We consider the system from.

  Alphabet:

    chain : [N -> N * list] --> list 
    cons : [N * list] --> list 
    false : [] --> B 
    from : [N * list] --> list 
    if : [B * list * list] --> list 
    incch : [list] --> list 
    lteq : [N * N] --> B 
    nil : [] --> list 
    o : [] --> N 
    s : [N] --> N 
    true : [] --> B 

  Rules:

    if(true, x, y) => x 
    if(false, x, y) => y 
    lteq(s(x), o) => false 
    lteq(o, x) => true 
    lteq(s(x), s(y)) => lteq(x, y) 
    from(x, nil) => nil 
    from(x, cons(y, z)) => if(lteq(x, y), cons(y, z), from(x, z)) 
    chain(f, nil) => nil 
    chain(f, cons(x, y)) => cons(f x, chain(f, from(f x, y))) 
    incch(x) => chain(/\y.s(y), x) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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

After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] lteq#(s(X), s(Y)) =#> lteq#(X, Y)   
    1] from#(X, cons(Y, Z)) =#> if#(lteq(X, Y), cons(Y, Z), from(X, Z))   
    2] from#(X, cons(Y, Z)) =#> lteq#(X, Y)   
    3] from#(X, cons(Y, Z)) =#> from#(X, Z)   
    4] chain#(F, cons(X, Y)) =#> F(X)   
    5] chain#(F, cons(X, Y)) =#> chain#(F, from(F X, Y))   
    6] chain#(F, cons(X, Y)) =#> from#(F X, Y)   
    7] chain#(F, cons(X, Y)) =#> F(X)   
    8] incch#(X) =#> chain#(/\x.s(x), X)   

  Rules R_0:

    if(true, X, Y) => X 
    if(false, X, Y) => Y 
    lteq(s(X), o) => false 
    lteq(o, X) => true 
    lteq(s(X), s(Y)) => lteq(X, Y) 
    from(X, nil) => nil 
    from(X, cons(Y, Z)) => if(lteq(X, Y), cons(Y, Z), from(X, Z)) 
    chain(F, nil) => nil 
    chain(F, cons(X, Y)) => cons(F X, chain(F, from(F X, Y))) 
    incch(X) => chain(/\x.s(x), 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 : 0 
    * 1 :  
    * 2 : 0 
    * 3 : 1, 2, 3 
    * 4 : 0, 1, 2, 3, 4, 5, 6, 7, 8 
    * 5 : 4, 5, 6, 7 
    * 6 : 1, 2, 3 
    * 7 : 0, 1, 2, 3, 4, 5, 6, 7, 8 
    * 8 : 4, 5, 6, 7 

This graph has the following strongly connected components:

  P_1:

    lteq#(s(X), s(Y)) =#> lteq#(X, Y)   

  P_2:

    from#(X, cons(Y, Z)) =#> from#(X, Z)   

  P_3:

    chain#(F, cons(X, Y)) =#> F(X)   
    chain#(F, cons(X, Y)) =#> chain#(F, from(F X, Y))   
    chain#(F, cons(X, Y)) =#> F(X)   
    incch#(X) =#> chain#(/\x.s(x), 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), (P_2, R_0, m, f) and (P_3, R_0, m, f).

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

The formative rules of (P_3, R_0) are R_1 ::=

  if(true, X, Y) => X 
  if(false, X, Y) => Y 
  lteq(s(X), o) => false 
  lteq(o, X) => true 
  lteq(s(X), s(Y)) => lteq(X, Y) 
  from(X, cons(Y, Z)) => if(lteq(X, Y), cons(Y, Z), from(X, Z)) 
  chain(F, cons(X, Y)) => cons(F X, chain(F, from(F X, Y))) 
  incch(X) => chain(/\x.s(x), X) 

