6220!6220odds >? odds 
  a!6220!6220incr(X) >? incr(X) 
  a!6220!6220tail(X) >? tail(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220head = \y0.y0 
  a!6220!6220incr = \y0.2y0 
  a!6220!6220nats = 0 
  a!6220!6220odds = 0 
  a!6220!6220pairs = 0 
  a!6220!6220tail = \y0.2 + y0 
  cons = \y0y1.y0 + 2y1 
  head = \y0.3 + 3y0 
  incr = \y0.2y0 
  mark = \y0.2y0 
  nats = 0 
  odds = 0 
  pairs = 0 
  s = \y0.y0 
  tail = \y0.1 + y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220nats]] = 0 >= 0 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 4x1 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[mark(nats)]] = 0 >= 0 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 0 >= 0 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(head(_x0))]] = 6 + 6x0 > 2x0 = [[a!6220!6220head(mark(_x0))]] 
  [[mark(tail(_x0))]] = 2 + 2x0 >= 2 + 2x0 = [[a!6220!6220tail(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] 
  [[mark(cons(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 2x1 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220nats]] = 0 >= 0 = [[nats]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = 2x0 >= 2x0 = [[incr(_x0)]] 
  [[a!6220!6220tail(_x0)]] = 2 + x0 > 1 + x0 = [[tail(_x0)]] 

We can thus remove the following rules:

  mark(head(X)) => a!6220!6220head(mark(X)) 
  a!6220!6220tail(X) => tail(X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  a!6220!6220nats >? cons(0, incr(nats)) 
  a!6220!6220pairs >? cons(0, incr(odds)) 
  a!6220!6220odds >? a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >? cons(s(mark(X)), incr(Y)) 
  mark(nats) >? a!6220!6220nats 
  mark(pairs) >? a!6220!6220pairs 
  mark(odds) >? a!6220!6220odds 
  mark(incr(X)) >? a!6220!6220incr(mark(X)) 
  mark(tail(X)) >? a!6220!6220tail(mark(X)) 
  mark(0) >? 0 
  mark(s(X)) >? s(mark(X)) 
  mark(cons(X, Y)) >? cons(mark(X), Y) 
  a!6220!6220nats >? nats 
  a!6220!6220pairs >? pairs 
  a!6220!6220odds >? odds 
  a!6220!6220incr(X) >? incr(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220incr = \y0.y0 
  a!6220!6220nats = 0 
  a!6220!6220odds = 0 
  a!6220!6220pairs = 0 
  a!6220!6220tail = \y0.y0 
  cons = \y0y1.y0 + y1 
  incr = \y0.y0 
  mark = \y0.y0 
  nats = 0 
  odds = 0 
  pairs = 0 
  s = \y0.y0 
  tail = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220nats]] = 0 >= 0 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[mark(nats)]] = 0 >= 0 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 0 >= 0 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = x0 >= x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(tail(_x0))]] = 3 + 3x0 > x0 = [[a!6220!6220tail(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[mark(cons(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220nats]] = 0 >= 0 = [[nats]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = x0 >= x0 = [[incr(_x0)]] 

We can thus remove the following rules:

