We consider the system h62.

  Alphabet:

    app : [list * list] --> list 
    cons : [nat * list] --> list 
    foldl : [list -> nat -> list * list * list] --> list 
    iconsc : [] --> list -> nat -> list 
    nil : [] --> list 
    reverse : [list] --> list 
    reverse1 : [list] --> list 
    xap : [list -> nat -> list * list] --> nat -> list 
    yap : [nat -> list * nat] --> list 

  Rules:

    app(nil, x) => x 
    app(cons(x, y), z) => cons(x, app(y, z)) 
    foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z 
    foldl(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => foldl(/\w./\x'.yap(xap(f, w), x'), yap(xap(f, z), u), v) 
    iconsc x y => cons(y, x) 
    reverse(x) => foldl(/\y./\z.yap(xap(iconsc, y), z), nil, x) 
    reverse1(x) => foldl(/\y./\z.app(cons(z, nil), y), nil, x) 
    xap(f, x) => f x 
    yap(f, x) => f x 

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

Symbol xap 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:

    app : [list * list] --> list 
    cons : [nat * list] --> list 
    foldl : [list -> nat -> list * list * list] --> list 
    iconsc : [list] --> nat -> list 
    nil : [] --> list 
    reverse : [list] --> list 
    reverse1 : [list] --> list 
    yap : [nat -> list * nat] --> list 

  Rules:

    app(nil, X) => X 
    app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
    iconsc(X) Y => cons(Y, X) 
    reverse(X) => foldl(/\x./\y.yap(iconsc(x), y), nil, X) 
    reverse1(X) => foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
    yap(F, X) => F X 

We observe that the rules contain a first-order subset:

  app(nil, X) => X 
  app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
  iconsc(X) Y => cons(Y, X) 

Moreover, the system is finitely branching.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS.

According to the external first-order termination prover, this system is indeed Ce-terminating:

 || proof of resources/system.trs
 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
 || 
 || 
 || Termination w.r.t. Q of the given QTRS could be proven:
 || 
 || (0) QTRS
 || (1) QTRSRRRProof [EQUIVALENT]
 || (2) QTRS
 || (3) RisEmptyProof [EQUIVALENT]
 || (4) YES
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    app(nil, %X) -> %X
 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
 ||    iconsc(%X, %Y) -> cons(%Y, %X)
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(app(x_1, x_2)) = 2 + 2*x_1 + x_2
 ||    POL(cons(x_1, x_2)) = 1 + x_1 + x_2
 ||    POL(iconsc(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
 ||    POL(nil) = 1
 ||    POL(~PAIR(x_1, x_2)) = 2 + x_1 + x_2
 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
 || 
 ||    app(nil, %X) -> %X
 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
 ||    iconsc(%X, %Y) -> cons(%Y, %X)
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (2)
 || Obligation:
 || Q restricted rewrite system:
 || R is empty.
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (3) RisEmptyProof (EQUIVALENT)
 || The TRS R is empty. Hence, termination is trivially proven.
 || ----------------------------------------
 || 
 || (4)
 || YES
 || 
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, all):

  Dependency Pairs P_0:

    0] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z)   
    1] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(z), u)   
    2] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(z)   
    3] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(X), Y)   
    4] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(X)   
    5] reverse#(X) =#> foldl#(/\x./\y.yap(iconsc(x), y), nil, X)   
    6] reverse#(X) =#> yap#(iconsc(x), y)   
    7] reverse#(X) =#> iconsc#(x)   
    8] reverse1#(X) =#> foldl#(/\x./\y.app(cons(y, nil), x), nil, X)   
    9] reverse1#(X) =#> app#(cons(x, nil), y)   
    10] yap#(F, X) =#> F(X)   

  Rules R_0:

    app(nil, X) => X 
    app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
    foldl(/\x./\y.yap(F(x), y), X, nil) => X 
    foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
    iconsc(X) Y => cons(Y, X) 
    reverse(X) => foldl(/\x./\y.yap(iconsc(x), y), nil, X) 
    reverse1(X) => foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
    yap(F, X) => F X 

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

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

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, 4 
    * 1 :  
    * 2 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 
    * 3 : 10 
    * 4 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 
    * 5 : 0, 1, 2, 3, 4 
    * 6 :  
    * 7 :  
    * 8 : 0, 1, 2, 3, 4 
    * 9 :  
    * 10 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 

This graph has the following strongly connected components:

  P_1:

    foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z)   
    foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(z)   
    foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> yap#(F(X), Y)   
    foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(X)   
    reverse#(X) =#> foldl#(/\x./\y.yap(iconsc(x), y), nil, X)   
    reverse1#(X) =#> foldl#(/\x./\y.app(cons(y, nil), x), nil, X)   
    yap#(F, X) =#> F(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, all) is finite.

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

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:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(~c1) ~c0 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? yap#(F(X), Y) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(X) ~c2 
  reverse#(X) >? foldl#(/\x./\y.yap(iconsc(x), y), nil, X) 
  reverse1#(X) >? foldl#(/\x./\y.app(cons(y, nil), x), nil, X) 
  yap#(F, X) >? F(X) 
  app(nil, X) >= X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  iconsc(X) Y >= cons(Y, X) 
  reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), nil, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
  yap(F, X) >= F X 
  foldl(F, X, Y) >= foldl#(F, X, Y) 
  reverse(X) >= reverse#(X) 
  reverse1(X) >= reverse1#(X) 
  yap(F, X) >= yap#(F, X) 

