We consider the system sort.

  Alphabet:

    0 : [] --> nat 
    ascending!6220sort : [list] --> list 
    cons : [nat * list] --> list 
    descending!6220sort : [list] --> list 
    insert : [nat * list * nat -> nat -> nat * nat -> nat -> nat] --> list 
    max : [nat * nat] --> nat 
    min : [nat * nat] --> nat 
    nil : [] --> list 
    s : [nat] --> nat 
    sort : [list * nat -> nat -> nat * nat -> nat -> nat] --> list 

  Rules:

    max(0, x) => x 
    max(x, 0) => x 
    max(s(x), s(y)) => s(max(x, y)) 
    min(0, x) => 0 
    min(x, 0) => 0 
    min(s(x), s(y)) => s(min(x, y)) 
    insert(x, nil, f, g) => cons(x, nil) 
    insert(x, cons(y, z), f, g) => cons(f x y, insert(g x y, z, f, g)) 
    sort(nil, f, g) => nil 
    sort(cons(x, y), f, g) => insert(x, sort(y, f, g), f, g) 
    ascending!6220sort(x) => sort(x, /\y./\z.min(y, z), /\u./\v.max(u, v)) 
    descending!6220sort(x) => sort(x, /\y./\z.max(y, z), /\u./\v.min(u, v)) 

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)) => s(max(X, Y)) 
  min(0, X) => 0 
  min(X, 0) => 0 
  min(s(X), s(Y)) => s(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)) -> s(max(%X, %Y))
 ||    min(0, %X) -> 0
 ||    min(%X, 0) -> 0
 ||    min(s(%X), s(%Y)) -> s(min(%X, %Y))
 ||    ~PAIR(%X, %Y) -> %X
 ||    ~PAIR(%X, %Y) -> %Y
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Knuth-Bendix order [KBO] with precedence:~PAIR_2 > 0 > min_2 > max_2 > s_1
 || 
 || and weight map:
 || 
 ||    0=1
 ||    s_1=1
 ||    max_2=0
 ||    min_2=0
 ||    ~PAIR_2=0
 || 
 || The variable weight is 1With 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)) -> s(max(%X, %Y))
 ||    min(0, %X) -> 0
 ||    min(%X, 0) -> 0
 ||    min(s(%X), s(%Y)) -> s(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#(X, cons(Y, Z), F, G) =#> F(X, Y)   
    1] insert#(X, cons(Y, Z), F, G) =#> insert#(G X Y, Z, F, G)   
    2] insert#(X, cons(Y, Z), F, G) =#> G(X, Y)   
    3] sort#(cons(X, Y), F, G) =#> insert#(X, sort(Y, F, G), F, G)   
    4] sort#(cons(X, Y), F, G) =#> sort#(Y, F, G)   
    5] ascending!6220sort#(X) =#> sort#(X, /\x./\y.min(x, y), /\z./\u.max(z, u))   
    6] ascending!6220sort#(X) =#> min#(x, y)   
    7] ascending!6220sort#(X) =#> max#(x, y)   
    8] descending!6220sort#(X) =#> sort#(X, /\x./\y.max(x, y), /\z./\u.min(z, u))   
    9] descending!6220sort#(X) =#> max#(x, y)   
    10] descending!6220sort#(X) =#> min#(x, y)   

  Rules R_0:

    max(0, X) => X 
    max(X, 0) => X 
    max(s(X), s(Y)) => s(max(X, Y)) 
    min(0, X) => 0 
    min(X, 0) => 0 
    min(s(X), s(Y)) => s(min(X, Y)) 
    insert(X, nil, F, G) => cons(X, nil) 
    insert(X, cons(Y, Z), F, G) => cons(F X Y, insert(G X Y, Z, F, G)) 
    sort(nil, F, G) => nil 
    sort(cons(X, Y), F, G) => insert(X, sort(Y, F, G), F, G) 
    ascending!6220sort(X) => sort(X, /\x./\y.min(x, y), /\z./\u.max(z, u)) 
    descending!6220sort(X) => sort(X, /\x./\y.max(x, y), /\z./\u.min(z, u)) 