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_3, R_0, minimal, formative) by (P_3, R_1, minimal, formative).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (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].  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:

  chain#(F, cons(X, Y)) >? F(X) 
  chain#(F, cons(X, Y)) >? chain#(F, from(F X, Y)) 
  chain#(F, cons(X, Y)) >? F(X) 
  incch#(X) >? chain#(/\x.s-(x), X) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  lteq(s(X), o) >= false 
  lteq(o, X) >= true 
  lteq(s(X), s(Y)) >= lteq(X, Y) 
  from(X, cons(Y, Z)) >= if(lteq(X, Y), cons(Y, Z), from(X, Z)) 
  chain(F, cons(X, Y)) >= cons(F X, chain(F, from(F X, Y))) 
  incch(X) >= chain(/\x.s-(x), X) 
  s-(X) >= s(X) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( incch(X) ) = #argfun-incch#(chain(/\x.s-(x), X)) 
  pi( incch#(X) ) = #argfun-incch##(chain#(/\x.s-(x), X)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[false]] = _|_ 
  [[from(x_1, x_2)]] = from(x_2) 
  [[incch(x_1)]] = x_1 
  [[incch#(x_1)]] = x_1 
  [[lteq(x_1, x_2)]] = _|_ 
  [[true]] = _|_ 

We choose Lex = {} and Mul = {#argfun-incch#, #argfun-incch##, @_{o -> o}, chain, chain#, cons, from, if, o, s, s-}, and the following precedence: #argfun-incch## > #argfun-incch# > s = s- > o > chain# > chain > @_{o -> o} > cons > from > if

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  chain#(F, cons(X, Y)) >= @_{o -> o}(F, X) 
  chain#(F, cons(X, Y)) >= chain#(F, from(Y)) 
  chain#(F, cons(X, Y)) >= @_{o -> o}(F, X) 
  #argfun-incch##(chain#(/\x.s-(x), X)) > chain#(/\x.s-(x), X) 
  if(_|_, X, Y) >= X 
  if(_|_, X, Y) >= Y 
  _|_ >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 
  from(cons(X, Y)) >= if(_|_, cons(X, Y), from(Y)) 
  chain(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y))) 
  #argfun-incch#(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X) 
  s-(X) >= s(X) 

With these choices, we have:

  1] chain#(F, cons(X, Y)) >= @_{o -> o}(F, X)  because [2], by (Star) 
  2] chain#*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain# > @_{o -> o}, [3] and [5], by (Copy) 
  3] chain#*(F, cons(X, Y)) >= F  because [4], by (Select) 
  4] F >= F  by (Meta) 
  5] chain#*(F, cons(X, Y)) >= X  because [6], by (Select) 
  6] cons(X, Y) >= X  because [7], by (Star) 
  7] cons*(X, Y) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

  9] chain#(F, cons(X, Y)) >= chain#(F, from(Y))  because chain# in Mul, [10] and [11], by (Fun) 
  10] F >= F  by (Meta) 
  11] cons(X, Y) >= from(Y)  because [12], by (Star) 
  12] cons*(X, Y) >= from(Y)  because cons > from and [13], by (Copy) 
  13] cons*(X, Y) >= Y  because [14], by (Select) 
  14] Y >= Y  by (Meta) 

  15] chain#(F, cons(X, Y)) >= @_{o -> o}(F, X)  because [2], by (Star) 

  16] #argfun-incch##(chain#(/\x.s-(x), X)) > chain#(/\x.s-(x), X)  because [17], by definition 
  17] #argfun-incch##*(chain#(/\x.s-(x), X)) >= chain#(/\x.s-(x), X)  because #argfun-incch## > chain#, [18] and [22], by (Copy) 
  18] #argfun-incch##*(chain#(/\x.s-(x), X)) >= /\x.s-(x)  because [19], by (F-Abs) 
  19] #argfun-incch##*(chain#(/\x.s-(x), X), y) >= s-(y)  because #argfun-incch## > s- and [20], by (Copy) 
  20] #argfun-incch##*(chain#(/\x.s-(x), X), y) >= y  because [21], by (Select) 
  21] y >= y  by (Var) 
  22] #argfun-incch##*(chain#(/\x.s-(x), X)) >= X  because [23], by (Select) 
  23] chain#(/\x.s-(x), X) >= X  because [24], by (Star) 
  24] chain#*(/\x.s-(x), X) >= X  because [25], by (Select) 
  25] X >= X  by (Meta) 

  26] if(_|_, X, Y) >= X  because [27], by (Star) 
  27] if*(_|_, X, Y) >= X  because [28], by (Select) 
  28] X >= X  by (Meta) 

  29] if(_|_, X, Y) >= Y  because [30], by (Star) 
  30] if*(_|_, X, Y) >= Y  because [31], by (Select) 
  31] Y >= Y  by (Meta) 

  32] _|_ >= _|_  by (Bot) 