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

  pi( reverse1#(X) ) = #argfun-reverse1##(foldl#(/\x./\y.app(cons(y, nil), x), nil, X)) 
  pi( reverse#(X) ) = #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), nil, X)) 
  pi( yap#(F, X) ) = #argfun-yap##(F 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-reverse1##(x_1)]] = x_1 
  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) 
  [[foldl#(x_1, x_2, x_3)]] = foldl#(x_3, x_2, x_1) 
  [[nil]] = _|_ 
  [[reverse1#(x_1)]] = x_1 
  [[reverse#(x_1)]] = x_1 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 
  [[~c2]] = _|_ 

We choose Lex = {foldl, foldl#} and Mul = {#argfun-reverse##, #argfun-yap##, @_{o -> o}, app, cons, iconsc, reverse, reverse1, yap, yap#}, and the following precedence: reverse > #argfun-reverse## > reverse1 > app > iconsc > cons > foldl > foldl# > yap > yap# > @_{o -> o} > #argfun-yap##

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

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl#(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > #argfun-yap##(@_{o -> o}(F(X), Y)) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > @_{o -> o}(F(X), _|_) 
  #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)) >= foldl#(/\x./\y.yap(iconsc(x), y), _|_, X) 
  foldl#(/\x./\y.app(cons(y, _|_), x), _|_, X) >= foldl#(/\x./\y.app(cons(y, _|_), x), _|_, X) 
  #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X) 
  app(_|_, X) >= X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, _|_) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
  @_{o -> o}(iconsc(X), Y) >= cons(Y, X) 
  reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X) 
  yap(F, X) >= @_{o -> o}(F, X) 
  foldl(F, X, Y) >= foldl#(F, X, Y) 
  reverse(X) >= #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)) 
  reverse1(X) >= foldl#(/\x./\y.app(cons(y, _|_), x), _|_, X) 
  yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X)) 

With these choices, we have:

  1] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl#(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [2], by (Star) 
  2] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl#(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [3], [6], [13] and [23], by (Stat) 
  3] cons(Y, Z) > Z  because [4], by definition 
  4] cons*(Y, Z) >= Z  because [5], by (Select) 
  5] Z >= Z  by (Meta) 
  6] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [7], by (Select) 
  7] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [8], by (Abs) 
  8] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [9], by (Abs) 
  9] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [10] and [12], by (Fun) 
  10] F(y) >= F(y)  because [11], by (Meta) 
  11] y >= y  by (Var) 
  12] x >= x  by (Var) 
  13] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl# > yap, [14] and [19], by (Copy) 
  14] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= F(X)  because [15], by (Select) 
  15] /\z.yap(F(foldl#*(/\u./\v.yap(F(u), v), X, cons(Y, Z))), z) >= F(X)  because [16], by (Eta)[Kop13:2] 
  16] F(foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [17], by (Meta) 
  17] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [18], by (Select) 
  18] X >= X  by (Meta) 
  19] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [20], by (Select) 
  20] cons(Y, Z) >= Y  because [21], by (Star) 
  21] cons*(Y, Z) >= Y  because [22], by (Select) 
  22] Y >= Y  by (Meta) 
  23] foldl#*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z  because [24], by (Select) 
  24] cons(Y, Z) >= Z  because [4], by (Star) 

  25] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_)  because [26], by (Star) 
  26] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(_|_), _|_)  because foldl# > @_{o -> o}, [27] and [31], by (Copy) 
  27] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(_|_)  because [28], by (Select) 
  28] /\x.yap(F(foldl#*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(_|_)  because [29], by (Eta)[Kop13:2] 
  29] F(foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(_|_)  because [30], by (Meta) 
  30] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 
  31] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 

  32] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > #argfun-yap##(@_{o -> o}(F(X), Y))  because [33], by definition 
  33] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= #argfun-yap##(@_{o -> o}(F(X), Y))  because foldl# > #argfun-yap## and [34], by (Copy) 
  34] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), Y)  because foldl# > @_{o -> o}, [14] and [19], by (Copy) 

  35] foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) > @_{o -> o}(F(X), _|_)  because [36], by definition 
  36] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= @_{o -> o}(F(X), _|_)  because foldl# > @_{o -> o}, [14] and [37], by (Copy) 
  37] foldl#*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= _|_  by (Bot) 

  38] #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)) >= foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)  because [39], by (Star) 
  39] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)) >= foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)  because #argfun-reverse## > foldl#, [40], [48] and [49], by (Copy) 
  40] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)) >= /\x./\y.yap(iconsc(x), y)  because [41], by (F-Abs) 
  41] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X), z) >= /\x.yap(iconsc(z), x)  because [42], by (F-Abs) 
  42] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X), z, u) >= yap(iconsc(z), u)  because #argfun-reverse## > yap, [43] and [46], by (Copy) 
  43] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X), z, u) >= iconsc(z)  because #argfun-reverse## > iconsc and [44], by (Copy) 
  44] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X), z, u) >= z  because [45], by (Select) 
  45] z >= z  by (Var) 
  46] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X), z, u) >= u  because [47], by (Select) 
  47] u >= u  by (Var) 
  48] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)) >= _|_  by (Bot) 
  49] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)) >= X  because [50], by (Select) 
  50] foldl#(/\x./\y.yap(iconsc(x), y), _|_, X) >= X  because [51], by (Star) 
  51] foldl#*(/\x./\y.yap(iconsc(x), y), _|_, X) >= X  because [52], by (Select) 
  52] X >= X  by (Meta) 

