We consider the system h61.

  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 

  Rules:

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

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:

    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 
    ~AP1 : [list -> nat -> list * list] --> nat -> list 

  Rules:

    app(nil, X) => X 
    app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
    foldl(/\x./\y.~AP1(F, x) y, X, nil) => X 
    foldl(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => foldl(/\z./\u.~AP1(F, z) u, ~AP1(F, X) Y, Z) 
    iconsc(X, Y) => cons(Y, X) 
    reverse(X) => foldl(/\x./\y.iconsc(x, y), nil, X) 
    reverse1(X) => foldl(/\x./\y.app(cons(y, nil), x), nil, X) 
    foldl(/\x./\y.iconsc(x, y), X, nil) => X 
    foldl(/\x./\y.iconsc(x, y), X, cons(Y, Z)) => foldl(/\z./\u.iconsc(z, u), iconsc(X, Y), Z) 
    ~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.  Additionally, we can remove some (now-)redundant rules.  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 

  Rules:

    app(nil, X) => X 
    app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
    foldl(/\x./\y.X(x, y), Y, nil) => Y 
    foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) 
    iconsc(X, Y) => cons(Y, X) 
    reverse(X) => foldl(/\x./\y.iconsc(x, y), nil, X) 
    reverse1(X) => foldl(/\x./\y.app(cons(y, nil), x), nil, 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 orthogonal.  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 terminating when seen as a many-sorted first-order TRS.

According to the external first-order termination prover, this system is indeed 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)
 || 
 || 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
 || 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)
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (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.

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

  Dependency Pairs P_0:

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

  Rules R_0:

    app(nil, X) => X 
    app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
    foldl(/\x./\y.X(x, y), Y, nil) => Y 
    foldl(/\x./\y.X(x, y), Y, cons(Z, U)) => foldl(/\z./\u.X(z, u), X(Y, Z), U) 
    iconsc(X, Y) => cons(Y, X) 
    reverse(X) => foldl(/\x./\y.iconsc(x, y), nil, X) 
    reverse1(X) => foldl(/\x./\y.app(cons(y, nil), x), nil, 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 
    * 1 : 0, 1, 2, 3, 4, 5, 6 
    * 2 : 0, 1, 2, 3, 4, 5, 6 
    * 3 : 0, 1, 2 
    * 4 :  
    * 5 : 0, 1, 2 
    * 6 :  

This graph has the following strongly connected components:

  P_1:

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

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

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

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

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:

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

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

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

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

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

  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > foldl#(/\x./\y.X(x, y), X(Y, Z), U) 
  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(_|_, _|_) 
  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > X(Y, Z) 
  #argfun-reverse##(foldl#(/\x./\y.iconsc-(x, y), nil, X)) >= foldl#(/\x./\y.iconsc-(x, y), nil, X) 
  foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X) >= foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X) 
  app(nil, X) >= X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.X(x, y), Y, nil) >= Y 
  foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) 
  cons(X, Y) >= cons(X, Y) 
  #argfun-reverse#(foldl(/\x./\y.iconsc-(x, y), nil, X)) >= foldl(/\x./\y.iconsc-(x, y), nil, X) 
  #argfun-reverse1#(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl(/\x./\y.app-(cons-(y, nil), x), nil, X) 
  app-(X, Y) >= app(X, Y) 
  cons-(X, Y) >= cons(X, Y) 
  iconsc-(X, Y) >= cons(Y, X) 

With these choices, we have:

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

  23] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(_|_, _|_)  because [24], by (Star) 
  24] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(_|_, _|_)  because [25], by (Select) 
  25] X(foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)), foldl#*(/\z./\u.X(z, u), Y, cons(Z, U))) >= X(_|_, _|_)  because [26] and [27], by (Meta) 
  26] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= _|_  by (Bot) 
  27] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= _|_  by (Bot) 

  28] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > X(Y, Z)  because [13], by definition 