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#(X, cons(Y, Z), F, G) =#> F(X, Y)   
    insert#(X, cons(Y, Z), F, G) =#> insert#(G X Y, Z, F, G)   
    insert#(X, cons(Y, Z), F, G) =#> G(X, Y)   
    sort#(cons(X, Y), F, G) =#> insert#(X, sort(Y, F, G), F, G)   
    sort#(cons(X, Y), F, G) =#> sort#(Y, F, G)   
    ascending!6220sort#(X) =#> sort#(X, /\x./\y.min(x, y), /\z./\u.max(z, u))   
    descending!6220sort#(X) =#> sort#(X, /\x./\y.max(x, y), /\z./\u.min(z, u))   

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(X, nil, F, G) => cons(X, nil) 
  insert(X, cons(Y, Z), F, G) => cons(F X Y, insert(G X Y, Z, F, G)) 
  sort(nil, F, G) => nil 
  sort(cons(X, Y), F, G) => insert(X, sort(Y, F, G), F, G) 
  ascending!6220sort(X) => sort(X, /\x./\y.min(x, y), /\z./\u.max(z, u)) 
  descending!6220sort(X) => sort(X, /\x./\y.max(x, y), /\z./\u.min(z, u)) 

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#(X, cons(Y, Z), F, G) >? F(X, Y) 
  insert#(X, cons(Y, Z), F, G) >? insert#(G X Y, Z, F, G) 
  insert#(X, cons(Y, Z), F, G) >? G(X, Y) 
  sort#(cons(X, Y), F, G) >? insert#(X, sort(Y, F, G), F, G) 
  sort#(cons(X, Y), F, G) >? sort#(Y, F, G) 
  ascending!6220sort#(X) >? sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) 
  descending!6220sort#(X) >? sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) 
  insert(X, nil, F, G) >= cons(X, nil) 
  insert(X, cons(Y, Z), F, G) >= cons(F X Y, insert(G X Y, Z, F, G)) 
  sort(nil, F, G) >= nil 
  sort(cons(X, Y), F, G) >= insert(X, sort(Y, F, G), F, G) 
  ascending!6220sort(X) >= sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) 
  descending!6220sort(X) >= sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) 
  max-(X, Y) >= max(X, Y) 
  min-(X, Y) >= min(X, Y) 

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

  pi( ascending!6220sort(X) ) = #argfun-ascending!6220sort#(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) 
  pi( ascending!6220sort#(X) ) = #argfun-ascending!6220sort##(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) 
  pi( descending!6220sort(X) ) = #argfun-descending!6220sort#(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) 
  pi( descending!6220sort#(X) ) = #argfun-descending!6220sort##(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) 

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!6220sort(x_1)]] = x_1 
  [[ascending!6220sort#(x_1)]] = x_1 
  [[descending!6220sort(x_1)]] = x_1 
  [[descending!6220sort#(x_1)]] = x_1 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_2, x_4, x_1, x_3) 
  [[insert#(x_1, x_2, x_3, x_4)]] = insert#(x_4, x_3, x_2, x_1) 
  [[min(x_1, x_2)]] = x_1 
  [[nil]] = _|_ 

We choose Lex = {insert, insert#} and Mul = {#argfun-ascending!6220sort#, #argfun-ascending!6220sort##, #argfun-descending!6220sort#, #argfun-descending!6220sort##, @_{o -> o -> o}, @_{o -> o}, cons, max, max-, min-, sort, sort#}, and the following precedence: #argfun-ascending!6220sort# > #argfun-ascending!6220sort## > #argfun-descending!6220sort# > #argfun-descending!6220sort## > max = max- > min- > sort# > insert# > sort > insert > @_{o -> o -> o} > cons > @_{o -> o}

