We consider the system prenex.

  Alphabet:

    and : [form * form] --> form 
    exists : [form -> form] --> form 
    forall : [form -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 

  Rules:

    and(x, forall(/\y.f y)) => forall(/\z.and(x, f z)) 
    or(x, forall(/\y.f y)) => forall(/\z.or(x, f z)) 
    and(forall(/\x.f x), y) => forall(/\z.and(f z, y)) 
    or(forall(/\x.f x), y) => forall(/\z.or(f z, y)) 
    not(forall(/\x.f x)) => exists(/\y.not(f y)) 
    and(x, exists(/\y.f y)) => exists(/\z.and(x, f z)) 
    or(x, exists(/\y.f y)) => exists(/\z.or(x, f z)) 
    and(exists(/\x.f x), y) => exists(/\z.and(f z, y)) 
    or(exists(/\x.f x), y) => exists(/\z.or(f z, y)) 
    not(exists(/\x.f x)) => forall(/\y.not(f y)) 

Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.

We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:

  Alphabet:

    and : [form * form] --> form 
    exists : [form -> form] --> form 
    forall : [form -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 
    ~AP1 : [form -> form * form] --> form 

  Rules:

    and(X, forall(/\x.~AP1(F, x))) => forall(/\y.and(X, ~AP1(F, y))) 
    or(X, forall(/\x.~AP1(F, x))) => forall(/\y.or(X, ~AP1(F, y))) 
    and(forall(/\x.~AP1(F, x)), X) => forall(/\y.and(~AP1(F, y), X)) 
    or(forall(/\x.~AP1(F, x)), X) => forall(/\y.or(~AP1(F, y), X)) 
    not(forall(/\x.~AP1(F, x))) => exists(/\y.not(~AP1(F, y))) 
    and(X, exists(/\x.~AP1(F, x))) => exists(/\y.and(X, ~AP1(F, y))) 
    or(X, exists(/\x.~AP1(F, x))) => exists(/\y.or(X, ~AP1(F, y))) 
    and(exists(/\x.~AP1(F, x)), X) => exists(/\y.and(~AP1(F, y), X)) 
    or(exists(/\x.~AP1(F, x)), X) => exists(/\y.or(~AP1(F, y), X)) 
    not(exists(/\x.~AP1(F, x))) => forall(/\y.not(~AP1(F, y))) 
    ~AP1(F, X) => F X 

Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:

  Alphabet:

    and : [form * form] --> form 
    exists : [form -> form] --> form 
    forall : [form -> form] --> form 
    not : [form] --> form 
    or : [form * form] --> form 

  Rules:

    and(X, forall(/\x.Y(x))) => forall(/\y.and(X, Y(y))) 
    or(X, forall(/\x.Y(x))) => forall(/\y.or(X, Y(y))) 
    and(forall(/\x.X(x)), Y) => forall(/\y.and(X(y), Y)) 
    or(forall(/\x.X(x)), Y) => forall(/\y.or(X(y), Y)) 
    not(forall(/\x.X(x))) => exists(/\y.not(X(y))) 
    and(X, exists(/\x.Y(x))) => exists(/\y.and(X, Y(y))) 
    or(X, exists(/\x.Y(x))) => exists(/\y.or(X, Y(y))) 
    and(exists(/\x.X(x)), Y) => exists(/\y.and(X(y), Y)) 
    or(exists(/\x.X(x)), Y) => exists(/\y.or(X(y), Y)) 
    not(exists(/\x.X(x))) => forall(/\y.not(X(y))) 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

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

  Dependency Pairs P_0:

    0] and#(X, forall(/\x.Y(x))) =#> and#(X, Y(y))   
    1] and#(X, forall(/\x.Y(x))) =#> Y(y)   
    2] or#(X, forall(/\x.Y(x))) =#> or#(X, Y(y))   
    3] or#(X, forall(/\x.Y(x))) =#> Y(y)   
    4] and#(forall(/\x.X(x)), Y) =#> and#(X(y), Y)   
    5] and#(forall(/\x.X(x)), Y) =#> X(y)   
    6] or#(forall(/\x.X(x)), Y) =#> or#(X(y), Y)   
    7] or#(forall(/\x.X(x)), Y) =#> X(y)   
    8] not#(forall(/\x.X(x))) =#> not#(X(y))   
    9] not#(forall(/\x.X(x))) =#> X(y)   
    10] and#(X, exists(/\x.Y(x))) =#> and#(X, Y(y))   
    11] and#(X, exists(/\x.Y(x))) =#> Y(y)   
    12] or#(X, exists(/\x.Y(x))) =#> or#(X, Y(y))   
    13] or#(X, exists(/\x.Y(x))) =#> Y(y)   
    14] and#(exists(/\x.X(x)), Y) =#> and#(X(y), Y)   
    15] and#(exists(/\x.X(x)), Y) =#> X(y)   
    16] or#(exists(/\x.X(x)), Y) =#> or#(X(y), Y)   
    17] or#(exists(/\x.X(x)), Y) =#> X(y)   
    18] not#(exists(/\x.X(x))) =#> not#(X(y))   
    19] not#(exists(/\x.X(x))) =#> X(y)   

