We consider the system Applicative_05__Ex5Sorting.

  Alphabet:

    0 : [] --> a 
    ascending!fac6220sort : [b] --> b 
    cons : [a * b] --> b 
    descending!fac6220sort : [b] --> b 
    insert : [a -> a -> a * a -> a -> a * b * a] --> b 
    max : [] --> a -> a -> a 
    min : [] --> a -> a -> a 
    nil : [] --> b 
    s : [a] --> a 
    sort : [a -> a -> a * a -> a -> a * b] --> b 

  Rules:

    max 0 x => x 
    max x 0 => x 
    max s(x) s(y) => max x y 
    min 0 x => 0 
    min x 0 => 0 
    min s(x) s(y) => min x y 
    insert(f, g, nil, x) => cons(x, nil) 
    insert(f, g, cons(x, y), z) => cons(f z x, insert(f, g, y, g z x)) 
    sort(f, g, nil) => nil 
    sort(f, g, cons(x, y)) => insert(f, g, sort(f, g, y), x) 
    ascending!fac6220sort(x) => sort(min, max, x) 
    descending!fac6220sort(x) => sort(max, min, 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:

  max 0 X => X 
  max X 0 => X 
  max s(X) s(Y) => max X Y 
  min 0 X => 0 
  min X 0 => 0 
  min s(X) s(Y) => min X Y 

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:
 || 
 ||    max(0, %X) -> %X
 ||    max(%X, 0) -> %X
 ||    max(s(%X), s(%Y)) -> max(%X, %Y)
 ||    min(0, %X) -> 0
 ||    min(%X, 0) -> 0
 ||    min(s(%X), s(%Y)) -> min(%X, %Y)
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(0) = 0
 ||    POL(max(x_1, x_2)) = 2 + 2*x_1 + x_2
 ||    POL(min(x_1, x_2)) = 1 + x_1 + x_2
 ||    POL(s(x_1)) = 1 + 2*x_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:
 || 
 ||    max(0, %X) -> %X
 ||    max(%X, 0) -> %X
 ||    max(s(%X), s(%Y)) -> max(%X, %Y)
 ||    min(0, %X) -> 0
 ||    min(%X, 0) -> 0
 ||    min(s(%X), s(%Y)) -> min(%X, %Y)
 ||    ~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, formative):

  Dependency Pairs P_0:

    0] insert#(F, G, cons(X, Y), Z) =#> F(Z, X)   
    1] insert#(F, G, cons(X, Y), Z) =#> insert#(F, G, Y, G Z X)   
    2] insert#(F, G, cons(X, Y), Z) =#> G(Z, X)   
    3] sort#(F, G, cons(X, Y)) =#> insert#(F, G, sort(F, G, Y), X)   
    4] sort#(F, G, cons(X, Y)) =#> sort#(F, G, Y)   
    5] ascending!fac6220sort#(X) =#> sort#(min, max, X)   
    6] ascending!fac6220sort#(X) =#> min#   
    7] ascending!fac6220sort#(X) =#> max#   
    8] descending!fac6220sort#(X) =#> sort#(max, min, X)   
    9] descending!fac6220sort#(X) =#> max#   
    10] descending!fac6220sort#(X) =#> min#   

  Rules R_0:

    max 0 X => X 
    max X 0 => X 
    max s(X) s(Y) => max X Y 
    min 0 X => 0 
    min X 0 => 0 
    min s(X) s(Y) => min X Y 
    insert(F, G, nil, X) => cons(X, nil) 
    insert(F, G, cons(X, Y), Z) => cons(F Z X, insert(F, G, Y, G Z X)) 
    sort(F, G, nil) => nil 
    sort(F, G, cons(X, Y)) => insert(F, G, sort(F, G, Y), X) 
    ascending!fac6220sort(X) => sort(min, max, X) 
    descending!fac6220sort(X) => sort(max, min, 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, 3, 4, 5, 6, 7, 8, 9, 10 
    * 1 : 0, 1, 2 
    * 2 : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 
    * 3 : 0, 1, 2 
    * 4 : 3, 4 
    * 5 : 3, 4 
    * 6 :  
    * 7 :  
    * 8 : 3, 4 
    * 9 :  
    * 10 :  