  33] _|_ >= _|_  by (Bot) 

  34] _|_ >= _|_  by (Bot) 

  35] from(cons(X, Y)) >= if(_|_, cons(X, Y), from(Y))  because [36], by (Star) 
  36] from*(cons(X, Y)) >= if(_|_, cons(X, Y), from(Y))  because from > if, [37], [38] and [42], by (Copy) 
  37] from*(cons(X, Y)) >= _|_  by (Bot) 
  38] from*(cons(X, Y)) >= cons(X, Y)  because [39], by (Select) 
  39] cons(X, Y) >= cons(X, Y)  because cons in Mul, [40] and [41], by (Fun) 
  40] X >= X  by (Meta) 
  41] Y >= Y  by (Meta) 
  42] from*(cons(X, Y)) >= from(Y)  because [43], by (Select) 
  43] cons(X, Y) >= from(Y)  because [44], by (Star) 
  44] cons*(X, Y) >= from(Y)  because cons > from and [45], by (Copy) 
  45] cons*(X, Y) >= Y  because [41], by (Select) 

  46] chain(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y)))  because [47], by (Star) 
  47] chain*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y)))  because chain > cons, [48] and [51], by (Copy) 
  48] chain*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain > @_{o -> o}, [49] and [50], by (Copy) 
  49] chain*(F, cons(X, Y)) >= F  because [10], by (Select) 
  50] chain*(F, cons(X, Y)) >= X  because [6], by (Select) 
  51] chain*(F, cons(X, Y)) >= chain(F, from(Y))  because chain in Mul, [10] and [52], by (Stat) 
  52] cons(X, Y) > from(Y)  because [53], by definition 
  53] cons*(X, Y) >= from(Y)  because cons > from and [13], by (Copy) 

  54] #argfun-incch#(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X)  because [55], by (Star) 
  55] #argfun-incch#*(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X)  because #argfun-incch# > chain, [56] and [60], by (Copy) 
  56] #argfun-incch#*(chain(/\x.s-(x), X)) >= /\x.s-(x)  because [57], by (F-Abs) 
  57] #argfun-incch#*(chain(/\x.s-(x), X), y) >= s-(y)  because #argfun-incch# > s- and [58], by (Copy) 
  58] #argfun-incch#*(chain(/\x.s-(x), X), y) >= y  because [59], by (Select) 
  59] y >= y  by (Var) 
  60] #argfun-incch#*(chain(/\x.s-(x), X)) >= X  because [61], by (Select) 
  61] chain(/\x.s-(x), X) >= X  because [62], by (Star) 
  62] chain*(/\x.s-(x), X) >= X  because [25], by (Select) 

  63] s-(X) >= s(X)  because s- = s, s- in Mul and [64], by (Fun) 
  64] X >= X  by (Meta) 

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:

  chain#(F, cons(X, Y)) =#> F(X)   
  chain#(F, cons(X, Y)) =#> chain#(F, from(F X, Y))   
  chain#(F, cons(X, Y)) =#> F(X)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_4, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_4, R_1, 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:

  chain#(F, cons(X, Y)) >? F(X) 
  chain#(F, cons(X, Y)) >? chain#(F, from(F X, Y)) 
  chain#(F, cons(X, Y)) >? F(X) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  lteq(s(X), o) >= false 
  lteq(o, X) >= true 
  lteq(s(X), s(Y)) >= lteq(X, Y) 
  from(X, cons(Y, Z)) >= if(lteq(X, Y), cons(Y, Z), from(X, Z)) 
  chain(F, cons(X, Y)) >= cons(F X, chain(F, from(F X, Y))) 
  incch(X) >= chain(/\x.s-(x), X) 
  s-(X) >= s(X) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( incch(X) ) = #argfun-incch#(chain(/\x.s-(x), X)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[false]] = _|_ 
  [[from(x_1, x_2)]] = from(x_2) 
  [[if(x_1, x_2, x_3)]] = if(x_2, x_3) 
  [[incch(x_1)]] = x_1 
  [[true]] = _|_ 

We choose Lex = {} and Mul = {#argfun-incch#, @_{o -> o}, chain, chain#, cons, from, if, lteq, o, s, s-}, and the following precedence: o > lteq > #argfun-incch# > s = s- > chain > @_{o -> o} = chain# > cons > from > if