  Rules R_0:

    and(X, forall(/\x.Y(x))) => forall(/\y.and(X, Y(y))) 
    or(X, forall(/\x.Y(x))) => forall(/\y.or(X, Y(y))) 
    and(forall(/\x.X(x)), Y) => forall(/\y.and(X(y), Y)) 
    or(forall(/\x.X(x)), Y) => forall(/\y.or(X(y), Y)) 
    not(forall(/\x.X(x))) => exists(/\y.not(X(y))) 
    and(X, exists(/\x.Y(x))) => exists(/\y.and(X, Y(y))) 
    or(X, exists(/\x.Y(x))) => exists(/\y.or(X, Y(y))) 
    and(exists(/\x.X(x)), Y) => exists(/\y.and(X(y), Y)) 
    or(exists(/\x.X(x)), Y) => exists(/\y.or(X(y), Y)) 
    not(exists(/\x.X(x))) => forall(/\y.not(X(y))) 

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 will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

  and#(X, forall(/\x.Y(x))) >? and#(X, Y(~c0)) 
  and#(X, forall(/\x.Y(x))) >? Y(~c1) 
  or#(X, forall(/\x.Y(x))) >? or#(X, Y(~c2)) 
  or#(X, forall(/\x.Y(x))) >? Y(~c3) 
  and#(forall(/\x.X(x)), Y) >? and#(X(~c4), Y) 
  and#(forall(/\x.X(x)), Y) >? X(~c5) 
  or#(forall(/\x.X(x)), Y) >? or#(X(~c6), Y) 
  or#(forall(/\x.X(x)), Y) >? X(~c7) 
  not#(forall(/\x.X(x))) >? not#(X(~c8)) 
  not#(forall(/\x.X(x))) >? X(~c9) 
  and#(X, exists(/\x.Y(x))) >? and#(X, Y(~c10)) 
  and#(X, exists(/\x.Y(x))) >? Y(~c11) 
  or#(X, exists(/\x.Y(x))) >? or#(X, Y(~c12)) 
  or#(X, exists(/\x.Y(x))) >? Y(~c13) 
  and#(exists(/\x.X(x)), Y) >? and#(X(~c14), Y) 
  and#(exists(/\x.X(x)), Y) >? X(~c15) 
  or#(exists(/\x.X(x)), Y) >? or#(X(~c16), Y) 
  or#(exists(/\x.X(x)), Y) >? X(~c17) 
  not#(exists(/\x.X(x))) >? not#(X(~c18)) 
  not#(exists(/\x.X(x))) >? X(~c19) 
  and(X, forall(/\x.Y(x))) >= forall(/\y.and-(X, Y(y))) 
  or(X, forall(/\x.Y(x))) >= forall(/\y.or-(X, Y(y))) 
  and(forall(/\x.X(x)), Y) >= forall(/\y.and-(X(y), Y)) 
  or(forall(/\x.X(x)), Y) >= forall(/\y.or-(X(y), Y)) 
  not(forall(/\x.X(x))) >= exists(/\y.not-(X(y))) 
  and(X, exists(/\x.Y(x))) >= exists(/\y.and-(X, Y(y))) 
  or(X, exists(/\x.Y(x))) >= exists(/\y.or-(X, Y(y))) 
  and(exists(/\x.X(x)), Y) >= exists(/\y.and-(X(y), Y)) 
  or(exists(/\x.X(x)), Y) >= exists(/\y.or-(X(y), Y)) 
  not(exists(/\x.X(x))) >= forall(/\y.not-(X(y))) 
  and-(X, Y) >= and(X, Y) 
  and-(X, Y) >= and#(X, Y) 
  not-(X) >= not(X) 
  not-(X) >= not#(X) 
  or-(X, Y) >= or(X, Y) 
  or-(X, Y) >= or#(X, Y) 

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

The following interpretation satisfies the requirements:

  and = \y0y1.1 + y0 + y1 
  and- = \y0y1.1 + y0 + y1 
  and# = \y0y1.1 + y0 + y1 
  exists = \G0.G0(0) 
  forall = \G0.G0(0) 
  not = \y0.y0 
  not- = \y0.y0 
  not# = \y0.y0 
  or = \y0y1.y1 + 2y0 
  or- = \y0y1.y1 + 2y0 
  or# = \y0y1.y0 + y1 
  ~c0 = 0 
  ~c1 = 0 
  ~c10 = 0 
  ~c11 = 0 
  ~c12 = 0 
  ~c13 = 0 
  ~c14 = 0 
  ~c15 = 0 
  ~c16 = 0 
  ~c17 = 0 
  ~c18 = 0 
  ~c19 = 0 
  ~c2 = 0 
  ~c3 = 0 
  ~c4 = 0 
  ~c5 = 0 
  ~c6 = 0 
  ~c7 = 0 
  ~c8 = 0 
  ~c9 = 0 

Using this interpretation, the requirements translate to:

  [[and#(_x0, forall(/\x._x1(x)))]] = 1 + x0 + F1(0) >= 1 + x0 + F1(0) = [[and#(_x0, _x1(~c0))]] 
  [[and#(_x0, forall(/\x._x1(x)))]] = 1 + x0 + F1(0) > F1(0) = [[_x1(~c1)]] 
  [[or#(_x0, forall(/\x._x1(x)))]] = x0 + F1(0) >= x0 + F1(0) = [[or#(_x0, _x1(~c2))]] 
  [[or#(_x0, forall(/\x._x1(x)))]] = x0 + F1(0) >= F1(0) = [[_x1(~c3)]] 
  [[and#(forall(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) >= 1 + x1 + F0(0) = [[and#(_x0(~c4), _x1)]] 
  [[and#(forall(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) > F0(0) = [[_x0(~c5)]] 
  [[or#(forall(/\x._x0(x)), _x1)]] = x1 + F0(0) >= x1 + F0(0) = [[or#(_x0(~c6), _x1)]] 
  [[or#(forall(/\x._x0(x)), _x1)]] = x1 + F0(0) >= F0(0) = [[_x0(~c7)]] 
  [[not#(forall(/\x._x0(x)))]] = F0(0) >= F0(0) = [[not#(_x0(~c8))]] 
  [[not#(forall(/\x._x0(x)))]] = F0(0) >= F0(0) = [[_x0(~c9)]] 
  [[and#(_x0, exists(/\x._x1(x)))]] = 1 + x0 + F1(0) >= 1 + x0 + F1(0) = [[and#(_x0, _x1(~c10))]] 
  [[and#(_x0, exists(/\x._x1(x)))]] = 1 + x0 + F1(0) > F1(0) = [[_x1(~c11)]] 
  [[or#(_x0, exists(/\x._x1(x)))]] = x0 + F1(0) >= x0 + F1(0) = [[or#(_x0, _x1(~c12))]] 
  [[or#(_x0, exists(/\x._x1(x)))]] = x0 + F1(0) >= F1(0) = [[_x1(~c13)]] 
  [[and#(exists(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) >= 1 + x1 + F0(0) = [[and#(_x0(~c14), _x1)]] 
  [[and#(exists(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) > F0(0) = [[_x0(~c15)]] 
  [[or#(exists(/\x._x0(x)), _x1)]] = x1 + F0(0) >= x1 + F0(0) = [[or#(_x0(~c16), _x1)]] 
  [[or#(exists(/\x._x0(x)), _x1)]] = x1 + F0(0) >= F0(0) = [[_x0(~c17)]] 
  [[not#(exists(/\x._x0(x)))]] = F0(0) >= F0(0) = [[not#(_x0(~c18))]] 
  [[not#(exists(/\x._x0(x)))]] = F0(0) >= F0(0) = [[_x0(~c19)]] 
  [[and(_x0, forall(/\x._x1(x)))]] = 1 + x0 + F1(0) >= 1 + x0 + F1(0) = [[forall(/\x.and-(_x0, _x1(x)))]] 
  [[or(_x0, forall(/\x._x1(x)))]] = 2x0 + F1(0) >= 2x0 + F1(0) = [[forall(/\x.or-(_x0, _x1(x)))]] 
  [[and(forall(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) >= 1 + x1 + F0(0) = [[forall(/\x.and-(_x0(x), _x1))]] 
  [[or(forall(/\x._x0(x)), _x1)]] = x1 + 2F0(0) >= x1 + 2F0(0) = [[forall(/\x.or-(_x0(x), _x1))]] 
  [[not(forall(/\x._x0(x)))]] = F0(0) >= F0(0) = [[exists(/\x.not-(_x0(x)))]] 
  [[and(_x0, exists(/\x._x1(x)))]] = 1 + x0 + F1(0) >= 1 + x0 + F1(0) = [[exists(/\x.and-(_x0, _x1(x)))]] 
  [[or(_x0, exists(/\x._x1(x)))]] = 2x0 + F1(0) >= 2x0 + F1(0) = [[exists(/\x.or-(_x0, _x1(x)))]] 
  [[and(exists(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) >= 1 + x1 + F0(0) = [[exists(/\x.and-(_x0(x), _x1))]] 
  [[or(exists(/\x._x0(x)), _x1)]] = x1 + 2F0(0) >= x1 + 2F0(0) = [[exists(/\x.or-(_x0(x), _x1))]] 
  [[not(exists(/\x._x0(x)))]] = F0(0) >= F0(0) = [[forall(/\x.not-(_x0(x)))]] 
  [[and-(_x0, _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[and(_x0, _x1)]] 
  [[and-(_x0, _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[and#(_x0, _x1)]] 
  [[not-(_x0)]] = x0 >= x0 = [[not(_x0)]] 
  [[not-(_x0)]] = x0 >= x0 = [[not#(_x0)]] 
  [[or-(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[or(_x0, _x1)]] 
  [[or-(_x0, _x1)]] = x1 + 2x0 >= x0 + x1 = [[or#(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_0, R_0, minimal, formative) by (P_1, R_0, minimal, formative), where P_1 consists of:

  and#(X, forall(/\x.Y(x))) =#> and#(X, Y(y))   
  or#(X, forall(/\x.Y(x))) =#> or#(X, Y(y))   
  or#(X, forall(/\x.Y(x))) =#> Y(y)   
  and#(forall(/\x.X(x)), Y) =#> and#(X(y), Y)   
  or#(forall(/\x.X(x)), Y) =#> or#(X(y), Y)   
  or#(forall(/\x.X(x)), Y) =#> X(y)   
  not#(forall(/\x.X(x))) =#> not#(X(y))   
  not#(forall(/\x.X(x))) =#> X(y)   
  and#(X, exists(/\x.Y(x))) =#> and#(X, Y(y))   
  or#(X, exists(/\x.Y(x))) =#> or#(X, Y(y))   
  or#(X, exists(/\x.Y(x))) =#> Y(y)   
  and#(exists(/\x.X(x)), Y) =#> and#(X(y), Y)   
  or#(exists(/\x.X(x)), Y) =#> or#(X(y), Y)   
  or#(exists(/\x.X(x)), Y) =#> X(y)   
  not#(exists(/\x.X(x))) =#> not#(X(y))   
  not#(exists(/\x.X(x))) =#> X(y)   

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).

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

    * 0 : 0, 3, 8, 11 
    * 1 : 1, 2, 4, 5, 9, 10, 12, 13 
    * 2 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 
    * 3 : 0, 3, 8, 11 
    * 4 : 1, 2, 4, 5, 9, 10, 12, 13 
    * 5 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 
    * 6 : 6, 7, 14, 15 
    * 7 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 
    * 8 : 0, 3, 8, 11 
    * 9 : 1, 2, 4, 5, 9, 10, 12, 13 
    * 10 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 
    * 11 : 0, 3, 8, 11 
    * 12 : 1, 2, 4, 5, 9, 10, 12, 13 
    * 13 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 
    * 14 : 6, 7, 14, 15 
    * 15 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 

This graph has the following strongly connected components:

  P_2:

    and#(X, forall(/\x.Y(x))) =#> and#(X, Y(y))   
    and#(forall(/\x.X(x)), Y) =#> and#(X(y), Y)   
    and#(X, exists(/\x.Y(x))) =#> and#(X, Y(y))   
    and#(exists(/\x.X(x)), Y) =#> and#(X(y), Y)   

  P_3:

    or#(X, forall(/\x.Y(x))) =#> or#(X, Y(y))   
    or#(X, forall(/\x.Y(x))) =#> Y(y)   
    or#(forall(/\x.X(x)), Y) =#> or#(X(y), Y)   
    or#(forall(/\x.X(x)), Y) =#> X(y)   
    not#(forall(/\x.X(x))) =#> not#(X(y))   
    not#(forall(/\x.X(x))) =#> X(y)   
    or#(X, exists(/\x.Y(x))) =#> or#(X, Y(y))   
    or#(X, exists(/\x.Y(x))) =#> Y(y)   
    or#(exists(/\x.X(x)), Y) =#> or#(X(y), Y)   
    or#(exists(/\x.X(x)), Y) =#> X(y)   
    not#(exists(/\x.X(x))) =#> not#(X(y))   
    not#(exists(/\x.X(x))) =#> X(y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_1, R_0, m, f) by (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_2, R_0, minimal, formative) and (P_3, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

  or#(X, forall(/\x.Y(x))) >? or#(X, Y(~c0)) 
  or#(X, forall(/\x.Y(x))) >? Y(~c1) 
  or#(forall(/\x.X(x)), Y) >? or#(X(~c2), Y) 
  or#(forall(/\x.X(x)), Y) >? X(~c3) 
  not#(forall(/\x.X(x))) >? not#(X(~c4)) 
  not#(forall(/\x.X(x))) >? X(~c5) 
  or#(X, exists(/\x.Y(x))) >? or#(X, Y(~c6)) 
  or#(X, exists(/\x.Y(x))) >? Y(~c7) 
  or#(exists(/\x.X(x)), Y) >? or#(X(~c8), Y) 
  or#(exists(/\x.X(x)), Y) >? X(~c9) 
  not#(exists(/\x.X(x))) >? not#(X(~c10)) 
  not#(exists(/\x.X(x))) >? X(~c11) 
  and(X, forall(/\x.Y(x))) >= forall(/\y.and-(X, Y(y))) 
  or(X, forall(/\x.Y(x))) >= forall(/\y.or-(X, Y(y))) 
  and(forall(/\x.X(x)), Y) >= forall(/\y.and-(X(y), Y)) 
  or(forall(/\x.X(x)), Y) >= forall(/\y.or-(X(y), Y)) 
  not(forall(/\x.X(x))) >= exists(/\y.not-(X(y))) 
  and(X, exists(/\x.Y(x))) >= exists(/\y.and-(X, Y(y))) 
  or(X, exists(/\x.Y(x))) >= exists(/\y.or-(X, Y(y))) 
  and(exists(/\x.X(x)), Y) >= exists(/\y.and-(X(y), Y)) 
  or(exists(/\x.X(x)), Y) >= exists(/\y.or-(X(y), Y)) 
  not(exists(/\x.X(x))) >= forall(/\y.not-(X(y))) 
  and-(X, Y) >= and(X, Y) 
  not-(X) >= not(X) 
  not-(X) >= not#(X) 
  or-(X, Y) >= or(X, Y) 
  or-(X, Y) >= or#(X, Y) 