This graph has the following strongly connected components:

  P_1:

    insert#(F, G, cons(X, Y), Z) =#> F(Z, X)   
    insert#(F, G, cons(X, Y), Z) =#> insert#(F, G, Y, G Z X)   
    insert#(F, G, cons(X, Y), Z) =#> G(Z, X)   
    sort#(F, G, cons(X, Y)) =#> insert#(F, G, sort(F, G, Y), X)   
    sort#(F, G, cons(X, Y)) =#> sort#(F, G, Y)   
    ascending!fac6220sort#(X) =#> sort#(min, max, X)   
    descending!fac6220sort#(X) =#> sort#(max, min, 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 ::=

  insert(F, G, nil, X) => cons(X, nil) 
  insert(F, G, cons(X, Y), Z) => cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) => nil 
  sort(F, G, cons(X, Y)) => insert(F, G, sort(F, G, Y), X) 
  ascending!fac6220sort(X) => sort(min, max, X) 
  descending!fac6220sort(X) => sort(max, min, 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:

  insert#(F, G, cons(X, Y), Z) >? F(Z, X) 
  insert#(F, G, cons(X, Y), Z) >? insert#(F, G, Y, G Z X) 
  insert#(F, G, cons(X, Y), Z) >? G(Z, X) 
  sort#(F, G, cons(X, Y)) >? insert#(F, G, sort(F, G, Y), X) 
  sort#(F, G, cons(X, Y)) >? sort#(F, G, Y) 
  ascending!fac6220sort#(X) >? sort#(min, max, X) 
  descending!fac6220sort#(X) >? sort#(max, min, X) 
  insert(F, G, nil, X) >= cons(X, nil) 
  insert(F, G, cons(X, Y), Z) >= cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) >= nil 
  sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X) 
  ascending!fac6220sort(X) >= sort(min, max, X) 
  descending!fac6220sort(X) >= sort(max, min, X) 

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

  pi( ascending!fac6220sort(X) ) = #argfun-ascending!fac6220sort#(sort(min, max, X)) 
  pi( ascending!fac6220sort#(X) ) = #argfun-ascending!fac6220sort##(sort#(min, max, X)) 
  pi( descending!fac6220sort(X) ) = #argfun-descending!fac6220sort#(sort(max, min, X)) 
  pi( descending!fac6220sort#(X) ) = #argfun-descending!fac6220sort##(sort#(max, min, 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:

  [[ascending!fac6220sort(x_1)]] = x_1 
  [[ascending!fac6220sort#(x_1)]] = x_1 
  [[descending!fac6220sort(x_1)]] = x_1 
  [[descending!fac6220sort#(x_1)]] = x_1 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_3, x_1, x_2, x_4) 
  [[insert#(x_1, x_2, x_3, x_4)]] = insert#(x_1, x_3, x_2, x_4) 
  [[max]] = _|_ 
  [[min]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {insert, insert#} and Mul = {#argfun-ascending!fac6220sort#, #argfun-ascending!fac6220sort##, #argfun-descending!fac6220sort#, #argfun-descending!fac6220sort##, @_{o -> o -> o}, @_{o -> o}, cons, sort, sort#}, and the following precedence: #argfun-ascending!fac6220sort# > #argfun-ascending!fac6220sort## > #argfun-descending!fac6220sort# > #argfun-descending!fac6220sort## > sort = sort# > insert > cons > insert# > @_{o -> o -> o} > @_{o -> o}

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

  insert#(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X) 
  insert#(F, G, cons(X, Y), Z) >= insert#(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)) 
  insert#(F, G, cons(X, Y), Z) > @_{o -> o}(@_{o -> o -> o}(G, Z), X) 
  sort#(F, G, cons(X, Y)) >= insert#(F, G, sort(F, G, Y), X) 
  sort#(F, G, cons(X, Y)) >= sort#(F, G, Y) 
  #argfun-ascending!fac6220sort##(sort#(_|_, _|_, X)) >= sort#(_|_, _|_, X) 
  #argfun-descending!fac6220sort##(sort#(_|_, _|_, X)) >= sort#(_|_, _|_, X) 
  insert(F, G, _|_, X) >= cons(X, _|_) 
  insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))) 
  sort(F, G, _|_) >= _|_ 
  sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X) 
  #argfun-ascending!fac6220sort#(sort(_|_, _|_, X)) >= sort(_|_, _|_, X) 
  #argfun-descending!fac6220sort#(sort(_|_, _|_, X)) >= sort(_|_, _|_, X) 