  mark(tail(X)) => a!6220!6220tail(mark(X)) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  a!6220!6220nats >? cons(0, incr(nats)) 
  a!6220!6220pairs >? cons(0, incr(odds)) 
  a!6220!6220odds >? a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >? cons(s(mark(X)), incr(Y)) 
  mark(nats) >? a!6220!6220nats 
  mark(pairs) >? a!6220!6220pairs 
  mark(odds) >? a!6220!6220odds 
  mark(incr(X)) >? a!6220!6220incr(mark(X)) 
  mark(0) >? 0 
  mark(s(X)) >? s(mark(X)) 
  mark(cons(X, Y)) >? cons(mark(X), Y) 
  a!6220!6220nats >? nats 
  a!6220!6220pairs >? pairs 
  a!6220!6220odds >? odds 
  a!6220!6220incr(X) >? incr(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220incr = \y0.2y0 
  a!6220!6220nats = 2 
  a!6220!6220odds = 0 
  a!6220!6220pairs = 0 
  cons = \y0y1.y1 + 2y0 
  incr = \y0.2y0 
  mark = \y0.2y0 
  nats = 1 
  odds = 0 
  pairs = 0 
  s = \y0.y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220nats]] = 2 >= 2 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = 2x1 + 4x0 >= 2x1 + 4x0 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[mark(nats)]] = 2 >= 2 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 0 >= 0 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = 4x0 >= 4x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[mark(s(_x0))]] = 2x0 >= 2x0 = [[s(mark(_x0))]] 
  [[mark(cons(_x0, _x1))]] = 2x1 + 4x0 >= x1 + 4x0 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220nats]] = 2 > 1 = [[nats]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 0 >= 0 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = 2x0 >= 2x0 = [[incr(_x0)]] 

We can thus remove the following rules:

  a!6220!6220nats => nats 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]).

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] a!6220!6220odds# =#> a!6220!6220incr#(a!6220!6220pairs)   
    1] a!6220!6220odds# =#> a!6220!6220pairs#   
    2] a!6220!6220incr#(cons(X, Y)) =#> mark#(X)   
    3] mark#(nats) =#> a!6220!6220nats#   
    4] mark#(pairs) =#> a!6220!6220pairs#   
    5] mark#(odds) =#> a!6220!6220odds#   
    6] mark#(incr(X)) =#> a!6220!6220incr#(mark(X))   
    7] mark#(incr(X)) =#> mark#(X)   
    8] mark#(s(X)) =#> mark#(X)   
    9] mark#(cons(X, Y)) =#> mark#(X)   

  Rules R_0:

    a!6220!6220nats => cons(0, incr(nats)) 
    a!6220!6220pairs => cons(0, incr(odds)) 
    a!6220!6220odds => a!6220!6220incr(a!6220!6220pairs) 
    a!6220!6220incr(cons(X, Y)) => cons(s(mark(X)), incr(Y)) 
    mark(nats) => a!6220!6220nats 
    mark(pairs) => a!6220!6220pairs 
    mark(odds) => a!6220!6220odds 
    mark(incr(X)) => a!6220!6220incr(mark(X)) 
    mark(0) => 0 
    mark(s(X)) => s(mark(X)) 
    mark(cons(X, Y)) => cons(mark(X), Y) 
    a!6220!6220pairs => pairs 
    a!6220!6220odds => odds 
    a!6220!6220incr(X) => incr(X) 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 2 
    * 1 :  
    * 2 : 3, 4, 5, 6, 7, 8, 9 
    * 3 :  
    * 4 :  
    * 5 : 0, 1 
    * 6 : 2 
    * 7 : 3, 4, 5, 6, 7, 8, 9 
    * 8 : 3, 4, 5, 6, 7, 8, 9 
    * 9 : 3, 4, 5, 6, 7, 8, 9 

This graph has the following strongly connected components:

  P_1:

    a!6220!6220odds# =#> a!6220!6220incr#(a!6220!6220pairs)   
    a!6220!6220incr#(cons(X, Y)) =#> mark#(X)   
    mark#(odds) =#> a!6220!6220odds#   
    mark#(incr(X)) =#> a!6220!6220incr#(mark(X))   
    mark#(incr(X)) =#> mark#(X)   
    mark#(s(X)) =#> mark#(X)   
    mark#(cons(X, Y)) =#> mark#(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).

Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

The formative rules of (P_1, R_0) are R_1 ::=

  a!6220!6220nats => cons(0, incr(nats)) 
  a!6220!6220pairs => cons(0, incr(odds)) 
  a!6220!6220odds => a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) => cons(s(mark(X)), incr(Y)) 
  mark(nats) => a!6220!6220nats 
  mark(pairs) => a!6220!6220pairs 
  mark(odds) => a!6220!6220odds 
  mark(incr(X)) => a!6220!6220incr(mark(X)) 
  mark(s(X)) => s(mark(X)) 
  mark(cons(X, Y)) => cons(mark(X), Y) 
  a!6220!6220pairs => pairs 
  a!6220!6220odds => odds 
  a!6220!6220incr(X) => incr(X) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative).