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

The following interpretation satisfies the requirements:

  and = \y0y1.y0 + y1 
  and- = \y0y1.y0 + y1 
  exists = \G0.G0(0) 
  forall = \G0.G0(0) 
  not = \y0.2 + 2y0 
  not- = \y0.2 + 2y0 
  not# = \y0.2 + y0 
  or = \y0y1.y1 + 2y0 
  or- = \y0y1.y1 + 2y0 
  or# = \y0y1.y0 + y1 
  ~c0 = 0 
  ~c1 = 0 
  ~c10 = 0 
  ~c11 = 0 
  ~c2 = 0 
  ~c3 = 0 
  ~c4 = 0 
  ~c5 = 0 
  ~c6 = 0 
  ~c7 = 0 
  ~c8 = 0 
  ~c9 = 0 

Using this interpretation, the requirements translate to:

  [[or#(_x0, forall(/\x._x1(x)))]] = x0 + F1(0) >= x0 + F1(0) = [[or#(_x0, _x1(~c0))]] 
  [[or#(_x0, forall(/\x._x1(x)))]] = x0 + F1(0) >= F1(0) = [[_x1(~c1)]] 
  [[or#(forall(/\x._x0(x)), _x1)]] = x1 + F0(0) >= x1 + F0(0) = [[or#(_x0(~c2), _x1)]] 
  [[or#(forall(/\x._x0(x)), _x1)]] = x1 + F0(0) >= F0(0) = [[_x0(~c3)]] 
  [[not#(forall(/\x._x0(x)))]] = 2 + F0(0) >= 2 + F0(0) = [[not#(_x0(~c4))]] 
  [[not#(forall(/\x._x0(x)))]] = 2 + F0(0) > F0(0) = [[_x0(~c5)]] 
  [[or#(_x0, exists(/\x._x1(x)))]] = x0 + F1(0) >= x0 + F1(0) = [[or#(_x0, _x1(~c6))]] 
  [[or#(_x0, exists(/\x._x1(x)))]] = x0 + F1(0) >= F1(0) = [[_x1(~c7)]] 
  [[or#(exists(/\x._x0(x)), _x1)]] = x1 + F0(0) >= x1 + F0(0) = [[or#(_x0(~c8), _x1)]] 
  [[or#(exists(/\x._x0(x)), _x1)]] = x1 + F0(0) >= F0(0) = [[_x0(~c9)]] 
  [[not#(exists(/\x._x0(x)))]] = 2 + F0(0) >= 2 + F0(0) = [[not#(_x0(~c10))]] 
  [[not#(exists(/\x._x0(x)))]] = 2 + F0(0) > F0(0) = [[_x0(~c11)]] 
  [[and(_x0, forall(/\x._x1(x)))]] = x0 + F1(0) >= x0 + F1(0) = [[forall(/\x.and-(_x0, _x1(x)))]] 
  [[or(_x0, forall(/\x._x1(x)))]] = 2x0 + F1(0) >= 2x0 + F1(0) = [[forall(/\x.or-(_x0, _x1(x)))]] 
  [[and(forall(/\x._x0(x)), _x1)]] = x1 + F0(0) >= x1 + F0(0) = [[forall(/\x.and-(_x0(x), _x1))]] 
  [[or(forall(/\x._x0(x)), _x1)]] = x1 + 2F0(0) >= x1 + 2F0(0) = [[forall(/\x.or-(_x0(x), _x1))]] 
  [[not(forall(/\x._x0(x)))]] = 2 + 2F0(0) >= 2 + 2F0(0) = [[exists(/\x.not-(_x0(x)))]] 
  [[and(_x0, exists(/\x._x1(x)))]] = x0 + F1(0) >= x0 + F1(0) = [[exists(/\x.and-(_x0, _x1(x)))]] 
  [[or(_x0, exists(/\x._x1(x)))]] = 2x0 + F1(0) >= 2x0 + F1(0) = [[exists(/\x.or-(_x0, _x1(x)))]] 
  [[and(exists(/\x._x0(x)), _x1)]] = x1 + F0(0) >= x1 + F0(0) = [[exists(/\x.and-(_x0(x), _x1))]] 
  [[or(exists(/\x._x0(x)), _x1)]] = x1 + 2F0(0) >= x1 + 2F0(0) = [[exists(/\x.or-(_x0(x), _x1))]] 
  [[not(exists(/\x._x0(x)))]] = 2 + 2F0(0) >= 2 + 2F0(0) = [[forall(/\x.not-(_x0(x)))]] 
  [[and-(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] 
  [[not-(_x0)]] = 2 + 2x0 >= 2 + 2x0 = [[not(_x0)]] 
  [[not-(_x0)]] = 2 + 2x0 >= 2 + x0 = [[not#(_x0)]] 
  [[or-(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[or(_x0, _x1)]] 
  [[or-(_x0, _x1)]] = x1 + 2x0 >= x0 + x1 = [[or#(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_0, minimal, formative) by (P_4, R_0, minimal, formative), where P_4 consists of:

  or#(X, forall(/\x.Y(x))) =#> or#(X, Y(y))   
  or#(X, forall(/\x.Y(x))) =#> Y(y)   
  or#(forall(/\x.X(x)), Y) =#> or#(X(y), Y)   
  or#(forall(/\x.X(x)), Y) =#> X(y)   
  not#(forall(/\x.X(x))) =#> not#(X(y))   
  or#(X, exists(/\x.Y(x))) =#> or#(X, Y(y))   
  or#(X, exists(/\x.Y(x))) =#> Y(y)   
  or#(exists(/\x.X(x)), Y) =#> or#(X(y), Y)   
  or#(exists(/\x.X(x)), Y) =#> X(y)   
  not#(exists(/\x.X(x))) =#> not#(X(y))   

Thus, the original system is terminating if each of (P_2, R_0, minimal, formative) and (P_4, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_4, 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 : 0, 1, 2, 3, 5, 6, 7, 8 
    * 1 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 
    * 2 : 0, 1, 2, 3, 5, 6, 7, 8 
    * 3 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 
    * 4 : 4, 9 
    * 5 : 0, 1, 2, 3, 5, 6, 7, 8 
    * 6 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 
    * 7 : 0, 1, 2, 3, 5, 6, 7, 8 
    * 8 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 
    * 9 : 4, 9 

This graph has the following strongly connected components:

  P_5:

    or#(X, forall(/\x.Y(x))) =#> or#(X, Y(y))   
    or#(X, forall(/\x.Y(x))) =#> Y(y)   
    or#(forall(/\x.X(x)), Y) =#> or#(X(y), Y)   
    or#(forall(/\x.X(x)), Y) =#> X(y)   
    or#(X, exists(/\x.Y(x))) =#> or#(X, Y(y))   
    or#(X, exists(/\x.Y(x))) =#> Y(y)   
    or#(exists(/\x.X(x)), Y) =#> or#(X(y), Y)   
    or#(exists(/\x.X(x)), Y) =#> X(y)   

  P_6:

    not#(forall(/\x.X(x))) =#> not#(X(y))   
    not#(exists(/\x.X(x))) =#> not#(X(y))   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_4, R_0, m, f) by (P_5, R_0, m, f) and (P_6, R_0, m, f).

Thus, the original system is terminating if each of (P_2, R_0, minimal, formative), (P_5, R_0, minimal, formative) and (P_6, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_6, R_0, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_6, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  not#(forall(/\x.X(x))) >? not#(X(~c0)) 
  not#(exists(/\x.X(x))) >? not#(X(~c1)) 

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

The following interpretation satisfies the requirements:

  exists = \G0.3 + G0(0) 
  forall = \G0.3 + G0(0) 
  not# = \y0.y0 
  ~c0 = 0 
  ~c1 = 0 

Using this interpretation, the requirements translate to:

  [[not#(forall(/\x._x0(x)))]] = 3 + F0(0) > F0(0) = [[not#(_x0(~c0))]] 
  [[not#(exists(/\x._x0(x)))]] = 3 + F0(0) > F0(0) = [[not#(_x0(~c1))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_6, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_2, R_0, minimal, formative) and (P_5, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:

  or#(X, forall(/\x.Y(x))) >? or#(X, Y(~c0)) 
  or#(X, forall(/\x.Y(x))) >? Y(~c1) 
  or#(forall(/\x.X(x)), Y) >? or#(X(~c2), Y) 
  or#(forall(/\x.X(x)), Y) >? X(~c3) 
  or#(X, exists(/\x.Y(x))) >? or#(X, Y(~c4)) 
  or#(X, exists(/\x.Y(x))) >? Y(~c5) 
  or#(exists(/\x.X(x)), Y) >? or#(X(~c6), Y) 
  or#(exists(/\x.X(x)), Y) >? X(~c7) 
  and(X, forall(/\x.Y(x))) >= forall(/\y.and-(X, Y(y))) 
  or(X, forall(/\x.Y(x))) >= forall(/\y.or-(X, Y(y))) 
  and(forall(/\x.X(x)), Y) >= forall(/\y.and-(X(y), Y)) 
  or(forall(/\x.X(x)), Y) >= forall(/\y.or-(X(y), Y)) 
  not(forall(/\x.X(x))) >= exists(/\y.not-(X(y))) 
  and(X, exists(/\x.Y(x))) >= exists(/\y.and-(X, Y(y))) 
  or(X, exists(/\x.Y(x))) >= exists(/\y.or-(X, Y(y))) 
  and(exists(/\x.X(x)), Y) >= exists(/\y.and-(X(y), Y)) 
  or(exists(/\x.X(x)), Y) >= exists(/\y.or-(X(y), Y)) 
  not(exists(/\x.X(x))) >= forall(/\y.not-(X(y))) 
  and-(X, Y) >= and(X, Y) 
  not-(X) >= not(X) 
  or-(X, Y) >= or(X, Y) 
  or-(X, Y) >= or#(X, Y) 