  53] foldl#(/\x./\y.app(cons(y, _|_), x), _|_, X) >= foldl#(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [54], [59] and [61], by (Fun) 
  54] /\x./\z.app(cons(z, _|_), x) >= /\x./\z.app(cons(z, _|_), x)  because [55], by (Abs) 
  55] /\z.app(cons(z, _|_), y) >= /\z.app(cons(z, _|_), y)  because [56], by (Abs) 
  56] app(cons(x, _|_), y) >= app(cons(x, _|_), y)  because app in Mul, [57] and [60], by (Fun) 
  57] cons(x, _|_) >= cons(x, _|_)  because cons in Mul, [58] and [59], by (Fun) 
  58] x >= x  by (Var) 
  59] _|_ >= _|_  by (Bot) 
  60] y >= y  by (Var) 
  61] X >= X  by (Meta) 

  62] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [63], by (Star) 
  63] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [64], by (Select) 
  64] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [65] and [66], by (Fun) 
  65] F >= F  by (Meta) 
  66] X >= X  by (Meta) 

  67] app(_|_, X) >= X  because [68], by (Star) 
  68] app*(_|_, X) >= X  because [69], by (Select) 
  69] X >= X  by (Meta) 

  70] app(cons(X, Y), Z) >= cons(X, app(Y, Z))  because [71], by (Star) 
  71] app*(cons(X, Y), Z) >= cons(X, app(Y, Z))  because app > cons, [72] and [76], by (Copy) 
  72] app*(cons(X, Y), Z) >= X  because [73], by (Select) 
  73] cons(X, Y) >= X  because [74], by (Star) 
  74] cons*(X, Y) >= X  because [75], by (Select) 
  75] X >= X  by (Meta) 
  76] app*(cons(X, Y), Z) >= app(Y, Z)  because app in Mul, [77] and [80], by (Stat) 
  77] cons(X, Y) > Y  because [78], by definition 
  78] cons*(X, Y) >= Y  because [79], by (Select) 
  79] Y >= Y  by (Meta) 
  80] Z >= Z  by (Meta) 

  81] foldl(/\x./\y.yap(F(x), y), X, _|_) >= X  because [82], by (Star) 
  82] foldl*(/\x./\y.yap(F(x), y), X, _|_) >= X  because [83], by (Select) 
  83] X >= X  by (Meta) 

  84] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [85], by (Star) 
  85] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [3], [86], [87] and [93], by (Stat) 
  86] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [7], by (Select) 
  87] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [88] and [92], by (Copy) 
  88] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [89], by (Select) 
  89] /\x.yap(F(foldl*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(X)  because [90], by (Eta)[Kop13:2] 
  90] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [91], by (Meta) 
  91] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [18], by (Select) 
  92] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [20], by (Select) 
  93] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [94], by (Select) 
  94] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [95], by (Star) 
  95] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [96], by (Select) 
  96] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [97], by (Select) 
  97] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [98], by (Star) 
  98] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [99], by (Select) 
  99] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [100], by (Select) 
  100] cons(Y, Z) >= Z  because [4], by (Star) 

  101] @_{o -> o}(iconsc(X), Y) >= cons(Y, X)  because [102], by (Star) 
  102] @_{o -> o}*(iconsc(X), Y) >= cons(Y, X)  because [103], by (Select) 
  103] iconsc(X) @_{o -> o}*(iconsc(X), Y) >= cons(Y, X)  because [104] 
  104] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= cons(Y, X)  because iconsc > cons, [105] and [108], by (Copy) 
  105] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= Y  because [106], by (Select) 
  106] @_{o -> o}*(iconsc(X), Y) >= Y  because [107], by (Select) 
  107] Y >= Y  by (Meta) 
  108] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= X  because [109], by (Select) 
  109] X >= X  by (Meta) 

  110] reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X)  because [111], by (Star) 
  111] reverse*(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X)  because reverse > foldl, [112], [120] and [121], by (Copy) 
  112] reverse*(X) >= /\y./\z.yap(iconsc(y), z)  because [113], by (F-Abs) 
  113] reverse*(X, x) >= /\z.yap(iconsc(x), z)  because [114], by (F-Abs) 
  114] reverse*(X, x, y) >= yap(iconsc(x), y)  because reverse > yap, [115] and [118], by (Copy) 
  115] reverse*(X, x, y) >= iconsc(x)  because reverse > iconsc and [116], by (Copy) 
  116] reverse*(X, x, y) >= x  because [117], by (Select) 
  117] x >= x  by (Var) 
  118] reverse*(X, x, y) >= y  because [119], by (Select) 
  119] y >= y  by (Var) 
  120] reverse*(X) >= _|_  by (Bot) 
  121] reverse*(X) >= X  because [52], by (Select) 

  122] reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [123], by (Star) 
  123] reverse1*(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl, [124], [133] and [134], by (Copy) 
  124] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [125], by (F-Abs) 
  125] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [126], by (F-Abs) 
  126] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [127] and [131], by (Copy) 
  127] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [128] and [130], by (Copy) 
  128] reverse1*(X, x, y) >= y  because [129], by (Select) 
  129] y >= y  by (Var) 
  130] reverse1*(X, x, y) >= _|_  by (Bot) 
  131] reverse1*(X, x, y) >= x  because [132], by (Select) 
  132] x >= x  by (Var) 
  133] reverse1*(X) >= _|_  by (Bot) 
  134] reverse1*(X) >= X  because [61], by (Select) 

  135] yap(F, X) >= @_{o -> o}(F, X)  because [136], by (Star) 
  136] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [137] and [138], by (Copy) 
  137] yap*(F, X) >= F  because [65], by (Select) 
  138] yap*(F, X) >= X  because [66], by (Select) 

