We consider the system reverse.

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

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

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

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

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))
 || 
 || 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 + 2*x_1 + 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))
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (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.

To start, the system is beta-saturated by adding the following rules:

  iconsc X => /\x.cons(x, X) 
  iconsc X Y => cons(Y, X) 

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

This graph has the following strongly connected components:

  P_1:

    foldl#(F, X, cons(Y, Z)) =#> foldl#(F, F X Y, Z)   
    foldl#(F, X, cons(Y, Z)) =#> F(X, Y)   
    reverse#(X) =#> foldl#(iconsc, 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).

The formative rules of (P_1, R_0) are R_1 ::=

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

By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative).

Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_1, 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:

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

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

  pi( iconsc ) = #argfun-iconsc#(/\x./\y.cons(y, x)) 
  pi( reverse(X) ) = #argfun-reverse#(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), 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#(#argfun-iconsc#(/\x./\y.cons(y, x)), 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:

  [[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 

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

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

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

With these choices, we have:

  1] foldl#(F, X, cons(Y, Z)) > foldl#(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [2], by definition 
  2] foldl#*(F, X, cons(Y, Z)) >= foldl#(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [3], [6], [8] and [16], 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#*(F, X, cons(Y, Z)) >= F  because [7], by (Select) 
  7] F >= F  by (Meta) 
  8] foldl#*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because foldl# > @_{o -> o}, [9] and [12], by (Copy) 
  9] foldl#*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, X)  because foldl# > @_{o -> o -> o}, [6] and [10], by (Copy) 
  10] foldl#*(F, X, cons(Y, Z)) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] foldl#*(F, X, cons(Y, Z)) >= Y  because [13], by (Select) 
  13] cons(Y, Z) >= Y  because [14], by (Star) 
  14] cons*(Y, Z) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 
  16] foldl#*(F, X, cons(Y, Z)) >= Z  because [17], by (Select) 
  17] cons(Y, Z) >= Z  because [4], by (Star) 

  18] foldl#(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because [8], by (Star) 

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

  34] #argfun-reverse1##(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)  because [35], by (Star) 
  35] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)  because #argfun-reverse1## > foldl#, [36], [45] and [46], by (Copy) 
  36] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= /\x./\y.app-(cons-(y, nil), x)  because [37], by (F-Abs) 
  37] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z) >= /\x.app-(cons-(x, nil), z)  because [38], by (F-Abs) 
  38] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= app-(cons-(u, nil), z)  because #argfun-reverse1## > app-, [39] and [43], by (Copy) 
  39] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= cons-(u, nil)  because #argfun-reverse1## > cons-, [40] and [42], by (Copy) 
  40] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= u  because [41], by (Select) 
  41] u >= u  by (Var) 
  42] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= nil  because #argfun-reverse1## > nil, by (Copy) 
  43] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= z  because [44], by (Select) 
  44] z >= z  by (Var) 
  45] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= nil  because #argfun-reverse1## > nil, by (Copy) 
  46] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= X  because [47], by (Select) 
  47] foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X) >= X  because [48], by (Star) 
  48] foldl#*(/\x./\y.app-(cons-(y, nil), x), nil, X) >= X  because [49], by (Select) 
  49] X >= X  by (Meta) 

  50] app(nil, X) >= X  because [51], by (Star) 
  51] app*(nil, X) >= X  because [52], by (Select) 
  52] X >= X  by (Meta) 

  53] app(cons(X, Y), Z) >= cons(X, app(Y, Z))  because [54], by (Star) 
  54] app*(cons(X, Y), Z) >= cons(X, app(Y, Z))  because app > cons, [55] and [59], by (Copy) 
  55] app*(cons(X, Y), Z) >= X  because [56], by (Select) 
  56] cons(X, Y) >= X  because [57], by (Star) 
  57] cons*(X, Y) >= X  because [58], by (Select) 
  58] X >= X  by (Meta) 
  59] app*(cons(X, Y), Z) >= app(Y, Z)  because app in Mul, [60] and [63], by (Stat) 
  60] cons(X, Y) > Y  because [61], by definition 
  61] cons*(X, Y) >= Y  because [62], by (Select) 
  62] Y >= Y  by (Meta) 
  63] Z >= Z  by (Meta) 