With these choices, we have:

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

  12] insert#(F, G, cons(X, Y), Z) >= insert#(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [13], by (Star) 
  13] insert#*(F, G, cons(X, Y), Z) >= insert#(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [14], [15], [4], [18], [20] and [22], by (Stat) 
  14] F >= F  by (Meta) 
  15] cons(X, Y) > Y  because [16], by definition 
  16] cons*(X, Y) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 
  18] insert#*(F, G, cons(X, Y), Z) >= G  because [19], by (Select) 
  19] G >= G  by (Meta) 
  20] insert#*(F, G, cons(X, Y), Z) >= Y  because [21], by (Select) 
  21] cons(X, Y) >= Y  because [16], by (Star) 
  22] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert# > @_{o -> o}, [23] and [8], by (Copy) 
  23] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert# > @_{o -> o -> o}, [18] and [6], by (Copy) 

  24] insert#(F, G, cons(X, Y), Z) > @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because [25], by definition 
  25] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert# > @_{o -> o}, [23] and [8], by (Copy) 

  26] sort#(F, G, cons(X, Y)) >= insert#(F, G, sort(F, G, Y), X)  because [27], by (Star) 
  27] sort#*(F, G, cons(X, Y)) >= insert#(F, G, sort(F, G, Y), X)  because sort# > insert#, [28], [30], [32] and [38], by (Copy) 
  28] sort#*(F, G, cons(X, Y)) >= F  because [29], by (Select) 
  29] F >= F  by (Meta) 
  30] sort#*(F, G, cons(X, Y)) >= G  because [31], by (Select) 
  31] G >= G  by (Meta) 
  32] sort#*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort# = sort, sort# in Mul, [33], [34] and [35], by (Stat) 
  33] F >= F  by (Meta) 
  34] G >= G  by (Meta) 
  35] cons(X, Y) > Y  because [36], by definition 
  36] cons*(X, Y) >= Y  because [37], by (Select) 
  37] Y >= Y  by (Meta) 
  38] sort#*(F, G, cons(X, Y)) >= X  because [39], by (Select) 
  39] cons(X, Y) >= X  because [40], by (Star) 
  40] cons*(X, Y) >= X  because [41], by (Select) 
  41] X >= X  by (Meta) 

  42] sort#(F, G, cons(X, Y)) >= sort#(F, G, Y)  because sort# in Mul, [33], [34] and [43], by (Fun) 
  43] cons(X, Y) >= Y  because [36], by (Star) 

  44] #argfun-ascending!fac6220sort##(sort#(_|_, _|_, X)) >= sort#(_|_, _|_, X)  because [45], by (Star) 
  45] #argfun-ascending!fac6220sort##*(sort#(_|_, _|_, X)) >= sort#(_|_, _|_, X)  because #argfun-ascending!fac6220sort## > sort#, [46], [47] and [48], by (Copy) 
  46] #argfun-ascending!fac6220sort##*(sort#(_|_, _|_, X)) >= _|_  by (Bot) 
  47] #argfun-ascending!fac6220sort##*(sort#(_|_, _|_, X)) >= _|_  by (Bot) 
  48] #argfun-ascending!fac6220sort##*(sort#(_|_, _|_, X)) >= X  because [49], by (Select) 
  49] sort#(_|_, _|_, X) >= X  because [50], by (Star) 
  50] sort#*(_|_, _|_, X) >= X  because [51], by (Select) 
  51] X >= X  by (Meta) 

  52] #argfun-descending!fac6220sort##(sort#(_|_, _|_, X)) >= sort#(_|_, _|_, X)  because [53], by (Star) 
  53] #argfun-descending!fac6220sort##*(sort#(_|_, _|_, X)) >= sort#(_|_, _|_, X)  because [54], by (Select) 
  54] sort#(_|_, _|_, X) >= sort#(_|_, _|_, X)  because sort# in Mul, [55], [56] and [57], by (Fun) 
  55] _|_ >= _|_  by (Bot) 
  56] _|_ >= _|_  by (Bot) 
  57] X >= X  by (Meta) 