  139] foldl(F, X, Y) >= foldl#(F, X, Y)  because [140], by (Star) 
  140] foldl*(F, X, Y) >= foldl#(F, X, Y)  because foldl > foldl#, [141], [143] and [145], by (Copy) 
  141] foldl*(F, X, Y) >= F  because [142], by (Select) 
  142] F >= F  by (Meta) 
  143] foldl*(F, X, Y) >= X  because [144], by (Select) 
  144] X >= X  by (Meta) 
  145] foldl*(F, X, Y) >= Y  because [146], by (Select) 
  146] Y >= Y  by (Meta) 

  147] reverse(X) >= #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X))  because [148], by (Star) 
  148] reverse*(X) >= #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), _|_, X))  because reverse > #argfun-reverse## and [149], by (Copy) 
  149] reverse*(X) >= foldl#(/\x./\y.yap(iconsc(x), y), _|_, X)  because reverse > foldl#, [150], [158] and [159], by (Copy) 
  150] reverse*(X) >= /\y./\z.yap(iconsc(y), z)  because [151], by (F-Abs) 
  151] reverse*(X, x) >= /\z.yap(iconsc(x), z)  because [152], by (F-Abs) 
  152] reverse*(X, x, y) >= yap(iconsc(x), y)  because reverse > yap, [153] and [156], by (Copy) 
  153] reverse*(X, x, y) >= iconsc(x)  because reverse > iconsc and [154], by (Copy) 
  154] reverse*(X, x, y) >= x  because [155], by (Select) 
  155] x >= x  by (Var) 
  156] reverse*(X, x, y) >= y  because [157], by (Select) 
  157] y >= y  by (Var) 
  158] reverse*(X) >= _|_  by (Bot) 
  159] reverse*(X) >= X  because [160], by (Select) 
  160] X >= X  by (Meta) 

  161] reverse1(X) >= foldl#(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [162], by (Star) 
  162] reverse1*(X) >= foldl#(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl#, [163], [172] and [173], by (Copy) 
  163] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [164], by (F-Abs) 
  164] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [165], by (F-Abs) 
  165] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [166] and [170], by (Copy) 
  166] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [167] and [169], by (Copy) 
  167] reverse1*(X, x, y) >= y  because [168], by (Select) 
  168] y >= y  by (Var) 
  169] reverse1*(X, x, y) >= _|_  by (Bot) 
  170] reverse1*(X, x, y) >= x  because [171], by (Select) 
  171] x >= x  by (Var) 
  172] reverse1*(X) >= _|_  by (Bot) 
  173] reverse1*(X) >= X  because [174], by (Select) 
  174] X >= X  by (Meta) 

  175] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [176], by (Star) 
  176] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [177], by (Copy) 
  177] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [178] and [180], by (Copy) 
  178] yap*(F, X) >= F  because [179], by (Select) 
  179] F >= F  by (Meta) 
  180] yap*(F, X) >= X  because [181], by (Select) 
  181] 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_1, R_0, minimal, all) by (P_2, R_0, minimal, all), where P_2 consists of:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z)   
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> F(z)   
  reverse#(X) =#> foldl#(/\x./\y.yap(iconsc(x), y), nil, X)   
  reverse1#(X) =#> foldl#(/\x./\y.app(cons(y, nil), x), nil, X)   
  yap#(F, X) =#> F(X)   

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

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

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:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? F(~c1) ~c0 
  reverse#(X) >? foldl#(/\x./\y.yap(iconsc(x), y), nil, X) 
  reverse1#(X) >? foldl#(/\x./\y.app(cons(y, nil), x), nil, X) 
  yap#(F, X) >? F(X) 
  app(nil, X) >= X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  iconsc(X) Y >= cons(Y, X) 
  reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), nil, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
  yap(F, X) >= F X 
  foldl(F, X, Y) >= foldl#(F, X, Y) 
  reverse(X) >= reverse#(X) 
  reverse1(X) >= reverse1#(X) 
  yap(F, X) >= yap#(F, X) 

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

  pi( reverse1#(X) ) = #argfun-reverse1##(foldl#(/\x./\y.app(cons(y, nil), x), nil, X)) 
  pi( reverse#(X) ) = #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), nil, X)) 
  pi( yap#(F, X) ) = #argfun-yap##(F 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:

  [[foldl(x_1, x_2, x_3)]] = foldl(x_1, x_3, x_2) 
  [[foldl#(x_1, x_2, x_3)]] = foldl#(x_1, x_3) 
  [[nil]] = _|_ 
  [[reverse1#(x_1)]] = x_1 
  [[reverse#(x_1)]] = x_1 
  [[~c0]] = _|_ 
  [[~c1]] = _|_ 

We choose Lex = {foldl} and Mul = {#argfun-reverse1##, #argfun-reverse##, #argfun-yap##, @_{o -> o}, app, cons, foldl#, iconsc, reverse, reverse1, yap, yap#}, and the following precedence: reverse > #argfun-reverse## > iconsc > reverse1 > #argfun-reverse1## > app > foldl > yap > #argfun-yap## > foldl# > @_{o -> o} > cons > yap#

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

  foldl#(/\x./\y.yap(F(x), y), cons(X, Y)) >= foldl#(/\x./\y.yap(F(x), y), Y) 
  foldl#(/\x./\y.yap(F(x), y), cons(X, Y)) > @_{o -> o}(F(_|_), _|_) 
  #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), X)) >= foldl#(/\x./\y.yap(iconsc(x), y), X) 
  #argfun-reverse1##(foldl#(/\x./\y.app(cons(y, _|_), x), X)) >= foldl#(/\x./\y.app(cons(y, _|_), x), X) 
  #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X) 
  app(_|_, X) >= X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, _|_) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
  @_{o -> o}(iconsc(X), Y) >= cons(Y, X) 
  reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X) 
  yap(F, X) >= @_{o -> o}(F, X) 
  foldl(F, X, Y) >= foldl#(F, Y) 
  reverse(X) >= #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), X)) 
  reverse1(X) >= #argfun-reverse1##(foldl#(/\x./\y.app(cons(y, _|_), x), X)) 
  yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X)) 