  64] foldl(F, X, nil) >= X  because [65], by (Star) 
  65] foldl*(F, X, nil) >= X  because [66], by (Select) 
  66] X >= X  by (Meta) 

  67] foldl(F, X, cons(Y, Z)) >= foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [68], by (Star) 
  68] foldl*(F, X, cons(Y, Z)) >= foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [69], [3], [70], [71] and [75], by (Stat) 
  69] F >= F  by (Meta) 
  70] foldl*(F, X, cons(Y, Z)) >= F  because [69], by (Select) 
  71] foldl*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because foldl > @_{o -> o}, [72] and [74], by (Copy) 
  72] foldl*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, X)  because foldl > @_{o -> o -> o}, [70] and [73], by (Copy) 
  73] foldl*(F, X, cons(Y, Z)) >= X  because [11], by (Select) 
  74] foldl*(F, X, cons(Y, Z)) >= Y  because [13], by (Select) 
  75] foldl*(F, X, cons(Y, Z)) >= Z  because [17], by (Select) 

  76] #argfun-reverse#(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)  because [77], by (Star) 
  77] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)  because #argfun-reverse# > foldl, [78], [86] and [87], by (Copy) 
  78] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= #argfun-iconsc#(/\x./\y.cons(y, x))  because #argfun-reverse# > #argfun-iconsc# and [79], by (Copy) 
  79] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= /\x./\y.cons(y, x)  because [80], by (F-Abs) 
  80] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z) >= /\x.cons(x, z)  because [81], by (F-Abs) 
  81] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= cons(u, z)  because #argfun-reverse# > cons, [82] and [84], by (Copy) 
  82] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= u  because [83], by (Select) 
  83] u >= u  by (Var) 
  84] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= z  because [85], by (Select) 
  85] z >= z  by (Var) 
  86] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= nil  because #argfun-reverse# > nil, by (Copy) 
  87] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= X  because [88], by (Select) 
  88] foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X) >= X  because [89], by (Star) 
  89] foldl*(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X) >= X  because [33], by (Select) 

  90] #argfun-reverse1#(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)  because [91], by (Star) 
  91] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)  because #argfun-reverse1# > foldl, [92], [101] and [102], by (Copy) 
  92] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= /\x./\y.app-(cons-(y, nil), x)  because [93], by (F-Abs) 
  93] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z) >= /\x.app-(cons-(x, nil), z)  because [94], by (F-Abs) 
  94] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= app-(cons-(u, nil), z)  because #argfun-reverse1# > app-, [95] and [99], by (Copy) 
  95] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= cons-(u, nil)  because #argfun-reverse1# > cons-, [96] and [98], by (Copy) 
  96] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= u  because [97], by (Select) 
  97] u >= u  by (Var) 
  98] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= nil  because #argfun-reverse1# > nil, by (Copy) 
  99] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= z  because [100], by (Select) 
  100] z >= z  by (Var) 
  101] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= nil  because #argfun-reverse1# > nil, by (Copy) 
  102] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= X  because [103], by (Select) 
  103] foldl(/\x./\y.app-(cons-(y, nil), x), nil, X) >= X  because [104], by (Star) 
  104] foldl*(/\x./\y.app-(cons-(y, nil), x), nil, X) >= X  because [49], by (Select) 