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

The following interpretation satisfies the requirements:

  and = \y0y1.y0 + y1 
  and- = \y0y1.y0 + y1 
  exists = \G0.G0(0) 
  forall = \G0.G0(0) 
  not = \y0.y0 
  not- = \y0.y0 
  or = \y0y1.1 + y0 + 2y1 
  or- = \y0y1.1 + y0 + 2y1 
  or# = \y0y1.1 + y0 + y1 
  ~c0 = 0 
  ~c1 = 0 
  ~c2 = 0 
  ~c3 = 0 
  ~c4 = 0 
  ~c5 = 0 
  ~c6 = 0 
  ~c7 = 0 

Using this interpretation, the requirements translate to:

  [[or#(_x0, forall(/\x._x1(x)))]] = 1 + x0 + F1(0) >= 1 + x0 + F1(0) = [[or#(_x0, _x1(~c0))]] 
  [[or#(_x0, forall(/\x._x1(x)))]] = 1 + x0 + F1(0) > F1(0) = [[_x1(~c1)]] 
  [[or#(forall(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) >= 1 + x1 + F0(0) = [[or#(_x0(~c2), _x1)]] 
  [[or#(forall(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) > F0(0) = [[_x0(~c3)]] 
  [[or#(_x0, exists(/\x._x1(x)))]] = 1 + x0 + F1(0) >= 1 + x0 + F1(0) = [[or#(_x0, _x1(~c4))]] 
  [[or#(_x0, exists(/\x._x1(x)))]] = 1 + x0 + F1(0) > F1(0) = [[_x1(~c5)]] 
  [[or#(exists(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) >= 1 + x1 + F0(0) = [[or#(_x0(~c6), _x1)]] 
  [[or#(exists(/\x._x0(x)), _x1)]] = 1 + x1 + F0(0) > F0(0) = [[_x0(~c7)]] 
  [[and(_x0, forall(/\x._x1(x)))]] = x0 + F1(0) >= x0 + F1(0) = [[forall(/\x.and-(_x0, _x1(x)))]] 
  [[or(_x0, forall(/\x._x1(x)))]] = 1 + x0 + 2F1(0) >= 1 + x0 + 2F1(0) = [[forall(/\x.or-(_x0, _x1(x)))]] 
  [[and(forall(/\x._x0(x)), _x1)]] = x1 + F0(0) >= x1 + F0(0) = [[forall(/\x.and-(_x0(x), _x1))]] 
  [[or(forall(/\x._x0(x)), _x1)]] = 1 + 2x1 + F0(0) >= 1 + 2x1 + F0(0) = [[forall(/\x.or-(_x0(x), _x1))]] 
  [[not(forall(/\x._x0(x)))]] = F0(0) >= F0(0) = [[exists(/\x.not-(_x0(x)))]] 
  [[and(_x0, exists(/\x._x1(x)))]] = x0 + F1(0) >= x0 + F1(0) = [[exists(/\x.and-(_x0, _x1(x)))]] 
  [[or(_x0, exists(/\x._x1(x)))]] = 1 + x0 + 2F1(0) >= 1 + x0 + 2F1(0) = [[exists(/\x.or-(_x0, _x1(x)))]] 
  [[and(exists(/\x._x0(x)), _x1)]] = x1 + F0(0) >= x1 + F0(0) = [[exists(/\x.and-(_x0(x), _x1))]] 
  [[or(exists(/\x._x0(x)), _x1)]] = 1 + 2x1 + F0(0) >= 1 + 2x1 + F0(0) = [[exists(/\x.or-(_x0(x), _x1))]] 
  [[not(exists(/\x._x0(x)))]] = F0(0) >= F0(0) = [[forall(/\x.not-(_x0(x)))]] 
  [[and-(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[and(_x0, _x1)]] 
  [[not-(_x0)]] = x0 >= x0 = [[not(_x0)]] 
  [[or-(_x0, _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[or(_x0, _x1)]] 
  [[or-(_x0, _x1)]] = 1 + x0 + 2x1 >= 1 + x0 + x1 = [[or#(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_0, minimal, formative) by (P_7, R_0, minimal, formative), where P_7 consists of:

  or#(X, forall(/\x.Y(x))) =#> or#(X, Y(y))   
  or#(forall(/\x.X(x)), Y) =#> or#(X(y), Y)   
  or#(X, exists(/\x.Y(x))) =#> or#(X, Y(y))   
  or#(exists(/\x.X(x)), Y) =#> or#(X(y), Y)   