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220odds# >? a!6220!6220incr#(a!6220!6220pairs) 
  a!6220!6220incr#(cons(X, Y)) >? mark#(X) 
  mark#(odds) >? a!6220!6220odds# 
  mark#(incr(X)) >? a!6220!6220incr#(mark(X)) 
  mark#(incr(X)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  mark#(cons(X, Y)) >? mark#(X) 
  a!6220!6220nats >= cons(0, incr(nats)) 
  a!6220!6220pairs >= cons(0, incr(odds)) 
  a!6220!6220odds >= a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >= cons(s(mark(X)), incr(Y)) 
  mark(nats) >= a!6220!6220nats 
  mark(pairs) >= a!6220!6220pairs 
  mark(odds) >= a!6220!6220odds 
  mark(incr(X)) >= a!6220!6220incr(mark(X)) 
  mark(s(X)) >= s(mark(X)) 
  mark(cons(X, Y)) >= cons(mark(X), Y) 
  a!6220!6220pairs >= pairs 
  a!6220!6220odds >= odds 
  a!6220!6220incr(X) >= incr(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220incr = \y0.y0 
  a!6220!6220incr# = \y0.2y0 
  a!6220!6220nats = 0 
  a!6220!6220odds = 1 
  a!6220!6220odds# = 0 
  a!6220!6220pairs = 0 
  cons = \y0y1.y0 
  incr = \y0.y0 
  mark = \y0.y0 
  mark# = \y0.2y0 
  nats = 0 
  odds = 1 
  pairs = 0 
  s = \y0.y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220odds#]] = 0 >= 0 = [[a!6220!6220incr#(a!6220!6220pairs)]] 
  [[a!6220!6220incr#(cons(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(odds)]] = 2 > 0 = [[a!6220!6220odds#]] 
  [[mark#(incr(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220incr#(mark(_x0))]] 
  [[mark#(incr(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(cons(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220nats]] = 0 >= 0 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 1 >= 0 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = x0 >= x0 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[mark(nats)]] = 0 >= 0 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 1 >= 1 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = x0 >= x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[mark(cons(_x0, _x1))]] = x0 >= x0 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 1 >= 1 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = x0 >= x0 = [[incr(_x0)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_1, minimal, formative) by (P_2, R_1, minimal, formative), where P_2 consists of:

  a!6220!6220odds# =#> a!6220!6220incr#(a!6220!6220pairs)   
  a!6220!6220incr#(cons(X, Y)) =#> mark#(X)   
  mark#(incr(X)) =#> a!6220!6220incr#(mark(X))   
  mark#(incr(X)) =#> mark#(X)   
  mark#(s(X)) =#> mark#(X)   
  mark#(cons(X, Y)) =#> mark#(X)   

Thus, the original system is terminating if (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_1, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 1 
    * 1 : 2, 3, 4, 5 
    * 2 : 1 
    * 3 : 2, 3, 4, 5 
    * 4 : 2, 3, 4, 5 
    * 5 : 2, 3, 4, 5 

This graph has the following strongly connected components:

  P_3:

    a!6220!6220incr#(cons(X, Y)) =#> mark#(X)   
    mark#(incr(X)) =#> a!6220!6220incr#(mark(X))   
    mark#(incr(X)) =#> mark#(X)   
    mark#(s(X)) =#> mark#(X)   
    mark#(cons(X, Y)) =#> mark#(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_2, R_1, m, f) by (P_3, R_1, m, f).