  29] #argfun-reverse##(foldl#(/\x./\y.iconsc-(x, y), nil, X)) >= foldl#(/\x./\y.iconsc-(x, y), nil, X)  because [30], by (Star) 
  30] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), nil, X)) >= foldl#(/\x./\y.iconsc-(x, y), nil, X)  because [31], by (Select) 
  31] foldl#(/\x./\y.iconsc-(x, y), nil, X) >= foldl#(/\x./\y.iconsc-(x, y), nil, X)  because [32], [37] and [38], by (Fun) 
  32] /\x./\z.iconsc-(x, z) >= /\x./\z.iconsc-(x, z)  because [33], by (Abs) 
  33] /\z.iconsc-(y, z) >= /\z.iconsc-(y, z)  because [34], by (Abs) 
  34] iconsc-(y, x) >= iconsc-(y, x)  because iconsc- in Mul, [35] and [36], by (Fun) 
  35] y >= y  by (Var) 
  36] x >= x  by (Var) 
  37] nil >= nil  by (Fun) 
  38] X >= X  by (Meta) 

  39] foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X) >= foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)  because [40], [37] and [46], by (Fun) 
  40] /\x./\z.app-(cons-(z, nil), x) >= /\x./\z.app-(cons-(z, nil), x)  because [41], by (Abs) 
  41] /\z.app-(cons-(z, nil), y) >= /\z.app-(cons-(z, nil), y)  because [42], by (Abs) 
  42] app-(cons-(x, nil), y) >= app-(cons-(x, nil), y)  because app- in Mul, [43] and [45], by (Fun) 
  43] cons-(x, nil) >= cons-(x, nil)  because cons- in Mul, [44] and [37], by (Fun) 
  44] x >= x  by (Var) 
  45] y >= y  by (Var) 
  46] X >= X  by (Meta) 

  47] app(nil, X) >= X  because [48], by (Star) 
  48] app*(nil, X) >= X  because [49], by (Select) 
  49] X >= X  by (Meta) 

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

  61] foldl(/\x./\y.X(x, y), Y, nil) >= Y  because [62], by (Star) 
  62] foldl*(/\x./\y.X(x, y), Y, nil) >= Y  because [63], by (Select) 
  63] Y >= Y  by (Meta) 

  64] foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U)  because [65], by (Star) 
  65] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U)  because [66], [3], [71], [72] and [76], by (Stat) 
  66] /\x./\z.X(x, z) >= /\x./\z.X(x, z)  because [67], by (Abs) 
  67] /\z.X(y, z) >= /\z.X(y, z)  because [68], by (Abs) 
  68] X(y, x) >= X(y, x)  because [69] and [70], by (Meta) 
  69] y >= y  by (Var) 
  70] x >= x  by (Var) 
  71] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= /\z./\u.X(z, u)  because [66], by (Select) 
  72] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= X(Y, Z)  because [73], by (Select) 
  73] X(foldl*(/\z./\u.X(z, u), Y, cons(Z, U)), foldl*(/\v./\w.X(v, w), Y, cons(Z, U))) >= X(Y, Z)  because [74] and [75], by (Meta) 
  74] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Y  because [16], by (Select) 
  75] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= Z  because [18], by (Select) 
  76] foldl*(/\z./\u.X(z, u), Y, cons(Z, U)) >= U  because [22], by (Select) 

  77] cons(X, Y) >= cons(X, Y)  because cons in Mul, [78] and [79], by (Fun) 
  78] X >= X  by (Meta) 
  79] Y >= Y  by (Meta) 

  80] #argfun-reverse#(foldl(/\x./\y.iconsc-(x, y), nil, X)) >= foldl(/\x./\y.iconsc-(x, y), nil, X)  because [81], by (Star) 
  81] #argfun-reverse#*(foldl(/\x./\y.iconsc-(x, y), nil, X)) >= foldl(/\x./\y.iconsc-(x, y), nil, X)  because [82], by (Select) 
  82] foldl(/\x./\y.iconsc-(x, y), nil, X) >= foldl(/\x./\y.iconsc-(x, y), nil, X)  because [32], [37] and [38], by (Fun) 

  83] #argfun-reverse1#(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)  because [84], by (Star) 
  84] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)  because [85], by (Select) 
  85] foldl(/\x./\y.app-(cons-(y, nil), x), nil, X) >= foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)  because [40], [37] and [46], by (Fun) 

  86] app-(X, Y) >= app(X, Y)  because app- = app, app- in Mul, [87] and [88], by (Fun) 
  87] X >= X  by (Meta) 
  88] Y >= Y  by (Meta) 