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

  insert#(X, cons(Y, Z), F, G) > @_{o -> o}(@_{o -> o -> o}(F, X), Y) 
  insert#(X, cons(Y, Z), F, G) >= insert#(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G) 
  insert#(X, cons(Y, Z), F, G) > @_{o -> o}(@_{o -> o -> o}(G, X), Y) 
  sort#(cons(X, Y), F, G) >= insert#(X, sort(Y, F, G), F, G) 
  sort#(cons(X, Y), F, G) >= sort#(Y, F, G) 
  #argfun-ascending!6220sort##(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) > sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) 
  #argfun-descending!6220sort##(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) 
  insert(X, _|_, F, G) >= cons(X, _|_) 
  insert(X, cons(Y, Z), F, G) >= cons(@_{o -> o}(@_{o -> o -> o}(F, X), Y), insert(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G)) 
  sort(_|_, F, G) >= _|_ 
  sort(cons(X, Y), F, G) >= insert(X, sort(Y, F, G), F, G) 
  #argfun-ascending!6220sort#(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) 
  #argfun-descending!6220sort#(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) 
  max-(X, Y) >= max(X, Y) 
  min-(X, Y) >= X 

With these choices, we have:

  1] insert#(X, cons(Y, Z), F, G) > @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because [2], by definition 
  2] insert#*(X, cons(Y, Z), F, G) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because insert# > @_{o -> o}, [3] and [8], by (Copy) 
  3] insert#*(X, cons(Y, Z), F, G) >= @_{o -> o -> o}(F, X)  because insert# > @_{o -> o -> o}, [4] and [6], by (Copy) 
  4] insert#*(X, cons(Y, Z), F, G) >= F  because [5], by (Select) 
  5] F >= F  by (Meta) 
  6] insert#*(X, cons(Y, Z), F, G) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] insert#*(X, cons(Y, Z), F, G) >= 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] insert#(X, cons(Y, Z), F, G) >= insert#(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G)  because [13], by (Star) 
  13] insert#*(X, cons(Y, Z), F, G) >= insert#(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G)  because [14], [17], [18], [19], [22], [4] and [21], by (Stat) 
  14] cons(Y, Z) > Z  because [15], by definition 
  15] cons*(Y, Z) >= Z  because [16], by (Select) 
  16] Z >= Z  by (Meta) 
  17] F >= F  by (Meta) 
  18] G >= G  by (Meta) 
  19] insert#*(X, cons(Y, Z), F, G) >= @_{o -> o}(@_{o -> o -> o}(G, X), Y)  because insert# > @_{o -> o}, [20] and [8], by (Copy) 
  20] insert#*(X, cons(Y, Z), F, G) >= @_{o -> o -> o}(G, X)  because insert# > @_{o -> o -> o}, [21] and [6], by (Copy) 
  21] insert#*(X, cons(Y, Z), F, G) >= G  because [18], by (Select) 
  22] insert#*(X, cons(Y, Z), F, G) >= Z  because [23], by (Select) 
  23] cons(Y, Z) >= Z  because [15], by (Star) 

  24] insert#(X, cons(Y, Z), F, G) > @_{o -> o}(@_{o -> o -> o}(G, X), Y)  because [25], by definition 
  25] insert#*(X, cons(Y, Z), F, G) >= @_{o -> o}(@_{o -> o -> o}(G, X), Y)  because insert# > @_{o -> o}, [20] and [8], by (Copy) 

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

  41] sort#(cons(X, Y), F, G) >= sort#(Y, F, G)  because sort# in Mul, [42], [43] and [44], by (Fun) 
  42] cons(X, Y) >= Y  because [35], by (Star) 
  43] F >= F  by (Meta) 
  44] G >= G  by (Meta) 

  45] #argfun-ascending!6220sort##(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) > sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))  because [46], by definition 
  46] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))  because #argfun-ascending!6220sort## > sort#, [47], [51] and [58], by (Copy) 
  47] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= X  because [48], by (Select) 
  48] sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) >= X  because [49], by (Star) 
  49] sort#*(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) >= X  because [50], by (Select) 
  50] X >= X  by (Meta) 
  51] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= /\x./\y.min-(x, y)  because [52], by (F-Abs) 
  52] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v) >= /\x.min-(v, x)  because [53], by (F-Abs) 
  53] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= min-(v, w)  because #argfun-ascending!6220sort## > min-, [54] and [56], by (Copy) 
  54] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= v  because [55], by (Select) 
  55] v >= v  by (Var) 
  56] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= w  because [57], by (Select) 
  57] w >= w  by (Var) 
  58] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= /\x./\y.max-(x, y)  because [59], by (F-Abs) 
  59] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x') >= /\x.max-(x', x)  because [60], by (F-Abs) 
  60] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= max-(x', y')  because #argfun-ascending!6220sort## > max-, [61] and [63], by (Copy) 
  61] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= x'  because [62], by (Select) 
  62] x' >= x'  by (Var) 
  63] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= y'  because [64], by (Select) 
  64] y' >= y'  by (Var) 