With these choices, we have:

  1] foldl#(/\x./\y.yap(F(x), y), cons(X, Y)) >= foldl#(/\x./\y.yap(F(x), y), Y)  because foldl# in Mul, [2] and [8], by (Fun) 
  2] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [3], by (Abs) 
  3] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [4], by (Abs) 
  4] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [5] and [7], by (Fun) 
  5] F(y) >= F(y)  because [6], by (Meta) 
  6] y >= y  by (Var) 
  7] x >= x  by (Var) 
  8] cons(X, Y) >= Y  because [9], by (Star) 
  9] cons*(X, Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 

  11] foldl#(/\x./\y.yap(F(x), y), cons(X, Y)) > @_{o -> o}(F(_|_), _|_)  because [12], by definition 
  12] foldl#*(/\x./\y.yap(F(x), y), cons(X, Y)) >= @_{o -> o}(F(_|_), _|_)  because foldl# > @_{o -> o}, [13] and [17], by (Copy) 
  13] foldl#*(/\x./\y.yap(F(x), y), cons(X, Y)) >= F(_|_)  because [14], by (Select) 
  14] /\x.yap(F(foldl#*(/\y./\z.yap(F(y), z), cons(X, Y))), x) >= F(_|_)  because [15], by (Eta)[Kop13:2] 
  15] F(foldl#*(/\x./\y.yap(F(x), y), cons(X, Y))) >= F(_|_)  because [16], by (Meta) 
  16] foldl#*(/\x./\y.yap(F(x), y), cons(X, Y)) >= _|_  by (Bot) 
  17] foldl#*(/\x./\y.yap(F(x), y), cons(X, Y)) >= _|_  by (Bot) 

  18] #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), X)) >= foldl#(/\x./\y.yap(iconsc(x), y), X)  because [19], by (Star) 
  19] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), X)) >= foldl#(/\x./\y.yap(iconsc(x), y), X)  because #argfun-reverse## > foldl#, [20] and [28], by (Copy) 
  20] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), X)) >= /\x./\y.yap(iconsc(x), y)  because [21], by (F-Abs) 
  21] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), X), z) >= /\x.yap(iconsc(z), x)  because [22], by (F-Abs) 
  22] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), X), z, u) >= yap(iconsc(z), u)  because #argfun-reverse## > yap, [23] and [26], by (Copy) 
  23] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), X), z, u) >= iconsc(z)  because #argfun-reverse## > iconsc and [24], by (Copy) 
  24] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), X), z, u) >= z  because [25], by (Select) 
  25] z >= z  by (Var) 
  26] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), X), z, u) >= u  because [27], by (Select) 
  27] u >= u  by (Var) 
  28] #argfun-reverse##*(foldl#(/\x./\y.yap(iconsc(x), y), X)) >= X  because [29], by (Select) 
  29] foldl#(/\x./\y.yap(iconsc(x), y), X) >= X  because [30], by (Star) 
  30] foldl#*(/\x./\y.yap(iconsc(x), y), X) >= X  because [31], by (Select) 
  31] X >= X  by (Meta) 

  32] #argfun-reverse1##(foldl#(/\x./\y.app(cons(y, _|_), x), X)) >= foldl#(/\x./\y.app(cons(y, _|_), x), X)  because [33], by (Star) 
  33] #argfun-reverse1##*(foldl#(/\x./\y.app(cons(y, _|_), x), X)) >= foldl#(/\x./\y.app(cons(y, _|_), x), X)  because [34], by (Select) 
  34] foldl#(/\x./\y.app(cons(y, _|_), x), X) >= foldl#(/\x./\y.app(cons(y, _|_), x), X)  because foldl# in Mul, [35] and [42], by (Fun) 
  35] /\x./\z.app(cons(z, _|_), x) >= /\x./\z.app(cons(z, _|_), x)  because [36], by (Abs) 
  36] /\z.app(cons(z, _|_), y) >= /\z.app(cons(z, _|_), y)  because [37], by (Abs) 
  37] app(cons(x, _|_), y) >= app(cons(x, _|_), y)  because app in Mul, [38] and [41], by (Fun) 
  38] cons(x, _|_) >= cons(x, _|_)  because cons in Mul, [39] and [40], by (Fun) 
  39] x >= x  by (Var) 
  40] _|_ >= _|_  by (Bot) 
  41] y >= y  by (Var) 
  42] X >= X  by (Meta) 

  43] #argfun-yap##(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [44], by (Star) 
  44] #argfun-yap##*(@_{o -> o}(F, X)) >= @_{o -> o}(F, X)  because [45], by (Select) 
  45] @_{o -> o}(F, X) >= @_{o -> o}(F, X)  because @_{o -> o} in Mul, [46] and [47], by (Fun) 
  46] F >= F  by (Meta) 
  47] X >= X  by (Meta) 

  48] app(_|_, X) >= X  because [49], by (Star) 
  49] app*(_|_, X) >= X  because [50], by (Select) 
  50] X >= X  by (Meta) 

  51] app(cons(X, Y), Z) >= cons(X, app(Y, Z))  because [52], by (Star) 
  52] app*(cons(X, Y), Z) >= cons(X, app(Y, Z))  because app > cons, [53] and [57], by (Copy) 
  53] app*(cons(X, Y), Z) >= X  because [54], by (Select) 
  54] cons(X, Y) >= X  because [55], by (Star) 
  55] cons*(X, Y) >= X  because [56], by (Select) 
  56] X >= X  by (Meta) 
  57] app*(cons(X, Y), Z) >= app(Y, Z)  because app in Mul, [58] and [61], by (Stat) 
  58] cons(X, Y) > Y  because [59], by definition 
  59] cons*(X, Y) >= Y  because [60], by (Select) 
  60] Y >= Y  by (Meta) 
  61] Z >= Z  by (Meta) 