  58] insert(F, G, _|_, X) >= cons(X, _|_)  because [59], by (Star) 
  59] insert*(F, G, _|_, X) >= cons(X, _|_)  because insert > cons, [60] and [62], by (Copy) 
  60] insert*(F, G, _|_, X) >= X  because [61], by (Select) 
  61] X >= X  by (Meta) 
  62] insert*(F, G, _|_, X) >= _|_  by (Bot) 

  63] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because [64], by (Star) 
  64] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because insert > cons, [65] and [70], by (Copy) 
  65] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [66] and [69], by (Copy) 
  66] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [67] and [68], by (Copy) 
  67] insert*(F, G, cons(X, Y), Z) >= F  because [14], by (Select) 
  68] insert*(F, G, cons(X, Y), Z) >= Z  because [7], by (Select) 
  69] insert*(F, G, cons(X, Y), Z) >= X  because [9], by (Select) 
  70] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [15], [67], [71], [72] and [73], by (Stat) 
  71] insert*(F, G, cons(X, Y), Z) >= G  because [19], by (Select) 
  72] insert*(F, G, cons(X, Y), Z) >= Y  because [21], by (Select) 
  73] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [74] and [69], by (Copy) 
  74] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [71] and [68], by (Copy) 

  75] sort(F, G, _|_) >= _|_  by (Bot) 

  76] sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because [77], by (Star) 
  77] sort*(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because sort > insert, [78], [79], [80] and [81], by (Copy) 
  78] sort*(F, G, cons(X, Y)) >= F  because [33], by (Select) 
  79] sort*(F, G, cons(X, Y)) >= G  because [34], by (Select) 
  80] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [33], [34] and [35], by (Stat) 
  81] sort*(F, G, cons(X, Y)) >= X  because [39], by (Select) 

  82] #argfun-ascending!fac6220sort#(sort(_|_, _|_, X)) >= sort(_|_, _|_, X)  because [83], by (Star) 
  83] #argfun-ascending!fac6220sort#*(sort(_|_, _|_, X)) >= sort(_|_, _|_, X)  because #argfun-ascending!fac6220sort# > sort, [84], [85] and [86], by (Copy) 
  84] #argfun-ascending!fac6220sort#*(sort(_|_, _|_, X)) >= _|_  by (Bot) 
  85] #argfun-ascending!fac6220sort#*(sort(_|_, _|_, X)) >= _|_  by (Bot) 
  86] #argfun-ascending!fac6220sort#*(sort(_|_, _|_, X)) >= X  because [87], by (Select) 
  87] sort(_|_, _|_, X) >= X  because [88], by (Star) 
  88] sort*(_|_, _|_, X) >= X  because [51], by (Select) 

  89] #argfun-descending!fac6220sort#(sort(_|_, _|_, X)) >= sort(_|_, _|_, X)  because [90], by (Star) 
  90] #argfun-descending!fac6220sort#*(sort(_|_, _|_, X)) >= sort(_|_, _|_, X)  because #argfun-descending!fac6220sort# > sort, [91], [92] and [93], by (Copy) 
  91] #argfun-descending!fac6220sort#*(sort(_|_, _|_, X)) >= _|_  by (Bot) 
  92] #argfun-descending!fac6220sort#*(sort(_|_, _|_, X)) >= _|_  by (Bot) 
  93] #argfun-descending!fac6220sort#*(sort(_|_, _|_, X)) >= X  because [94], by (Select) 
  94] sort(_|_, _|_, X) >= X  because [95], by (Star) 
  95] sort*(_|_, _|_, X) >= X  because [57], by (Select) 