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  chain#(F, cons(X, Y)) >= @_{o -> o}(F, X) 
  chain#(F, cons(X, Y)) > chain#(F, from(Y)) 
  chain#(F, cons(X, Y)) >= @_{o -> o}(F, X) 
  if(X, Y) >= X 
  if(X, Y) >= Y 
  lteq(s(X), o) >= _|_ 
  lteq(o, X) >= _|_ 
  lteq(s(X), s(Y)) >= lteq(X, Y) 
  from(cons(X, Y)) >= if(cons(X, Y), from(Y)) 
  chain(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y))) 
  #argfun-incch#(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X) 
  s-(X) >= s(X) 

With these choices, we have:

  1] chain#(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain# = @_{o -> o}, chain# in Mul, [2] and [3], by (Fun) 
  2] F >= F  by (Meta) 
  3] cons(X, Y) >= X  because [4], by (Star) 
  4] cons*(X, Y) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] chain#(F, cons(X, Y)) > chain#(F, from(Y))  because [7], by definition 
  7] chain#*(F, cons(X, Y)) >= chain#(F, from(Y))  because chain# in Mul, [2] and [8], by (Stat) 
  8] cons(X, Y) > from(Y)  because [9], by definition 
  9] cons*(X, Y) >= from(Y)  because cons > from and [10], by (Copy) 
  10] cons*(X, Y) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 

  12] chain#(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain# = @_{o -> o}, chain# in Mul, [2] and [3], by (Fun) 

  13] if(X, Y) >= X  because [14], by (Star) 
  14] if*(X, Y) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 

  16] if(X, Y) >= Y  because [17], by (Star) 
  17] if*(X, Y) >= Y  because [18], by (Select) 
  18] Y >= Y  by (Meta) 

  19] lteq(s(X), o) >= _|_  by (Bot) 

  20] lteq(o, X) >= _|_  by (Bot) 

  21] lteq(s(X), s(Y)) >= lteq(X, Y)  because [22], by (Star) 
  22] lteq*(s(X), s(Y)) >= lteq(X, Y)  because lteq in Mul, [23] and [26], by (Stat) 
  23] s(X) > X  because [24], by definition 
  24] s*(X) >= X  because [25], by (Select) 
  25] X >= X  by (Meta) 
  26] s(Y) >= Y  because [27], by (Star) 
  27] s*(Y) >= Y  because [28], by (Select) 
  28] Y >= Y  by (Meta) 

  29] from(cons(X, Y)) >= if(cons(X, Y), from(Y))  because [30], by (Star) 
  30] from*(cons(X, Y)) >= if(cons(X, Y), from(Y))  because from > if, [31] and [35], by (Copy) 
  31] from*(cons(X, Y)) >= cons(X, Y)  because [32], by (Select) 
  32] cons(X, Y) >= cons(X, Y)  because cons in Mul, [33] and [34], by (Fun) 
  33] X >= X  by (Meta) 
  34] Y >= Y  by (Meta) 
  35] from*(cons(X, Y)) >= from(Y)  because from in Mul and [36], by (Stat) 
  36] cons(X, Y) > Y  because [37], by definition 
  37] cons*(X, Y) >= Y  because [34], by (Select) 

  38] chain(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y)))  because [39], by (Star) 
  39] chain*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y)))  because chain > cons, [40] and [43], by (Copy) 
  40] chain*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain > @_{o -> o}, [41] and [42], by (Copy) 
  41] chain*(F, cons(X, Y)) >= F  because [2], by (Select) 
  42] chain*(F, cons(X, Y)) >= X  because [3], by (Select) 
  43] chain*(F, cons(X, Y)) >= chain(F, from(Y))  because chain in Mul, [2] and [8], by (Stat) 

  44] #argfun-incch#(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X)  because [45], by (Star) 
  45] #argfun-incch#*(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X)  because #argfun-incch# > chain, [46] and [50], by (Copy) 
  46] #argfun-incch#*(chain(/\x.s-(x), X)) >= /\x.s-(x)  because [47], by (F-Abs) 
  47] #argfun-incch#*(chain(/\x.s-(x), X), y) >= s-(y)  because #argfun-incch# > s- and [48], by (Copy) 
  48] #argfun-incch#*(chain(/\x.s-(x), X), y) >= y  because [49], by (Select) 
  49] y >= y  by (Var) 
  50] #argfun-incch#*(chain(/\x.s-(x), X)) >= X  because [51], by (Select) 
  51] chain(/\x.s-(x), X) >= X  because [52], by (Star) 
  52] chain*(/\x.s-(x), X) >= X  because [53], by (Select) 
  53] X >= X  by (Meta) 