  105] @_{o -> o}(@_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X), Y) >= cons(Y, X)  because [106], by (Star) 
  106] @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X), Y) >= cons(Y, X)  because [107], by (Select) 
  107] @_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X) @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y) >= cons(Y, X)  because [108] 
  108] @_{o -> o -> o}*(#argfun-iconsc#(/\x./\y.cons(y, x)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y)) >= cons(Y, X)  because [109], by (Select) 
  109] #argfun-iconsc#(/\x./\y.cons(y, x)) @_{o -> o -> o}*(#argfun-iconsc#(/\z./\u.cons(u, z)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\v./\w.cons(w, v)), X), Y)) @_{o -> o -> o}*(#argfun-iconsc#(/\x'./\y'.cons(y', x')), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z'./\u'.cons(u', z')), X), Y)) >= cons(Y, X)  because [110] 
  110] #argfun-iconsc#*(/\x./\y.cons(y, x), @_{o -> o -> o}*(#argfun-iconsc#(/\z./\u.cons(u, z)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\v./\w.cons(w, v)), X), Y)), @_{o -> o -> o}*(#argfun-iconsc#(/\x'./\y'.cons(y', x')), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z'./\u'.cons(u', z')), X), Y))) >= cons(Y, X)  because [111], by (Select) 
  111] @_{o -> o -> o}*(#argfun-iconsc#(/\x./\y.cons(y, x)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y)) >= cons(Y, X)  because @_{o -> o -> o} > cons, [112] and [115], by (Copy) 
  112] @_{o -> o -> o}*(#argfun-iconsc#(/\x./\y.cons(y, x)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y)) >= Y  because [113], by (Select) 
  113] @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X), Y) >= Y  because [114], by (Select) 
  114] Y >= Y  by (Meta) 
  115] @_{o -> o -> o}*(#argfun-iconsc#(/\x./\y.cons(y, x)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y)) >= X  because [116], by (Select) 
  116] @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X), Y) >= X  because [117], by (Select) 
  117] @_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X) @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y) >= X  because [118] 
  118] @_{o -> o -> o}*(#argfun-iconsc#(/\x./\y.cons(y, x)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y)) >= X  because [119], by (Select) 
  119] #argfun-iconsc#(/\x./\y.cons(y, x)) @_{o -> o -> o}*(#argfun-iconsc#(/\z./\u.cons(u, z)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\v./\w.cons(w, v)), X), Y)) @_{o -> o -> o}*(#argfun-iconsc#(/\x'./\y'.cons(y', x')), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z'./\u'.cons(u', z')), X), Y)) >= X  because [120] 
  120] #argfun-iconsc#*(/\x./\y.cons(y, x), @_{o -> o -> o}*(#argfun-iconsc#(/\z./\u.cons(u, z)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\v./\w.cons(w, v)), X), Y)), @_{o -> o -> o}*(#argfun-iconsc#(/\x'./\y'.cons(y', x')), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z'./\u'.cons(u', z')), X), Y))) >= X  because [121], by (Select) 
  121] @_{o -> o -> o}*(#argfun-iconsc#(/\x./\y.cons(y, x)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y)) >= X  because [122], by (Select) 
  122] X >= X  by (Meta) 

  123] app-(X, Y) >= app(X, Y)  because [124], by (Star) 
  124] app-*(X, Y) >= app(X, Y)  because app- > app, [125] and [127], by (Copy) 
  125] app-*(X, Y) >= X  because [126], by (Select) 
  126] X >= X  by (Meta) 
  127] app-*(X, Y) >= Y  because [128], by (Select) 
  128] Y >= Y  by (Meta) 

  129] cons-(X, Y) >= cons(X, Y)  because [130], by (Star) 
  130] cons-*(X, Y) >= cons(X, Y)  because cons- > cons, [131] and [133], by (Copy) 
  131] cons-*(X, Y) >= X  because [132], by (Select) 
  132] X >= X  by (Meta) 
  133] cons-*(X, Y) >= Y  because [134], by (Select) 
  134] Y >= Y  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_1, minimal, formative) by (P_2, R_1, minimal, formative), where P_2 consists of:

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

Thus, the original system is terminating if (P_2, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_2, 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:

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

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

  pi( iconsc ) = #argfun-iconsc#(/\x./\y.cons(y, x)) 
  pi( reverse(X) ) = #argfun-reverse#(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), 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#(#argfun-iconsc#(/\x./\y.cons(y, x)), 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:

  [[foldl(x_1, x_2, x_3)]] = foldl(x_1, x_3, x_2) 
  [[reverse(x_1)]] = x_1 
  [[reverse1(x_1)]] = x_1 
  [[reverse1#(x_1)]] = x_1 
  [[reverse#(x_1)]] = x_1 

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

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

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

With these choices, we have:

  1] foldl#(F, X, cons(Y, Z)) > @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because [2], by definition 
  2] foldl#*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because foldl# > @_{o -> o}, [3] and [8], by (Copy) 
  3] foldl#*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, X)  because foldl# > @_{o -> o -> o}, [4] and [6], by (Copy) 
  4] foldl#*(F, X, cons(Y, Z)) >= F  because [5], by (Select) 
  5] F >= F  by (Meta) 
  6] foldl#*(F, X, cons(Y, Z)) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] foldl#*(F, X, cons(Y, Z)) >= Y  because [9], by (Select) 
  9] cons(Y, Z) >= Y  because [10], by (Star) 
  10] cons*(Y, Z) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 