  89] cons-(X, Y) >= cons(X, Y)  because [90], by (Star) 
  90] cons-*(X, Y) >= cons(X, Y)  because cons- > cons, [91] and [93], by (Copy) 
  91] cons-*(X, Y) >= X  because [92], by (Select) 
  92] X >= X  by (Meta) 
  93] cons-*(X, Y) >= Y  because [94], by (Select) 
  94] Y >= Y  by (Meta) 

  95] iconsc-(X, Y) >= cons(Y, X)  because [96], by (Star) 
  96] iconsc-*(X, Y) >= cons(Y, X)  because iconsc- > cons, [97] and [99], by (Copy) 
  97] iconsc-*(X, Y) >= Y  because [98], by (Select) 
  98] Y >= Y  by (Meta) 
  99] iconsc-*(X, Y) >= X  because [100], by (Select) 
  100] 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, formative) by (P_2, R_0, minimal, formative), where P_2 consists of:

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

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 [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:

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

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

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

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

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

  foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > X(_|_, _|_) 
  #argfun-reverse##(foldl#(/\x./\y.iconsc-(x, y), _|_, X)) >= foldl#(/\x./\y.iconsc-(x, y), _|_, X) 
  #argfun-reverse1##(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X) 
  app(_|_, X) >= X 
  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
  foldl(/\x./\y.X(x, y), Y, _|_) >= Y 
  foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U) 
  #argfun-iconsc#(cons(X, Y)) >= cons(X, Y) 
  foldl(/\x./\y.iconsc-(x, y), _|_, X) >= foldl(/\x./\y.iconsc-(x, y), _|_, X) 
  #argfun-reverse1#(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X) 
  app-(X, Y) >= app(X, Y) 
  cons-(X, Y) >= cons(X, Y) 
  iconsc-(X, Y) >= #argfun-iconsc#(cons(Y, X)) 

With these choices, we have:

  1] foldl#(/\x./\y.X(x, y), Y, cons(Z, U)) > X(_|_, _|_)  because [2], by definition 
  2] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(_|_, _|_)  because [3], by (Select) 
  3] X(foldl#*(/\x./\y.X(x, y), Y, cons Z U), foldl#*(/\z./\u.X(z, u), Y, cons)) >= X(_|_, _|_)  because [4] and [5], by (Meta) 
  4] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= _|_  by (Bot) 
  5] foldl#*(/\x./\y.X(x, y), Y, cons(Z, U)) >= _|_  by (Bot) 

  6] #argfun-reverse##(foldl#(/\x./\y.iconsc-(x, y), _|_, X)) >= foldl#(/\x./\y.iconsc-(x, y), _|_, X)  because [7], by (Star) 
  7] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), _|_, X)) >= foldl#(/\x./\y.iconsc-(x, y), _|_, X)  because #argfun-reverse## > foldl#, [8], [15] and [16], by (Copy) 
  8] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), _|_, X)) >= /\x./\y.iconsc-(x, y)  because [9], by (F-Abs) 
  9] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), _|_, X), z) >= /\x.iconsc-(z, x)  because [10], by (F-Abs) 
  10] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), _|_, X), z, u) >= iconsc-(z, u)  because #argfun-reverse## > iconsc-, [11] and [13], by (Copy) 
  11] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), _|_, X), z, u) >= z  because [12], by (Select) 
  12] z >= z  by (Var) 
  13] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), _|_, X), z, u) >= u  because [14], by (Select) 
  14] u >= u  by (Var) 
  15] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), _|_, X)) >= _|_  by (Bot) 
  16] #argfun-reverse##*(foldl#(/\x./\y.iconsc-(x, y), _|_, X)) >= X  because [17], by (Select) 
  17] foldl#(/\x./\y.iconsc-(x, y), _|_, X) >= X  because [18], by (Star) 
  18] foldl#*(/\x./\y.iconsc-(x, y), _|_, X) >= X  because [19], by (Select) 
  19] X >= X  by (Meta) 