  54] s-(X) >= s(X)  because s- = s, s- in Mul and [55], by (Fun) 
  55] X >= X  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_4, R_1, minimal, formative) by (P_5, R_1, minimal, formative), where P_5 consists of:

  chain#(F, cons(X, Y)) =#> F(X)   
  chain#(F, cons(X, Y)) =#> F(X)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_5, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_1, 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:

  chain#(F, cons(X, Y)) >? F(X) 
  chain#(F, cons(X, Y)) >? F(X) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  lteq(s(X), o) >= false 
  lteq(o, X) >= true 
  lteq(s(X), s(Y)) >= lteq(X, Y) 
  from(X, cons(Y, Z)) >= if(lteq(X, Y), cons(Y, Z), from(X, Z)) 
  chain(F, cons(X, Y)) >= cons(F X, chain(F, from(F X, Y))) 
  incch(X) >= chain(/\x.s-(x), X) 
  s-(X) >= s(X) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( incch(X) ) = #argfun-incch#(chain(/\x.s-(x), X)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[false]] = _|_ 
  [[from(x_1, x_2)]] = from(x_2) 
  [[if(x_1, x_2, x_3)]] = if(x_2, x_3) 
  [[incch(x_1)]] = x_1 
  [[lteq(x_1, x_2)]] = lteq(x_1) 
  [[true]] = _|_ 

We choose Lex = {} and Mul = {#argfun-incch#, @_{o -> o}, chain, chain#, cons, from, if, lteq, o, s, s-}, and the following precedence: #argfun-incch# > chain# > o > s = s- > lteq > chain > @_{o -> o} > cons = from > if

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  chain#(F, cons(X, Y)) >= @_{o -> o}(F, X) 
  chain#(F, cons(X, Y)) > @_{o -> o}(F, X) 
  if(X, Y) >= X 
  if(X, Y) >= Y 
  lteq(s(X)) >= _|_ 
  lteq(o) >= _|_ 
  lteq(s(X)) >= lteq(X) 
  from(cons(X, Y)) >= if(cons(X, Y), from(Y)) 
  chain(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y))) 
  #argfun-incch#(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X) 
  s-(X) >= s(X) 

With these choices, we have:

  1] chain#(F, cons(X, Y)) >= @_{o -> o}(F, X)  because [2], by (Star) 
  2] chain#*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain# > @_{o -> o}, [3] and [5], by (Copy) 
  3] chain#*(F, cons(X, Y)) >= F  because [4], by (Select) 
  4] F >= F  by (Meta) 
  5] chain#*(F, cons(X, Y)) >= X  because [6], by (Select) 
  6] cons(X, Y) >= X  because [7], by (Star) 
  7] cons*(X, Y) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

  9] chain#(F, cons(X, Y)) > @_{o -> o}(F, X)  because [10], by definition 
  10] chain#*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain# > @_{o -> o}, [3] and [5], by (Copy) 

  11] if(X, Y) >= X  because [12], by (Star) 
  12] if*(X, Y) >= X  because [13], by (Select) 
  13] X >= X  by (Meta) 

  14] if(X, Y) >= Y  because [15], by (Star) 
  15] if*(X, Y) >= Y  because [16], by (Select) 
  16] Y >= Y  by (Meta) 

  17] lteq(s(X)) >= _|_  by (Bot) 

  18] lteq(o) >= _|_  by (Bot) 

  19] lteq(s(X)) >= lteq(X)  because [20], by (Star) 
  20] lteq*(s(X)) >= lteq(X)  because [21], by (Select) 
  21] s(X) >= lteq(X)  because [22], by (Star) 
  22] s*(X) >= lteq(X)  because s > lteq and [23], by (Copy) 
  23] s*(X) >= X  because [24], by (Select) 
  24] X >= X  by (Meta) 