Thus, the original system is terminating if each of (P_2, R_0, minimal, formative) and (P_7, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_7, R_0, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_7, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  or#(X, forall(/\x.Y(x))) >? or#(X, Y(~c0)) 
  or#(forall(/\x.X(x)), Y) >? or#(X(~c1), Y) 
  or#(X, exists(/\x.Y(x))) >? or#(X, Y(~c2)) 
  or#(exists(/\x.X(x)), Y) >? or#(X(~c3), Y) 

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

The following interpretation satisfies the requirements:

  exists = \G0.3 + G0(0) 
  forall = \G0.3 + 2G0(0) 
  or# = \y0y1.y1 
  ~c0 = 0 
  ~c1 = 0 
  ~c2 = 0 
  ~c3 = 0 

Using this interpretation, the requirements translate to:

  [[or#(_x0, forall(/\x._x1(x)))]] = 3 + 2F1(0) > F1(0) = [[or#(_x0, _x1(~c0))]] 
  [[or#(forall(/\x._x0(x)), _x1)]] = x1 >= x1 = [[or#(_x0(~c1), _x1)]] 
  [[or#(_x0, exists(/\x._x1(x)))]] = 3 + F1(0) > F1(0) = [[or#(_x0, _x1(~c2))]] 
  [[or#(exists(/\x._x0(x)), _x1)]] = x1 >= x1 = [[or#(_x0(~c3), _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_7, R_0, minimal, formative) by (P_8, R_0, minimal, formative), where P_8 consists of:

  or#(forall(/\x.X(x)), Y) =#> or#(X(y), Y)   
  or#(exists(/\x.X(x)), Y) =#> or#(X(y), Y)   

Thus, the original system is terminating if each of (P_2, R_0, minimal, formative) and (P_8, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_8, R_0, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_8, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  or#(forall(/\x.X(x)), Y) >? or#(X(~c0), Y) 
  or#(exists(/\x.X(x)), Y) >? or#(X(~c1), Y) 

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

The following interpretation satisfies the requirements:

  exists = \G0.3 + G0(0) 
  forall = \G0.3 + G0(0) 
  or# = \y0y1.y0 
  ~c0 = 0 
  ~c1 = 0 

Using this interpretation, the requirements translate to:

  [[or#(forall(/\x._x0(x)), _x1)]] = 3 + F0(0) > F0(0) = [[or#(_x0(~c0), _x1)]] 
  [[or#(exists(/\x._x0(x)), _x1)]] = 3 + F0(0) > F0(0) = [[or#(_x0(~c1), _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_8, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_2, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  and#(X, forall(/\x.Y(x))) >? and#(X, Y(~c0)) 
  and#(forall(/\x.X(x)), Y) >? and#(X(~c1), Y) 
  and#(X, exists(/\x.Y(x))) >? and#(X, Y(~c2)) 
  and#(exists(/\x.X(x)), Y) >? and#(X(~c3), Y) 

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

The following interpretation satisfies the requirements:

  and# = \y0y1.y1 
  exists = \G0.3 + G0(0) 
  forall = \G0.3 + 2G0(0) 
  ~c0 = 0 
  ~c1 = 0 
  ~c2 = 0 
  ~c3 = 0 

Using this interpretation, the requirements translate to:

  [[and#(_x0, forall(/\x._x1(x)))]] = 3 + 2F1(0) > F1(0) = [[and#(_x0, _x1(~c0))]] 
  [[and#(forall(/\x._x0(x)), _x1)]] = x1 >= x1 = [[and#(_x0(~c1), _x1)]] 
  [[and#(_x0, exists(/\x._x1(x)))]] = 3 + F1(0) > F1(0) = [[and#(_x0, _x1(~c2))]] 
  [[and#(exists(/\x._x0(x)), _x1)]] = x1 >= x1 = [[and#(_x0(~c3), _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_0, minimal, formative) by (P_9, R_0, minimal, formative), where P_9 consists of:

  and#(forall(/\x.X(x)), Y) =#> and#(X(y), Y)   
  and#(exists(/\x.X(x)), Y) =#> and#(X(y), Y)   

Thus, the original system is terminating if (P_9, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_9, R_0, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  (P_9, R_0) has no usable rules.

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  and#(forall(/\x.X(x)), Y) >? and#(X(~c0), Y) 
  and#(exists(/\x.X(x)), Y) >? and#(X(~c1), Y) 

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

The following interpretation satisfies the requirements:

  and# = \y0y1.y0 
  exists = \G0.3 + G0(0) 
  forall = \G0.3 + G0(0) 
  ~c0 = 0 
  ~c1 = 0 

Using this interpretation, the requirements translate to:

  [[and#(forall(/\x._x0(x)), _x1)]] = 3 + F0(0) > F0(0) = [[and#(_x0(~c0), _x1)]] 
  [[and#(exists(/\x._x0(x)), _x1)]] = 3 + F0(0) > F0(0) = [[and#(_x0(~c1), _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_9, R_0) by ({}, R_0).  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 +++

[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.