  62] foldl(/\x./\y.yap(F(x), y), X, _|_) >= X  because [63], by (Star) 
  63] foldl*(/\x./\y.yap(F(x), y), X, _|_) >= X  because [64], by (Select) 
  64] X >= X  by (Meta) 

  65] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [66], by (Star) 
  66] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [67], by (Select) 
  67] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [68], by (Star) 
  68] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [69], by (Select) 
  69] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [2], [70], [72], [73] and [83], by (Stat) 
  70] cons(Y, Z) > Z  because [71], by definition 
  71] cons*(Y, Z) >= Z  because [10], by (Select) 
  72] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [2], by (Select) 
  73] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [74] and [79], by (Copy) 
  74] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X)  because [75], by (Select) 
  75] /\x.yap(F(foldl*(/\y./\z.yap(F(y), z), X, cons(Y, Z))), x) >= F(X)  because [76], by (Eta)[Kop13:2] 
  76] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X)  because [77], by (Meta) 
  77] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X  because [78], by (Select) 
  78] X >= X  by (Meta) 
  79] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y  because [80], by (Select) 
  80] cons(Y, Z) >= Y  because [81], by (Star) 
  81] cons*(Y, Z) >= Y  because [82], by (Select) 
  82] Y >= Y  by (Meta) 
  83] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [84], by (Select) 
  84] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [85], by (Star) 
  85] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [86], by (Select) 
  86] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [87], by (Select) 
  87] yap(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [88], by (Star) 
  88] yap*(F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))), foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= Z  because [89], by (Select) 
  89] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z  because [90], by (Select) 
  90] cons(Y, Z) >= Z  because [71], by (Star) 

  91] @_{o -> o}(iconsc(X), Y) >= cons(Y, X)  because [92], by (Star) 
  92] @_{o -> o}*(iconsc(X), Y) >= cons(Y, X)  because @_{o -> o} > cons, [93] and [95], by (Copy) 
  93] @_{o -> o}*(iconsc(X), Y) >= Y  because [94], by (Select) 
  94] Y >= Y  by (Meta) 
  95] @_{o -> o}*(iconsc(X), Y) >= X  because [96], by (Select) 
  96] iconsc(X) @_{o -> o}*(iconsc(X), Y) >= X  because [97] 
  97] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= X  because [98], by (Select) 
  98] @_{o -> o}*(iconsc(X), Y) >= X  because [99], by (Select) 
  99] iconsc(X) @_{o -> o}*(iconsc(X), Y) >= X  because [100] 
  100] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= X  because [101], by (Select) 
  101] X >= X  by (Meta) 

  102] reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X)  because [103], by (Star) 
  103] reverse*(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X)  because reverse > foldl, [104], [112] and [113], by (Copy) 
  104] reverse*(X) >= /\y./\z.yap(iconsc(y), z)  because [105], by (F-Abs) 
  105] reverse*(X, x) >= /\z.yap(iconsc(x), z)  because [106], by (F-Abs) 
  106] reverse*(X, x, y) >= yap(iconsc(x), y)  because reverse > yap, [107] and [110], by (Copy) 
  107] reverse*(X, x, y) >= iconsc(x)  because reverse > iconsc and [108], by (Copy) 
  108] reverse*(X, x, y) >= x  because [109], by (Select) 
  109] x >= x  by (Var) 
  110] reverse*(X, x, y) >= y  because [111], by (Select) 
  111] y >= y  by (Var) 
  112] reverse*(X) >= _|_  by (Bot) 
  113] reverse*(X) >= X  because [31], by (Select) 

  114] reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [115], by (Star) 
  115] reverse1*(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl, [116], [125] and [126], by (Copy) 
  116] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [117], by (F-Abs) 
  117] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [118], by (F-Abs) 
  118] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [119] and [123], by (Copy) 
  119] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [120] and [122], by (Copy) 
  120] reverse1*(X, x, y) >= y  because [121], by (Select) 
  121] y >= y  by (Var) 
  122] reverse1*(X, x, y) >= _|_  by (Bot) 
  123] reverse1*(X, x, y) >= x  because [124], by (Select) 
  124] x >= x  by (Var) 
  125] reverse1*(X) >= _|_  by (Bot) 
  126] reverse1*(X) >= X  because [42], by (Select) 

  127] yap(F, X) >= @_{o -> o}(F, X)  because [128], by (Star) 
  128] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [129] and [130], by (Copy) 
  129] yap*(F, X) >= F  because [46], by (Select) 
  130] yap*(F, X) >= X  because [47], by (Select) 

  131] foldl(F, X, Y) >= foldl#(F, Y)  because [132], by (Star) 
  132] foldl*(F, X, Y) >= foldl#(F, Y)  because foldl > foldl#, [133] and [135], by (Copy) 
  133] foldl*(F, X, Y) >= F  because [134], by (Select) 
  134] F >= F  by (Meta) 
  135] foldl*(F, X, Y) >= Y  because [136], by (Select) 
  136] Y >= Y  by (Meta) 