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:

  insert#(F, G, cons(X, Y), Z) =#> F(Z, X)   
  insert#(F, G, cons(X, Y), Z) =#> insert#(F, G, Y, G Z X)   
  sort#(F, G, cons(X, Y)) =#> insert#(F, G, sort(F, G, Y), X)   
  sort#(F, G, cons(X, Y)) =#> sort#(F, G, Y)   
  ascending!fac6220sort#(X) =#> sort#(min, max, X)   
  descending!fac6220sort#(X) =#> sort#(max, min, 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:

  insert#(F, G, cons(X, Y), Z) >? F(Z, X) 
  insert#(F, G, cons(X, Y), Z) >? insert#(F, G, Y, G Z X) 
  sort#(F, G, cons(X, Y)) >? insert#(F, G, sort(F, G, Y), X) 
  sort#(F, G, cons(X, Y)) >? sort#(F, G, Y) 
  ascending!fac6220sort#(X) >? sort#(min, max, X) 
  descending!fac6220sort#(X) >? sort#(max, min, X) 
  insert(F, G, nil, X) >= cons(X, nil) 
  insert(F, G, cons(X, Y), Z) >= cons(F Z X, insert(F, G, Y, G Z X)) 
  sort(F, G, nil) >= nil 
  sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X) 
  ascending!fac6220sort(X) >= sort(min, max, X) 
  descending!fac6220sort(X) >= sort(max, min, X) 

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

  pi( ascending!fac6220sort(X) ) = #argfun-ascending!fac6220sort#(sort(min, max, X)) 
  pi( ascending!fac6220sort#(X) ) = #argfun-ascending!fac6220sort##(sort#(min, max, X)) 
  pi( descending!fac6220sort(X) ) = #argfun-descending!fac6220sort#(sort(max, min, X)) 
  pi( descending!fac6220sort#(X) ) = #argfun-descending!fac6220sort##(sort#(max, min, 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-descending!fac6220sort#(x_1)]] = x_1 
  [[ascending!fac6220sort(x_1)]] = x_1 
  [[ascending!fac6220sort#(x_1)]] = x_1 
  [[descending!fac6220sort(x_1)]] = x_1 
  [[descending!fac6220sort#(x_1)]] = x_1 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_2, x_1, x_3, x_4) 
  [[insert#(x_1, x_2, x_3, x_4)]] = insert#(x_1, x_3, x_4, x_2) 
  [[nil]] = _|_ 

We choose Lex = {insert, insert#} and Mul = {#argfun-ascending!fac6220sort#, #argfun-ascending!fac6220sort##, #argfun-descending!fac6220sort##, @_{o -> o -> o}, @_{o -> o}, cons, max, min, sort, sort#}, and the following precedence: #argfun-ascending!fac6220sort# > #argfun-ascending!fac6220sort## > #argfun-descending!fac6220sort## > max = min > sort# > insert# > sort > insert > @_{o -> o -> o} > @_{o -> o} > cons

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

  insert#(F, G, cons(X, Y), Z) > @_{o -> o}(@_{o -> o -> o}(F, Z), X) 
  insert#(F, G, cons(X, Y), Z) >= insert#(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)) 
  sort#(F, G, cons(X, Y)) >= insert#(F, G, sort(F, G, Y), X) 
  sort#(F, G, cons(X, Y)) >= sort#(F, G, Y) 
  #argfun-ascending!fac6220sort##(sort#(min, max, X)) >= sort#(min, max, X) 
  #argfun-descending!fac6220sort##(sort#(max, min, X)) >= sort#(max, min, X) 
  insert(F, G, _|_, X) >= cons(X, _|_) 
  insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))) 
  sort(F, G, _|_) >= _|_ 
  sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X) 
  #argfun-ascending!fac6220sort#(sort(min, max, X)) >= sort(min, max, X) 
  sort(max, min, X) >= sort(max, min, X) 

With these choices, we have:

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

  12] insert#(F, G, cons(X, Y), Z) >= insert#(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [13], by (Star) 
  13] insert#*(F, G, cons(X, Y), Z) >= insert#(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [14], [15], [4], [18], [20] and [22], by (Stat) 
  14] F >= F  by (Meta) 
  15] cons(X, Y) > Y  because [16], by definition 
  16] cons*(X, Y) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 
  18] insert#*(F, G, cons(X, Y), Z) >= G  because [19], by (Select) 
  19] G >= G  by (Meta) 
  20] insert#*(F, G, cons(X, Y), Z) >= Y  because [21], by (Select) 
  21] cons(X, Y) >= Y  because [16], by (Star) 
  22] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert# > @_{o -> o}, [23] and [8], by (Copy) 
  23] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert# > @_{o -> o -> o}, [18] and [6], by (Copy) 