  25] from(cons(X, Y)) >= if(cons(X, Y), from(Y))  because [26], by (Star) 
  26] from*(cons(X, Y)) >= if(cons(X, Y), from(Y))  because from > if, [27] and [31], by (Copy) 
  27] from*(cons(X, Y)) >= cons(X, Y)  because [28], by (Select) 
  28] cons(X, Y) >= cons(X, Y)  because cons in Mul, [29] and [30], by (Fun) 
  29] X >= X  by (Meta) 
  30] Y >= Y  by (Meta) 
  31] from*(cons(X, Y)) >= from(Y)  because [32], by (Select) 
  32] cons(X, Y) >= from(Y)  because [33], by (Star) 
  33] cons*(X, Y) >= from(Y)  because cons = from, cons in Mul and [30], by (Stat) 

  34] chain(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y)))  because [35], by (Star) 
  35] chain*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y)))  because chain > cons, [36] and [39], by (Copy) 
  36] chain*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain > @_{o -> o}, [37] and [38], by (Copy) 
  37] chain*(F, cons(X, Y)) >= F  because [4], by (Select) 
  38] chain*(F, cons(X, Y)) >= X  because [6], by (Select) 
  39] chain*(F, cons(X, Y)) >= chain(F, from(Y))  because chain in Mul, [40] and [41], by (Stat) 
  40] F >= F  by (Meta) 
  41] cons(X, Y) > from(Y)  because [42], by definition 
  42] cons*(X, Y) >= from(Y)  because cons = from, cons in Mul and [43], by (Stat) 
  43] Y >= Y  by (Meta) 

  44] #argfun-incch#(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X)  because [45], by (Star) 
  45] #argfun-incch#*(chain(/\x.s-(x), X)) >= chain(/\x.s-(x), X)  because #argfun-incch# > chain, [46] and [50], by (Copy) 
  46] #argfun-incch#*(chain(/\x.s-(x), X)) >= /\x.s-(x)  because [47], by (F-Abs) 
  47] #argfun-incch#*(chain(/\x.s-(x), X), y) >= s-(y)  because #argfun-incch# > s- and [48], by (Copy) 
  48] #argfun-incch#*(chain(/\x.s-(x), X), y) >= y  because [49], by (Select) 
  49] y >= y  by (Var) 
  50] #argfun-incch#*(chain(/\x.s-(x), X)) >= X  because [51], by (Select) 
  51] chain(/\x.s-(x), X) >= X  because [52], by (Star) 
  52] chain*(/\x.s-(x), X) >= X  because [53], by (Select) 
  53] X >= X  by (Meta) 

  54] s-(X) >= s(X)  because s- = s, s- in Mul and [55], by (Fun) 
  55] X >= X  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_1, minimal, formative) by (P_6, R_1, minimal, formative), where P_6 consists of:

  chain#(F, cons(X, Y)) =#> F(X)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_6, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_6, R_1, 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:

  chain#(F, cons(X, Y)) >? F(X) 
  if(true, X, Y) >= X 
  if(false, X, Y) >= Y 
  lteq(s(X), o) >= false 
  lteq(o, X) >= true 
  lteq(s(X), s(Y)) >= lteq(X, Y) 
  from(X, cons(Y, Z)) >= if(lteq(X, Y), cons(Y, Z), from(X, Z)) 
  chain(F, cons(X, Y)) >= cons(F X, chain(F, from(F X, Y))) 
  incch(X) >= chain(/\x.s-(x), X) 
  s-(X) >= s(X) 

We apply [Kop12, Thm. 6.75] and use the following argument functions:

  pi( incch(X) ) = #argfun-incch#(chain(/\x.s-(x), X)) 

Since this representation is not advantageous for the higher-order recursive path ordering, we present the strict requirements in their unextended form, which is not problematic since for any F, s and substituion gamma: (F s)gamma beta-reduces to F(s)gamma.)

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[#argfun-incch#(x_1)]] = x_1 
  [[false]] = _|_ 
  [[from(x_1, x_2)]] = from(x_2) 
  [[if(x_1, x_2, x_3)]] = if(x_2, x_3) 
  [[incch(x_1)]] = x_1 
  [[true]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, chain, chain#, cons, from, if, lteq, o, s, s-}, and the following precedence: chain > chain# > @_{o -> o} > cons > from > if > lteq > o > s = s-