  137] reverse(X) >= #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), X))  because [138], by (Star) 
  138] reverse*(X) >= #argfun-reverse##(foldl#(/\x./\y.yap(iconsc(x), y), X))  because reverse > #argfun-reverse## and [139], by (Copy) 
  139] reverse*(X) >= foldl#(/\x./\y.yap(iconsc(x), y), X)  because reverse > foldl#, [140] and [148], by (Copy) 
  140] reverse*(X) >= /\y./\z.yap(iconsc(y), z)  because [141], by (F-Abs) 
  141] reverse*(X, x) >= /\z.yap(iconsc(x), z)  because [142], by (F-Abs) 
  142] reverse*(X, x, y) >= yap(iconsc(x), y)  because reverse > yap, [143] and [146], by (Copy) 
  143] reverse*(X, x, y) >= iconsc(x)  because reverse > iconsc and [144], by (Copy) 
  144] reverse*(X, x, y) >= x  because [145], by (Select) 
  145] x >= x  by (Var) 
  146] reverse*(X, x, y) >= y  because [147], by (Select) 
  147] y >= y  by (Var) 
  148] reverse*(X) >= X  because [149], by (Select) 
  149] X >= X  by (Meta) 

  150] reverse1(X) >= #argfun-reverse1##(foldl#(/\x./\y.app(cons(y, _|_), x), X))  because [151], by (Star) 
  151] reverse1*(X) >= #argfun-reverse1##(foldl#(/\x./\y.app(cons(y, _|_), x), X))  because reverse1 > #argfun-reverse1## and [152], by (Copy) 
  152] reverse1*(X) >= foldl#(/\x./\y.app(cons(y, _|_), x), X)  because reverse1 > foldl#, [153] and [162], by (Copy) 
  153] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [154], by (F-Abs) 
  154] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [155], by (F-Abs) 
  155] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [156] and [160], by (Copy) 
  156] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [157] and [159], by (Copy) 
  157] reverse1*(X, x, y) >= y  because [158], by (Select) 
  158] y >= y  by (Var) 
  159] reverse1*(X, x, y) >= _|_  by (Bot) 
  160] reverse1*(X, x, y) >= x  because [161], by (Select) 
  161] x >= x  by (Var) 
  162] reverse1*(X) >= X  because [163], by (Select) 
  163] X >= X  by (Meta) 

  164] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [165], by (Star) 
  165] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [166], by (Copy) 
  166] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [167] and [169], by (Copy) 
  167] yap*(F, X) >= F  because [168], by (Select) 
  168] F >= F  by (Meta) 
  169] yap*(F, X) >= X  because [170], by (Select) 
  170] 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_2, R_0, minimal, all) by (P_3, R_0, minimal, all), where P_3 consists of:

  foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z)   
  reverse#(X) =#> foldl#(/\x./\y.yap(iconsc(x), y), nil, X)   
  reverse1#(X) =#> foldl#(/\x./\y.app(cons(y, nil), x), nil, X)   
  yap#(F, X) =#> F(X)   

Thus, the original system is terminating if (P_3, R_0, minimal, all) is finite.

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

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 : 0 
    * 2 : 0 
    * 3 : 0, 1, 2, 3 

This graph has the following strongly connected components:

  P_4:

    foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z)) =#> foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z)   

  P_5:

    yap#(F, X) =#> F(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_3, R_0, m, f) by (P_4, R_0, m, f) and (P_5, R_0, m, f).

Thus, the original system is terminating if each of (P_4, R_0, minimal, all) and (P_5, R_0, minimal, all) is finite.

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

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:

  yap#(F, X) >? F(X) 
  app(nil, X) >= X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, nil) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 
  iconsc(X) Y >= cons(Y, X) 
  reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), nil, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
  yap(F, X) >= F X 
  yap(F, X) >= yap#(F, X) 

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

  pi( yap#(F, X) ) = #argfun-yap##(F 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:

  [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_2, x_1) 
  [[nil]] = _|_ 

We choose Lex = {foldl} and Mul = {#argfun-yap##, @_{o -> o}, app, cons, iconsc, reverse, reverse1, yap, yap#}, and the following precedence: reverse > iconsc > reverse1 > app > foldl > yap > #argfun-yap## > cons > @_{o -> o} > yap#

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

  #argfun-yap##(@_{o -> o}(F, X)) > @_{o -> o}(F, X) 
  app(_|_, X) >= X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.yap(F(x), y), X, _|_) >= X 
  foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) 
  @_{o -> o}(iconsc(X), Y) >= cons(Y, X) 
  reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X) 
  reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X) 
  yap(F, X) >= @_{o -> o}(F, X) 
  yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X)) 

With these choices, we have:

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

  6] app(_|_, X) >= X  because [7], by (Star) 
  7] app*(_|_, X) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 

  9] app(cons(X, Y), Z) >= cons(X, app(Y, Z))  because [10], by (Star) 
  10] app*(cons(X, Y), Z) >= cons(X, app(Y, Z))  because app > cons, [11] and [15], by (Copy) 
  11] app*(cons(X, Y), Z) >= X  because [12], by (Select) 
  12] cons(X, Y) >= X  because [13], by (Star) 
  13] cons*(X, Y) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 
  15] app*(cons(X, Y), Z) >= app(Y, Z)  because app in Mul, [16] and [19], by (Stat) 
  16] cons(X, Y) > Y  because [17], by definition 
  17] cons*(X, Y) >= Y  because [18], by (Select) 
  18] Y >= Y  by (Meta) 
  19] Z >= Z  by (Meta) 

  20] foldl(/\x./\y.yap(F(x), y), X, _|_) >= X  because [21], by (Star) 
  21] foldl*(/\x./\y.yap(F(x), y), X, _|_) >= X  because [22], by (Select) 
  22] X >= X  by (Meta) 