Thus, the original system is terminating if (P_3, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_1, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220incr#(cons(X, Y)) >? mark#(X) 
  mark#(incr(X)) >? a!6220!6220incr#(mark(X)) 
  mark#(incr(X)) >? mark#(X) 
  mark#(s(X)) >? mark#(X) 
  mark#(cons(X, Y)) >? mark#(X) 
  a!6220!6220nats >= cons(0, incr(nats)) 
  a!6220!6220pairs >= cons(0, incr(odds)) 
  a!6220!6220odds >= a!6220!6220incr(a!6220!6220pairs) 
  a!6220!6220incr(cons(X, Y)) >= cons(s(mark(X)), incr(Y)) 
  mark(nats) >= a!6220!6220nats 
  mark(pairs) >= a!6220!6220pairs 
  mark(odds) >= a!6220!6220odds 
  mark(incr(X)) >= a!6220!6220incr(mark(X)) 
  mark(s(X)) >= s(mark(X)) 
  mark(cons(X, Y)) >= cons(mark(X), Y) 
  a!6220!6220pairs >= pairs 
  a!6220!6220odds >= odds 
  a!6220!6220incr(X) >= incr(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  a!6220!6220incr = \y0.2 + 3y0 
  a!6220!6220incr# = \y0.y0 
  a!6220!6220nats = 0 
  a!6220!6220odds = 2 
  a!6220!6220pairs = 0 
  cons = \y0y1.3y0 
  incr = \y0.2 + 3y0 
  mark = \y0.y0 
  mark# = \y0.2y0 
  nats = 0 
  odds = 2 
  pairs = 0 
  s = \y0.y0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220incr#(cons(_x0, _x1))]] = 3x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(incr(_x0))]] = 4 + 6x0 > x0 = [[a!6220!6220incr#(mark(_x0))]] 
  [[mark#(incr(_x0))]] = 4 + 6x0 > 2x0 = [[mark#(_x0)]] 
  [[mark#(s(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(cons(_x0, _x1))]] = 6x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220nats]] = 0 >= 0 = [[cons(0, incr(nats))]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[cons(0, incr(odds))]] 
  [[a!6220!6220odds]] = 2 >= 2 = [[a!6220!6220incr(a!6220!6220pairs)]] 
  [[a!6220!6220incr(cons(_x0, _x1))]] = 2 + 9x0 >= 3x0 = [[cons(s(mark(_x0)), incr(_x1))]] 
  [[mark(nats)]] = 0 >= 0 = [[a!6220!6220nats]] 
  [[mark(pairs)]] = 0 >= 0 = [[a!6220!6220pairs]] 
  [[mark(odds)]] = 2 >= 2 = [[a!6220!6220odds]] 
  [[mark(incr(_x0))]] = 2 + 3x0 >= 2 + 3x0 = [[a!6220!6220incr(mark(_x0))]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[mark(cons(_x0, _x1))]] = 3x0 >= 3x0 = [[cons(mark(_x0), _x1)]] 
  [[a!6220!6220pairs]] = 0 >= 0 = [[pairs]] 
  [[a!6220!6220odds]] = 2 >= 2 = [[odds]] 
  [[a!6220!6220incr(_x0)]] = 2 + 3x0 >= 2 + 3x0 = [[incr(_x0)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_1, minimal, formative) by (P_4, R_1, minimal, formative), where P_4 consists of:

  a!6220!6220incr#(cons(X, Y)) =#> mark#(X)   
  mark#(s(X)) =#> mark#(X)   
  mark#(cons(X, Y)) =#> mark#(X)   

Thus, the original system is terminating if (P_4, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_4, R_1, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 1, 2 
    * 1 : 1, 2 
    * 2 : 1, 2 

This graph has the following strongly connected components:

  P_5:

    mark#(s(X)) =#> mark#(X)   
    mark#(cons(X, Y)) =#> mark#(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_4, R_1, m, f) by (P_5, R_1, m, f).

Thus, the original system is terminating if (P_5, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_1, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(mark#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(mark#(s(X))) = s(X) |> X = nu(mark#(X)) 
  nu(mark#(cons(X, Y))) = cons(X, Y) |> X = nu(mark#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_5, R_1, minimal, f) by ({}, R_1, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.