  12] #argfun-reverse##(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)  because [13], by (Star) 
  13] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)  because #argfun-reverse## > foldl#, [14], [22] and [23], by (Copy) 
  14] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= #argfun-iconsc#(/\x./\y.cons(y, x))  because #argfun-reverse## > #argfun-iconsc# and [15], by (Copy) 
  15] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= /\x./\y.cons(y, x)  because [16], by (F-Abs) 
  16] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z) >= /\x.cons(x, z)  because [17], by (F-Abs) 
  17] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= cons(u, z)  because #argfun-reverse## > cons, [18] and [20], by (Copy) 
  18] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= u  because [19], by (Select) 
  19] u >= u  by (Var) 
  20] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= z  because [21], by (Select) 
  21] z >= z  by (Var) 
  22] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= nil  because #argfun-reverse## > nil, by (Copy) 
  23] #argfun-reverse##*(foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= X  because [24], by (Select) 
  24] foldl#(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X) >= X  because [25], by (Star) 
  25] foldl#*(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X) >= X  because [26], by (Select) 
  26] X >= X  by (Meta) 

  27] #argfun-reverse1##(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)  because [28], by (Star) 
  28] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)  because #argfun-reverse1## > foldl#, [29], [38] and [39], by (Copy) 
  29] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= /\x./\y.app-(cons-(y, nil), x)  because [30], by (F-Abs) 
  30] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z) >= /\x.app-(cons-(x, nil), z)  because [31], by (F-Abs) 
  31] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= app-(cons-(u, nil), z)  because #argfun-reverse1## > app-, [32] and [36], by (Copy) 
  32] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= cons-(u, nil)  because #argfun-reverse1## > cons-, [33] and [35], by (Copy) 
  33] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= u  because [34], by (Select) 
  34] u >= u  by (Var) 
  35] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= nil  because #argfun-reverse1## > nil, by (Copy) 
  36] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= z  because [37], by (Select) 
  37] z >= z  by (Var) 
  38] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= nil  because #argfun-reverse1## > nil, by (Copy) 
  39] #argfun-reverse1##*(foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= X  because [40], by (Select) 
  40] foldl#(/\x./\y.app-(cons-(y, nil), x), nil, X) >= X  because [41], by (Star) 
  41] foldl#*(/\x./\y.app-(cons-(y, nil), x), nil, X) >= X  because [42], by (Select) 
  42] X >= X  by (Meta) 

  43] app(nil, X) >= X  because [44], by (Star) 
  44] app*(nil, X) >= X  because [45], by (Select) 
  45] X >= X  by (Meta) 

  46] app(cons(X, Y), Z) >= cons(X, app(Y, Z))  because [47], by (Star) 
  47] app*(cons(X, Y), Z) >= cons(X, app(Y, Z))  because app > cons, [48] and [52], by (Copy) 
  48] app*(cons(X, Y), Z) >= X  because [49], by (Select) 
  49] cons(X, Y) >= X  because [50], by (Star) 
  50] cons*(X, Y) >= X  because [51], by (Select) 
  51] X >= X  by (Meta) 
  52] app*(cons(X, Y), Z) >= app(Y, Z)  because app in Mul, [53] and [56], by (Stat) 
  53] cons(X, Y) > Y  because [54], by definition 
  54] cons*(X, Y) >= Y  because [55], by (Select) 
  55] Y >= Y  by (Meta) 
  56] Z >= Z  by (Meta) 

  57] foldl(F, X, nil) >= X  because [58], by (Star) 
  58] foldl*(F, X, nil) >= X  because [59], by (Select) 
  59] X >= X  by (Meta) 