  20] #argfun-reverse1##(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X)  because [21], by (Star) 
  21] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X)  because #argfun-reverse1## > foldl#, [22], [31] and [32], by (Copy) 
  22] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= /\x./\y.app-(cons-(y, _|_), x)  because [23], by (F-Abs) 
  23] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z) >= /\x.app-(cons-(x, _|_), z)  because [24], by (F-Abs) 
  24] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= app-(cons-(u, _|_), z)  because #argfun-reverse1## > app-, [25] and [29], by (Copy) 
  25] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= cons-(u, _|_)  because #argfun-reverse1## > cons-, [26] and [28], by (Copy) 
  26] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= u  because [27], by (Select) 
  27] u >= u  by (Var) 
  28] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= _|_  by (Bot) 
  29] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= z  because [30], by (Select) 
  30] z >= z  by (Var) 
  31] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= _|_  by (Bot) 
  32] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= X  because [33], by (Select) 
  33] foldl#(/\x./\y.app-(cons-(y, _|_), x), _|_, X) >= X  because [34], by (Star) 
  34] foldl#*(/\x./\y.app-(cons-(y, _|_), x), _|_, X) >= X  because [35], by (Select) 
  35] X >= X  by (Meta) 

  36] app(_|_, X) >= X  because [37], by (Star) 
  37] app*(_|_, X) >= X  because [38], by (Select) 
  38] X >= X  by (Meta) 

  39] app(cons(X, Y), Z) >= cons(X, app(Y, Z))  because [40], by (Star) 
  40] app*(cons(X, Y), Z) >= cons(X, app(Y, Z))  because app > cons, [41] and [45], by (Copy) 
  41] app*(cons(X, Y), Z) >= X  because [42], by (Select) 
  42] cons(X, Y) >= X  because [43], by (Star) 
  43] cons*(X, Y) >= X  because [44], by (Select) 
  44] X >= X  by (Meta) 
  45] app*(cons(X, Y), Z) >= app(Y, Z)  because app in Mul, [46] and [49], by (Stat) 
  46] cons(X, Y) > Y  because [47], by definition 
  47] cons*(X, Y) >= Y  because [48], by (Select) 
  48] Y >= Y  by (Meta) 
  49] Z >= Z  by (Meta) 

  50] foldl(/\x./\y.X(x, y), Y, _|_) >= Y  because [51], by (Star) 
  51] foldl*(/\x./\y.X(x, y), Y, _|_) >= Y  because [52], by (Select) 
  52] Y >= Y  by (Meta) 

  53] foldl(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U)  because [54], by (Star) 
  54] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldl(/\x./\y.X(x, y), X(Y, Z), U)  because [55], [58], [66] and [74], by (Stat) 
  55] cons(Z, U) > U  because [56], by definition 
  56] cons*(Z, U) >= U  because [57], by (Select) 
  57] U >= U  by (Meta) 
  58] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= /\x./\y.X(x, y)  because [59], by (F-Abs) 
  59] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z) >= /\x.X(z, x)  because [60], by (F-Abs) 
  60] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z, u) >= X(z, u)  because [61], by (Select) 
  61] X(foldl*(/\x./\y.X(x, y), Y, cons Z U, z, u), foldl*(/\v./\w.X(v, w), Y, cons, z, u)) >= X(z, u)  because [62] and [64], by (Meta) 
  62] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z, u) >= z  because [63], by (Select) 
  63] z >= z  by (Var) 
  64] foldl*(/\x./\y.X(x, y), Y, cons(Z, U), z, u) >= u  because [65], by (Select) 
  65] u >= u  by (Var) 
  66] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(Y, Z)  because [67], by (Select) 
  67] X(foldl*(/\x./\y.X(x, y), Y, cons Z U), foldl*(/\v./\w.X(v, w), Y, cons)) >= X(Y, Z)  because [68] and [70], by (Meta) 
  68] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Y  because [69], by (Select) 
  69] Y >= Y  by (Meta) 
  70] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Z  because [71], by (Select) 
  71] cons(Z, U) >= Z  because [72], by (Star) 
  72] cons*(Z, U) >= Z  because [73], by (Select) 
  73] Z >= Z  by (Meta) 
  74] foldl*(/\x./\y.X(x, y), Y, cons(Z, U)) >= U  because [75], by (Select) 
  75] cons(Z, U) >= U  because [56], by (Star) 