  24] sort#(F, G, cons(X, Y)) >= insert#(F, G, sort(F, G, Y), X)  because [25], by (Star) 
  25] sort#*(F, G, cons(X, Y)) >= insert#(F, G, sort(F, G, Y), X)  because sort# > insert#, [26], [28], [30] and [35], by (Copy) 
  26] sort#*(F, G, cons(X, Y)) >= F  because [27], by (Select) 
  27] F >= F  by (Meta) 
  28] sort#*(F, G, cons(X, Y)) >= G  because [29], by (Select) 
  29] G >= G  by (Meta) 
  30] sort#*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort# > sort, [26], [28] and [31], by (Copy) 
  31] sort#*(F, G, cons(X, Y)) >= Y  because [32], by (Select) 
  32] cons(X, Y) >= Y  because [33], by (Star) 
  33] cons*(X, Y) >= Y  because [34], by (Select) 
  34] Y >= Y  by (Meta) 
  35] sort#*(F, G, cons(X, Y)) >= X  because [36], by (Select) 
  36] cons(X, Y) >= X  because [37], by (Star) 
  37] cons*(X, Y) >= X  because [38], by (Select) 
  38] X >= X  by (Meta) 

  39] sort#(F, G, cons(X, Y)) >= sort#(F, G, Y)  because [40], by (Star) 
  40] sort#*(F, G, cons(X, Y)) >= sort#(F, G, Y)  because sort# in Mul, [41], [42] and [43], by (Stat) 
  41] F >= F  by (Meta) 
  42] G >= G  by (Meta) 
  43] cons(X, Y) > Y  because [44], by definition 
  44] cons*(X, Y) >= Y  because [34], by (Select) 

  45] #argfun-ascending!fac6220sort##(sort#(min, max, X)) >= sort#(min, max, X)  because [46], by (Star) 
  46] #argfun-ascending!fac6220sort##*(sort#(min, max, X)) >= sort#(min, max, X)  because [47], by (Select) 
  47] sort#(min, max, X) >= sort#(min, max, X)  because sort# in Mul, [48], [49] and [50], by (Fun) 
  48] min >= max  because min = max, by (Fun) 
  49] max >= min  because max = min, by (Fun) 
  50] X >= X  by (Meta) 

  51] #argfun-descending!fac6220sort##(sort#(max, min, X)) >= sort#(max, min, X)  because [52], by (Star) 
  52] #argfun-descending!fac6220sort##*(sort#(max, min, X)) >= sort#(max, min, X)  because #argfun-descending!fac6220sort## > sort#, [53], [54] and [55], by (Copy) 
  53] #argfun-descending!fac6220sort##*(sort#(max, min, X)) >= max  because #argfun-descending!fac6220sort## > max, by (Copy) 
  54] #argfun-descending!fac6220sort##*(sort#(max, min, X)) >= min  because #argfun-descending!fac6220sort## > min, by (Copy) 
  55] #argfun-descending!fac6220sort##*(sort#(max, min, X)) >= X  because [56], by (Select) 
  56] sort#(max, min, X) >= X  because [57], by (Star) 
  57] sort#*(max, min, X) >= X  because [58], by (Select) 
  58] X >= X  by (Meta) 

  59] insert(F, G, _|_, X) >= cons(X, _|_)  because [60], by (Star) 
  60] insert*(F, G, _|_, X) >= cons(X, _|_)  because insert > cons, [61] and [63], by (Copy) 
  61] insert*(F, G, _|_, X) >= X  because [62], by (Select) 
  62] X >= X  by (Meta) 
  63] insert*(F, G, _|_, X) >= _|_  by (Bot) 