  60] foldl(F, X, cons(Y, Z)) >= foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [61], by (Star) 
  61] foldl*(F, X, cons(Y, Z)) >= foldl(F, @_{o -> o}(@_{o -> o -> o}(F, X), Y), Z)  because [62], [63], [66], [67] and [71], by (Stat) 
  62] F >= F  by (Meta) 
  63] cons(Y, Z) > Z  because [64], by definition 
  64] cons*(Y, Z) >= Z  because [65], by (Select) 
  65] Z >= Z  by (Meta) 
  66] foldl*(F, X, cons(Y, Z)) >= F  because [62], by (Select) 
  67] foldl*(F, X, cons(Y, Z)) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because foldl > @_{o -> o}, [68] and [70], by (Copy) 
  68] foldl*(F, X, cons(Y, Z)) >= @_{o -> o -> o}(F, X)  because foldl > @_{o -> o -> o}, [66] and [69], by (Copy) 
  69] foldl*(F, X, cons(Y, Z)) >= X  because [7], by (Select) 
  70] foldl*(F, X, cons(Y, Z)) >= Y  because [9], by (Select) 
  71] foldl*(F, X, cons(Y, Z)) >= Z  because [72], by (Select) 
  72] cons(Y, Z) >= Z  because [64], by (Star) 

  73] #argfun-reverse#(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)  because [74], by (Star) 
  74] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)  because #argfun-reverse# > foldl, [75], [83] and [84], by (Copy) 
  75] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= #argfun-iconsc#(/\x./\y.cons(y, x))  because #argfun-reverse# > #argfun-iconsc# and [76], by (Copy) 
  76] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= /\x./\y.cons(y, x)  because [77], by (F-Abs) 
  77] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z) >= /\x.cons(x, z)  because [78], by (F-Abs) 
  78] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= cons(u, z)  because #argfun-reverse# > cons, [79] and [81], by (Copy) 
  79] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= u  because [80], by (Select) 
  80] u >= u  by (Var) 
  81] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X), z, u) >= z  because [82], by (Select) 
  82] z >= z  by (Var) 
  83] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= nil  because #argfun-reverse# > nil, by (Copy) 
  84] #argfun-reverse#*(foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X)) >= X  because [85], by (Select) 
  85] foldl(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X) >= X  because [86], by (Star) 
  86] foldl*(#argfun-iconsc#(/\x./\y.cons(y, x)), nil, X) >= X  because [26], by (Select) 

  87] #argfun-reverse1#(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)  because [88], by (Star) 
  88] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)  because #argfun-reverse1# > foldl, [89], [98] and [99], by (Copy) 
  89] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= /\x./\y.app-(cons-(y, nil), x)  because [90], by (F-Abs) 
  90] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z) >= /\x.app-(cons-(x, nil), z)  because [91], by (F-Abs) 
  91] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= app-(cons-(u, nil), z)  because #argfun-reverse1# > app-, [92] and [96], by (Copy) 
  92] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= cons-(u, nil)  because #argfun-reverse1# > cons-, [93] and [95], by (Copy) 
  93] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= u  because [94], by (Select) 
  94] u >= u  by (Var) 
  95] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= nil  because #argfun-reverse1# > nil, by (Copy) 
  96] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X), z, u) >= z  because [97], by (Select) 
  97] z >= z  by (Var) 
  98] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= nil  because #argfun-reverse1# > nil, by (Copy) 
  99] #argfun-reverse1#*(foldl(/\x./\y.app-(cons-(y, nil), x), nil, X)) >= X  because [100], by (Select) 
  100] foldl(/\x./\y.app-(cons-(y, nil), x), nil, X) >= X  because [101], by (Star) 
  101] foldl*(/\x./\y.app-(cons-(y, nil), x), nil, X) >= X  because [42], by (Select) 

  102] @_{o -> o}(@_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X), Y) >= cons(Y, X)  because [103], by (Star) 
  103] @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X), Y) >= cons(Y, X)  because @_{o -> o} > cons, [104] and [106], by (Copy) 
  104] @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X), Y) >= Y  because [105], by (Select) 
  105] Y >= Y  by (Meta) 
  106] @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X), Y) >= X  because [107], by (Select) 
  107] @_{o -> o -> o}(#argfun-iconsc#(/\x./\y.cons(y, x)), X) @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y) >= X  because [108] 
  108] @_{o -> o -> o}*(#argfun-iconsc#(/\x./\y.cons(y, x)), X, @_{o -> o}*(@_{o -> o -> o}(#argfun-iconsc#(/\z./\u.cons(u, z)), X), Y)) >= X  because [109], by (Select) 
  109] X >= X  by (Meta) 

  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) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_1, minimal, formative) by (P_3, R_1, minimal, formative), where P_3 consists of:

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

Thus, the original system is terminating if (P_3, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_1, 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 +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.