We consider termination of the STRS with no additional rule schemes:

  Signature: and     :: a -> a -> a
             cons    :: c -> d -> d
             false   :: b
             filter  :: (c -> b) -> d -> d
             filter2 :: b -> (c -> b) -> c -> d -> d
             map     :: (c -> c) -> d -> d
             nil     :: d
             not     :: a -> a
             or      :: a -> a -> a
             true    :: b

  Rules: not(not(X)) -> X
         not(or(Y, U)) -> and(not(Y), not(U))
         not(and(V, W)) -> or(not(V), not(W))
         and(P, or(X1, Y1)) -> or(and(P, X1), and(P, Y1))
         and(or(V1, W1), U1) -> or(and(U1, V1), and(U1, W1))
         map(J1, nil) -> nil
         map(F2, cons(Y2, U2)) -> cons(F2(Y2), map(F2, U2))
         filter(H2, nil) -> nil
         filter(I2, cons(P2, X3)) -> filter2(I2(P2), I2, P2, X3)
         filter2(true, Z3, U3, V3) -> cons(U3, filter(Z3, V3))
         filter2(false, I3, P3, X4) -> filter(I3, X4)

The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the initial DP problem D1 = (P1, R, i, c), where:

  P1. (1)  not#(or(Y, U)) => not#(Y)
      (2)  not#(or(Y, U)) => not#(U)
      (3)  not#(or(Y, U)) => and#(not(Y), not(U))
      (4)  not#(and(V, W)) => not#(V)
      (5)  not#(and(V, W)) => not#(W)
      (6)  and#(P, or(X1, Y1)) => and#(P, X1)
      (7)  and#(P, or(X1, Y1)) => and#(P, Y1)
      (8)  and#(or(V1, W1), U1) => and#(U1, V1)
      (9)  and#(or(V1, W1), U1) => and#(U1, W1)
      (10) map#(F2, cons(Y2, U2)) => map#(F2, U2)
      (11) filter#(I2, cons(P2, X3)) => filter2#(I2(P2), I2, P2, X3)
      (12) filter2#(true, Z3, U3, V3) => filter#(Z3, V3)
      (13) filter2#(false, I3, P3, X4) => filter#(I3, X4)

***** We apply the Graph Processor on D1 = (P1, R, i, c).

We compute a graph approximation with the following edges:

  1:  1 2 3 4 5 
  2:  1 2 3 4 5 
  3:  6 7 8 9 
  4:  1 2 3 4 5 
  5:  1 2 3 4 5 
  6:  6 7 8 9 
  7:  6 7 8 9 
  8:  6 7 8 9 
  9:  6 7 8 9 
  10: 10 
  11: 12 13 
  12: 11 
  13: 11 

There are 4 SCCs.

Processor output: { D2 = (P2, R, i, c) ; D3 = (P3, R, i, c) ; D4 = (P4, R, i, c) ; D5 = (P5, R, i, c) }, where:

  P2. (1) and#(P, or(X1, Y1)) => and#(P, X1)
      (2) and#(P, or(X1, Y1)) => and#(P, Y1)
      (3) and#(or(V1, W1), U1) => and#(U1, V1)
      (4) and#(or(V1, W1), U1) => and#(U1, W1)

  P3. (1) not#(or(Y, U)) => not#(Y)
      (2) not#(or(Y, U)) => not#(U)
      (3) not#(and(V, W)) => not#(V)
      (4) not#(and(V, W)) => not#(W)

  P4. (1) map#(F2, cons(Y2, U2)) => map#(F2, U2)

  P5. (1) filter#(I2, cons(P2, X3)) => filter2#(I2(P2), I2, P2, X3)
      (2) filter2#(true, Z3, U3, V3) => filter#(Z3, V3)
      (3) filter2#(false, I3, P3, X4) => filter#(I3, X4)

***** We apply the Usable Rules Processor on D2 = (P2, R, i, c).

We obtain 0 usable rules (out of 11 rules in the input problem).

Processor output: { D6 = (P2, {}, i, c) }.

***** We apply the Subterm Criterion Processor on D3 = (P3, R, i, c).

We use the following projection function:

  nu(not#) = 1

We thus have:

  (1) or(Y, U)  |>| Y
  (2) or(Y, U)  |>| U
  (3) and(V, W) |>| V
  (4) and(V, W) |>| W

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.

***** We apply the Subterm Criterion Processor on D4 = (P4, R, i, c).

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(Y2, U2) |>| U2

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.

***** We apply the Subterm Criterion Processor on D5 = (P5, R, i, c).

We use the following projection function:

  nu(filter#)  = 2
  nu(filter2#) = 4

We thus have:

  (1) cons(P2, X3) |>|  X3
  (2) V3           |>=| V3
  (3) X4           |>=| X4

We may remove the strictly oriented DPs.

Processor output: { D7 = (P6, R, i, c) }, where:

  P6. (1) filter2#(true, Z3, U3, V3) => filter#(Z3, V3)
      (2) filter2#(false, I3, P3, X4) => filter#(I3, X4)

***** We apply the Usable rules with respect to an argument wiltering [with HORPO] Processor on D6 = (P2, {}, i, c).

Constrained HORPO yields:

  and#2(P, or2(X1, Y1)) (>) and#2(P, X1)
  and#2(P, or2(X1, Y1)) (>) and#2(P, Y1)
  and#2(or2(V1, W1), U1) (>) and#2(U1, V1)
  and#2(or2(V1, W1), U1) (>) and#2(U1, W1)

We do this using the following settings:

* Monotonicity requirements: this is a strongly monotonic reduction pair (all arguments of function symbols were regarded).

* Precedence and permutation:

    or2    { 1 }   2
  > and#2  { 1 2 }

* Well-founded theory orderings:

  [>]_{Bool} = {(true,false)}
  [>]_{Int}  = {(x,y) | x < 1000 /\ x < y }

Processor output: { }.

***** We apply the Graph Processor on D7 = (P6, R, i, c).

We compute a graph approximation with the following edges:

  1:
  2:

As there are no SCCs, this DP problem is removed.

Processor output: { }.