  23] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [24], by (Star) 
  24] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z)  because [25], [28], [35] and [45], by (Stat) 
  25] cons(Y, Z) > Z  because [26], by definition 
  26] cons*(Y, Z) >= Z  because [27], by (Select) 
  27] Z >= Z  by (Meta) 
  28] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y)  because [29], by (Select) 
  29] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [30], by (Abs) 
  30] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [31], by (Abs) 
  31] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [32] and [34], by (Fun) 
  32] F(y) >= F(y)  because [33], by (Meta) 
  33] y >= y  by (Var) 
  34] x >= x  by (Var) 
  35] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= yap(F(X), Y)  because foldl > yap, [36] and [41], by (Copy) 
  36] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= F(X)  because [37], by (Select) 
  37] /\z.yap(F(foldl*(/\u./\v.yap(F(u), v), X, cons(Y, Z))), z) >= F(X)  because [38], by (Eta)[Kop13:2] 
  38] F(foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z))) >= F(X)  because [39], by (Meta) 
  39] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= X  because [40], by (Select) 
  40] X >= X  by (Meta) 
  41] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Y  because [42], by (Select) 
  42] cons(Y, Z) >= Y  because [43], by (Star) 
  43] cons*(Y, Z) >= Y  because [44], by (Select) 
  44] Y >= Y  by (Meta) 
  45] foldl*(/\z./\u.yap(F(z), u), X, cons(Y, Z)) >= Z  because [46], by (Select) 
  46] cons(Y, Z) >= Z  because [26], by (Star) 

  47] @_{o -> o}(iconsc(X), Y) >= cons(Y, X)  because [48], by (Star) 
  48] @_{o -> o}*(iconsc(X), Y) >= cons(Y, X)  because [49], by (Select) 
  49] iconsc(X) @_{o -> o}*(iconsc(X), Y) >= cons(Y, X)  because [50] 
  50] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= cons(Y, X)  because iconsc > cons, [51] and [54], by (Copy) 
  51] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= Y  because [52], by (Select) 
  52] @_{o -> o}*(iconsc(X), Y) >= Y  because [53], by (Select) 
  53] Y >= Y  by (Meta) 
  54] iconsc*(X, @_{o -> o}*(iconsc(X), Y)) >= X  because [55], by (Select) 
  55] X >= X  by (Meta) 

  56] reverse(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X)  because [57], by (Star) 
  57] reverse*(X) >= foldl(/\x./\y.yap(iconsc(x), y), _|_, X)  because reverse > foldl, [58], [66] and [67], by (Copy) 
  58] reverse*(X) >= /\y./\z.yap(iconsc(y), z)  because [59], by (F-Abs) 
  59] reverse*(X, x) >= /\z.yap(iconsc(x), z)  because [60], by (F-Abs) 
  60] reverse*(X, x, y) >= yap(iconsc(x), y)  because reverse > yap, [61] and [64], by (Copy) 
  61] reverse*(X, x, y) >= iconsc(x)  because reverse > iconsc and [62], by (Copy) 
  62] reverse*(X, x, y) >= x  because [63], by (Select) 
  63] x >= x  by (Var) 
  64] reverse*(X, x, y) >= y  because [65], by (Select) 
  65] y >= y  by (Var) 
  66] reverse*(X) >= _|_  by (Bot) 
  67] reverse*(X) >= X  because [68], by (Select) 
  68] X >= X  by (Meta) 

  69] reverse1(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because [70], by (Star) 
  70] reverse1*(X) >= foldl(/\x./\y.app(cons(y, _|_), x), _|_, X)  because reverse1 > foldl, [71], [80] and [81], by (Copy) 
  71] reverse1*(X) >= /\y./\z.app(cons(z, _|_), y)  because [72], by (F-Abs) 
  72] reverse1*(X, x) >= /\z.app(cons(z, _|_), x)  because [73], by (F-Abs) 
  73] reverse1*(X, x, y) >= app(cons(y, _|_), x)  because reverse1 > app, [74] and [78], by (Copy) 
  74] reverse1*(X, x, y) >= cons(y, _|_)  because reverse1 > cons, [75] and [77], by (Copy) 
  75] reverse1*(X, x, y) >= y  because [76], by (Select) 
  76] y >= y  by (Var) 
  77] reverse1*(X, x, y) >= _|_  by (Bot) 
  78] reverse1*(X, x, y) >= x  because [79], by (Select) 
  79] x >= x  by (Var) 
  80] reverse1*(X) >= _|_  by (Bot) 
  81] reverse1*(X) >= X  because [82], by (Select) 
  82] X >= X  by (Meta) 

  83] yap(F, X) >= @_{o -> o}(F, X)  because [84], by (Star) 
  84] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [85] and [86], by (Copy) 
  85] yap*(F, X) >= F  because [4], by (Select) 
  86] yap*(F, X) >= X  because [5], by (Select) 

  87] yap(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because [88], by (Star) 
  88] yap*(F, X) >= #argfun-yap##(@_{o -> o}(F, X))  because yap > #argfun-yap## and [89], by (Copy) 
  89] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [90] and [92], by (Copy) 
  90] yap*(F, X) >= F  because [91], by (Select) 
  91] F >= F  by (Meta) 
  92] yap*(F, X) >= X  because [93], by (Select) 
  93] 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_5, 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_4, R_0, minimal, all) is finite.

We consider the dependency pair problem (P_4, R_0, minimal, all).

We apply the subterm criterion with the following projection function:

  nu(foldl#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(foldl#(/\x./\y.yap(F(x), y), X, cons(Y, Z))) = cons(Y, Z) |> Z = nu(foldl#(/\z./\u.yap(F(z), u), yap(F(X), Y), Z)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_4, 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 +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[Kop13:2]  C. Kop.  StarHorpo with an Eta-Rule.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/etahorpo.pdf, 2013.