  65] #argfun-descending!6220sort##(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))  because [66], by (Star) 
  66] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))  because #argfun-descending!6220sort## > sort#, [67], [71] and [78], by (Copy) 
  67] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= X  because [68], by (Select) 
  68] sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) >= X  because [69], by (Star) 
  69] sort#*(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) >= X  because [70], by (Select) 
  70] X >= X  by (Meta) 
  71] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= /\x./\y.max-(x, y)  because [72], by (F-Abs) 
  72] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v) >= /\x.max-(v, x)  because [73], by (F-Abs) 
  73] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= max-(v, w)  because #argfun-descending!6220sort## > max-, [74] and [76], by (Copy) 
  74] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= v  because [75], by (Select) 
  75] v >= v  by (Var) 
  76] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= w  because [77], by (Select) 
  77] w >= w  by (Var) 
  78] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= /\x./\y.min-(x, y)  because [79], by (F-Abs) 
  79] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x') >= /\x.min-(x', x)  because [80], by (F-Abs) 
  80] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= min-(x', y')  because #argfun-descending!6220sort## > min-, [81] and [83], by (Copy) 
  81] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= x'  because [82], by (Select) 
  82] x' >= x'  by (Var) 
  83] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= y'  because [84], by (Select) 
  84] y' >= y'  by (Var) 

  85] insert(X, _|_, F, G) >= cons(X, _|_)  because [86], by (Star) 
  86] insert*(X, _|_, F, G) >= cons(X, _|_)  because insert > cons, [87] and [89], by (Copy) 
  87] insert*(X, _|_, F, G) >= X  because [88], by (Select) 
  88] X >= X  by (Meta) 
  89] insert*(X, _|_, F, G) >= _|_  by (Bot) 

  90] insert(X, cons(Y, Z), F, G) >= cons(@_{o -> o}(@_{o -> o -> o}(F, X), Y), insert(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G))  because [91], by (Star) 
  91] insert*(X, cons(Y, Z), F, G) >= cons(@_{o -> o}(@_{o -> o -> o}(F, X), Y), insert(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G))  because insert > cons, [92] and [97], by (Copy) 
  92] insert*(X, cons(Y, Z), F, G) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because insert > @_{o -> o}, [93] and [96], by (Copy) 
  93] insert*(X, cons(Y, Z), F, G) >= @_{o -> o -> o}(F, X)  because insert > @_{o -> o -> o}, [94] and [95], by (Copy) 
  94] insert*(X, cons(Y, Z), F, G) >= F  because [17], by (Select) 
  95] insert*(X, cons(Y, Z), F, G) >= X  because [7], by (Select) 
  96] insert*(X, cons(Y, Z), F, G) >= Y  because [9], by (Select) 
  97] insert*(X, cons(Y, Z), F, G) >= insert(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G)  because [14], [98], [101], [94] and [100], by (Stat) 
  98] insert*(X, cons(Y, Z), F, G) >= @_{o -> o}(@_{o -> o -> o}(G, X), Y)  because insert > @_{o -> o}, [99] and [96], by (Copy) 
  99] insert*(X, cons(Y, Z), F, G) >= @_{o -> o -> o}(G, X)  because insert > @_{o -> o -> o}, [100] and [95], by (Copy) 
  100] insert*(X, cons(Y, Z), F, G) >= G  because [18], by (Select) 
  101] insert*(X, cons(Y, Z), F, G) >= Z  because [23], by (Select) 

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

  103] sort(cons(X, Y), F, G) >= insert(X, sort(Y, F, G), F, G)  because [104], by (Star) 
  104] sort*(cons(X, Y), F, G) >= insert(X, sort(Y, F, G), F, G)  because sort > insert, [105], [106], [109] and [110], by (Copy) 
  105] sort*(cons(X, Y), F, G) >= X  because [29], by (Select) 
  106] sort*(cons(X, Y), F, G) >= sort(Y, F, G)  because sort in Mul, [107], [43] and [44], by (Stat) 
  107] cons(X, Y) > Y  because [108], by definition 
  108] cons*(X, Y) >= Y  because [36], by (Select) 
  109] sort*(cons(X, Y), F, G) >= F  because [43], by (Select) 
  110] sort*(cons(X, Y), F, G) >= G  because [44], by (Select) 