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  chain#(F, cons(X, Y)) > @_{o -> o}(F, X) 
  if(X, Y) >= X 
  if(X, Y) >= Y 
  lteq(s(X), o) >= _|_ 
  lteq(o, X) >= _|_ 
  lteq(s(X), s(Y)) >= lteq(X, Y) 
  from(cons(X, Y)) >= if(cons(X, Y), from(Y)) 
  chain(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y))) 
  chain(/\x.s-(x), X) >= chain(/\x.s-(x), X) 
  s-(X) >= s(X) 

With these choices, we have:

  1] chain#(F, cons(X, Y)) > @_{o -> o}(F, X)  because [2], by definition 
  2] chain#*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain# > @_{o -> o}, [3] and [5], by (Copy) 
  3] chain#*(F, cons(X, Y)) >= F  because [4], by (Select) 
  4] F >= F  by (Meta) 
  5] chain#*(F, cons(X, Y)) >= X  because [6], by (Select) 
  6] cons(X, Y) >= X  because [7], by (Star) 
  7] cons*(X, Y) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

  9] if(X, Y) >= X  because [10], by (Star) 
  10] if*(X, Y) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 

  12] if(X, Y) >= Y  because [13], by (Star) 
  13] if*(X, Y) >= Y  because [14], by (Select) 
  14] Y >= Y  by (Meta) 

  15] lteq(s(X), o) >= _|_  by (Bot) 

  16] lteq(o, X) >= _|_  by (Bot) 

  17] lteq(s(X), s(Y)) >= lteq(X, Y)  because [18], by (Star) 
  18] lteq*(s(X), s(Y)) >= lteq(X, Y)  because lteq in Mul, [19] and [22], by (Stat) 
  19] s(X) >= X  because [20], by (Star) 
  20] s*(X) >= X  because [21], by (Select) 
  21] X >= X  by (Meta) 
  22] s(Y) > Y  because [23], by definition 
  23] s*(Y) >= Y  because [24], by (Select) 
  24] Y >= Y  by (Meta) 

  25] from(cons(X, Y)) >= if(cons(X, Y), from(Y))  because [26], by (Star) 
  26] from*(cons(X, Y)) >= if(cons(X, Y), from(Y))  because from > if, [27] and [31], by (Copy) 
  27] from*(cons(X, Y)) >= cons(X, Y)  because [28], by (Select) 
  28] cons(X, Y) >= cons(X, Y)  because cons in Mul, [29] and [30], by (Fun) 
  29] X >= X  by (Meta) 
  30] Y >= Y  by (Meta) 
  31] from*(cons(X, Y)) >= from(Y)  because from in Mul and [32], by (Stat) 
  32] cons(X, Y) > Y  because [33], by definition 
  33] cons*(X, Y) >= Y  because [30], by (Select) 

  34] chain(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y)))  because [35], by (Star) 
  35] chain*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), chain(F, from(Y)))  because chain > cons, [36] and [39], by (Copy) 
  36] chain*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because chain > @_{o -> o}, [37] and [38], by (Copy) 
  37] chain*(F, cons(X, Y)) >= F  because [4], by (Select) 
  38] chain*(F, cons(X, Y)) >= X  because [6], by (Select) 
  39] chain*(F, cons(X, Y)) >= chain(F, from(Y))  because chain in Mul, [40] and [41], by (Stat) 
  40] F >= F  by (Meta) 
  41] cons(X, Y) > from(Y)  because [42], by definition 
  42] cons*(X, Y) >= from(Y)  because cons > from and [43], by (Copy) 
  43] cons*(X, Y) >= Y  because [44], by (Select) 
  44] Y >= Y  by (Meta) 

  45] chain(/\x.s-(x), X) >= chain(/\x.s-(x), X)  because chain in Mul, [46] and [49], by (Fun) 
  46] /\y.s-(y) >= /\y.s-(y)  because [47], by (Abs) 
  47] s-(x) >= s-(x)  because s- in Mul and [48], by (Fun) 
  48] x >= x  by (Var) 
  49] X >= X  by (Meta) 

  50] s-(X) >= s(X)  because s- = s, s- in Mul and [51], by (Fun) 
  51] X >= X  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_6, R_1) by ({}, R_1).  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_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite.

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

We apply the subterm criterion with the following projection function:

  nu(from#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(from#(X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(from#(X, Z)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

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 apply the subterm criterion with the following projection function:

  nu(lteq#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(lteq#(s(X), s(Y))) = s(X) |> X = nu(lteq#(X, Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by ({}, R_0, 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.