  64] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because [65], by (Star) 
  65] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, Z), X), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X)))  because insert > cons, [66] and [71], by (Copy) 
  66] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, Z), X)  because insert > @_{o -> o}, [67] and [70], by (Copy) 
  67] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, Z)  because insert > @_{o -> o -> o}, [68] and [69], by (Copy) 
  68] insert*(F, G, cons(X, Y), Z) >= F  because [14], by (Select) 
  69] insert*(F, G, cons(X, Y), Z) >= Z  because [7], by (Select) 
  70] insert*(F, G, cons(X, Y), Z) >= X  because [9], by (Select) 
  71] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, Z), X))  because [14], [72], [15], [68], [73], [74] and [75], by (Stat) 
  72] G >= G  by (Meta) 
  73] insert*(F, G, cons(X, Y), Z) >= G  because [72], by (Select) 
  74] insert*(F, G, cons(X, Y), Z) >= Y  because [21], by (Select) 
  75] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, Z), X)  because insert > @_{o -> o}, [76] and [70], by (Copy) 
  76] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, Z)  because insert > @_{o -> o -> o}, [73] and [69], by (Copy) 

  77] sort(F, G, _|_) >= _|_  by (Bot) 

  78] sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because [79], by (Star) 
  79] sort*(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because sort > insert, [80], [81], [82] and [83], by (Copy) 
  80] sort*(F, G, cons(X, Y)) >= F  because [41], by (Select) 
  81] sort*(F, G, cons(X, Y)) >= G  because [42], by (Select) 
  82] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [41], [42] and [43], by (Stat) 
  83] sort*(F, G, cons(X, Y)) >= X  because [36], by (Select) 

  84] #argfun-ascending!fac6220sort#(sort(min, max, X)) >= sort(min, max, X)  because [85], by (Star) 
  85] #argfun-ascending!fac6220sort#*(sort(min, max, X)) >= sort(min, max, X)  because #argfun-ascending!fac6220sort# > sort, [86], [87] and [88], by (Copy) 
  86] #argfun-ascending!fac6220sort#*(sort(min, max, X)) >= min  because #argfun-ascending!fac6220sort# > min, by (Copy) 
  87] #argfun-ascending!fac6220sort#*(sort(min, max, X)) >= max  because #argfun-ascending!fac6220sort# > max, by (Copy) 
  88] #argfun-ascending!fac6220sort#*(sort(min, max, X)) >= X  because [89], by (Select) 
  89] sort(min, max, X) >= X  because [90], by (Star) 
  90] sort*(min, max, X) >= X  because [50], by (Select) 

  91] sort(max, min, X) >= sort(max, min, X)  because sort in Mul, [49], [48] and [92], by (Fun) 
  92] 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_1, minimal, formative) by (P_3, R_1, minimal, formative), where P_3 consists of:

  insert#(F, G, cons(X, Y), Z) =#> insert#(F, G, Y, G Z X)   
  sort#(F, G, cons(X, Y)) =#> insert#(F, G, sort(F, G, Y), X)   
  sort#(F, G, cons(X, Y)) =#> sort#(F, G, Y)   
  ascending!fac6220sort#(X) =#> sort#(min, max, X)   
  descending!fac6220sort#(X) =#> sort#(max, min, 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 : 0 
    * 1 : 0 
    * 2 : 1, 2 
    * 3 : 1, 2 
    * 4 : 1, 2 

This graph has the following strongly connected components:

  P_4:

    insert#(F, G, cons(X, Y), Z) =#> insert#(F, G, Y, G Z X)   

  P_5:

    sort#(F, G, cons(X, Y)) =#> sort#(F, G, Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_3, R_1, m, f) by (P_4, R_1, m, f) and (P_5, R_1, m, f).

Thus, the original system is terminating if each of (P_4, R_1, minimal, formative) and (P_5, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_1, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(sort#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(sort#(F, G, cons(X, Y))) = cons(X, Y) |> Y = nu(sort#(F, G, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_5, R_1, minimal, f) by ({}, R_1, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_4, R_1, minimal, formative) is finite.

We consider the dependency pair problem (P_4, R_1, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(insert#) = 3 

Thus, we can orient the dependency pairs as follows:

  nu(insert#(F, G, cons(X, Y), Z)) = cons(X, Y) |> Y = nu(insert#(F, G, Y, G Z X)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_4, R_1, minimal, f) by ({}, R_1, 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.