  111] #argfun-ascending!6220sort#(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))  because [112], by (Star) 
  112] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))  because #argfun-ascending!6220sort# > sort, [113], [116] and [123], by (Copy) 
  113] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= X  because [114], by (Select) 
  114] sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) >= X  because [115], by (Star) 
  115] sort*(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) >= X  because [50], by (Select) 
  116] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= /\x./\y.min-(x, y)  because [117], by (F-Abs) 
  117] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v) >= /\x.min-(v, x)  because [118], by (F-Abs) 
  118] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= min-(v, w)  because #argfun-ascending!6220sort# > min-, [119] and [121], by (Copy) 
  119] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= v  because [120], by (Select) 
  120] v >= v  by (Var) 
  121] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= w  because [122], by (Select) 
  122] w >= w  by (Var) 
  123] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= /\x./\y.max-(x, y)  because [124], by (F-Abs) 
  124] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x') >= /\x.max-(x', x)  because [125], by (F-Abs) 
  125] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= max-(x', y')  because #argfun-ascending!6220sort# > max-, [126] and [128], by (Copy) 
  126] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= x'  because [127], by (Select) 
  127] x' >= x'  by (Var) 
  128] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= y'  because [129], by (Select) 
  129] y' >= y'  by (Var) 

  130] #argfun-descending!6220sort#(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))  because [131], by (Star) 
  131] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))  because #argfun-descending!6220sort# > sort, [132], [135] and [142], by (Copy) 
  132] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= X  because [133], by (Select) 
  133] sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) >= X  because [134], by (Star) 
  134] sort*(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) >= X  because [70], by (Select) 
  135] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= /\x./\y.max-(x, y)  because [136], by (F-Abs) 
  136] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v) >= /\x.max-(v, x)  because [137], by (F-Abs) 
  137] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= max-(v, w)  because #argfun-descending!6220sort# > max-, [138] and [140], by (Copy) 
  138] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= v  because [139], by (Select) 
  139] v >= v  by (Var) 
  140] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= w  because [141], by (Select) 
  141] w >= w  by (Var) 
  142] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= /\x./\y.min-(x, y)  because [143], by (F-Abs) 
  143] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x') >= /\x.min-(x', x)  because [144], by (F-Abs) 
  144] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= min-(x', y')  because #argfun-descending!6220sort# > min-, [145] and [147], by (Copy) 
  145] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= x'  because [146], by (Select) 
  146] x' >= x'  by (Var) 
  147] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= y'  because [148], by (Select) 
  148] y' >= y'  by (Var) 

  149] max-(X, Y) >= max(X, Y)  because max- = max, max- in Mul, [150] and [151], by (Fun) 
  150] X >= X  by (Meta) 
  151] Y >= Y  by (Meta) 

  152] min-(X, Y) >= X  because [153], by (Star) 
  153] min-*(X, Y) >= X  because [154], by (Select) 
  154] 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_1, minimal, formative) by (P_2, R_1, minimal, formative), where P_2 consists of:

  insert#(X, cons(Y, Z), F, G) =#> insert#(G X Y, Z, F, G)   
  sort#(cons(X, Y), F, G) =#> insert#(X, sort(Y, F, G), F, G)   
  sort#(cons(X, Y), F, G) =#> sort#(Y, F, G)   
  descending!6220sort#(X) =#> sort#(X, /\x./\y.max(x, y), /\z./\u.min(z, u))   

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 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 

This graph has the following strongly connected components:

  P_3:

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

  P_4:

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

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_2, R_1, m, f) by (P_3, R_1, m, f) and (P_4, R_1, m, f).

Thus, the original system is terminating if each of (P_3, R_1, minimal, formative) and (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(sort#) = 1 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35], 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.

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 apply the subterm criterion with the following projection function:

  nu(insert#) = 2 

Thus, we can orient the dependency pairs as follows:

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

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_3, 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 +++

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