  76] #argfun-iconsc#(cons(X, Y)) >= cons(X, Y)  because [77], by (Star) 
  77] #argfun-iconsc#*(cons(X, Y)) >= cons(X, Y)  because #argfun-iconsc# = cons, [78], [81] and [85], by (Stat) 
  78] cons(X, Y) > Y  because [79], by definition 
  79] cons*(X, Y) >= Y  because [80], by (Select) 
  80] Y >= Y  by (Meta) 
  81] #argfun-iconsc#*(cons(X, Y)) >= X  because [82], by (Select) 
  82] cons(X, Y) >= X  because [83], by (Star) 
  83] cons*(X, Y) >= X  because [84], by (Select) 
  84] X >= X  by (Meta) 
  85] #argfun-iconsc#*(cons(X, Y)) >= Y  because [86], by (Select) 
  86] cons(X, Y) >= Y  because [79], by (Star) 

  87] foldl(/\x./\y.iconsc-(x, y), _|_, X) >= foldl(/\x./\y.iconsc-(x, y), _|_, X)  because [88], [93] and [94], by (Fun) 
  88] /\x./\z.iconsc-(x, z) >= /\x./\z.iconsc-(x, z)  because [89], by (Abs) 
  89] /\z.iconsc-(y, z) >= /\z.iconsc-(y, z)  because [90], by (Abs) 
  90] iconsc-(y, x) >= iconsc-(y, x)  because iconsc- in Mul, [91] and [92], by (Fun) 
  91] y >= y  by (Var) 
  92] x >= x  by (Var) 
  93] _|_ >= _|_  by (Bot) 
  94] X >= X  by (Meta) 

  95] #argfun-reverse1#(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X)  because [96], by (Star) 
  96] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X)  because #argfun-reverse1# > foldl, [97], [106] and [107], by (Copy) 
  97] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= /\x./\y.app-(cons-(y, _|_), x)  because [98], by (F-Abs) 
  98] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z) >= /\x.app-(cons-(x, _|_), z)  because [99], by (F-Abs) 
  99] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= app-(cons-(u, _|_), z)  because #argfun-reverse1# > app-, [100] and [104], by (Copy) 
  100] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= cons-(u, _|_)  because #argfun-reverse1# > cons-, [101] and [103], by (Copy) 
  101] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= u  because [102], by (Select) 
  102] u >= u  by (Var) 
  103] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= _|_  by (Bot) 
  104] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X), z, u) >= z  because [105], by (Select) 
  105] z >= z  by (Var) 
  106] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= _|_  by (Bot) 
  107] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X)) >= X  because [108], by (Select) 
  108] foldl(/\x./\y.app-(cons-(y, _|_), x), _|_, X) >= X  because [109], by (Star) 
  109] foldl*(/\x./\y.app-(cons-(y, _|_), x), _|_, X) >= X  because [35], by (Select) 

  110] app-(X, Y) >= app(X, Y)  because [111], by (Star) 
  111] app-*(X, Y) >= app(X, Y)  because app- > app, [112] and [114], by (Copy) 
  112] app-*(X, Y) >= X  because [113], by (Select) 
  113] X >= X  by (Meta) 
  114] app-*(X, Y) >= Y  because [115], by (Select) 
  115] Y >= Y  by (Meta) 

  116] cons-(X, Y) >= cons(X, Y)  because [117], by (Star) 
  117] cons-*(X, Y) >= cons(X, Y)  because cons- > cons, [118] and [120], by (Copy) 
  118] cons-*(X, Y) >= X  because [119], by (Select) 
  119] X >= X  by (Meta) 
  120] cons-*(X, Y) >= Y  because [121], by (Select) 
  121] Y >= Y  by (Meta) 

  122] iconsc-(X, Y) >= #argfun-iconsc#(cons(Y, X))  because [123], by (Star) 
  123] iconsc-*(X, Y) >= #argfun-iconsc#(cons(Y, X))  because iconsc- > #argfun-iconsc# and [124], by (Copy) 
  124] iconsc-*(X, Y) >= cons(Y, X)  because iconsc- > cons, [125] and [127], by (Copy) 
  125] iconsc-*(X, Y) >= Y  because [126], by (Select) 
  126] Y >= Y  by (Meta) 
  127] iconsc-*(X, Y) >= X  because [128], by (Select) 
  128] 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, formative) by (P_3, R_0, minimal, formative), where P_3 consists of:

  reverse#(X) =#> foldl#(/\x./\y.iconsc(x, y), nil, X)   
  reverse1#(X) =#> foldl#(/\x./\y.app(cons(y, nil), x), nil, X)   

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

We consider the dependency pair problem (P_3, 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 :  
    * 1 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

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.