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) QTRSRRRProof [EQUIVALENT]
 || (4) QTRS
 || (5) RisEmptyProof [EQUIVALENT]
 || (6) 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:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(0) = 0
 ||    POL(max(x_1, x_2)) = 2*x_1 + 2*x_2
 ||    POL(min(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
 ||    POL(s(x_1)) = 2 + 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(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:
 || The TRS R consists of the following rules:
 || 
 ||    max(0, %X) -> %X
 ||    max(%X, 0) -> %X
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (3) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Quasi precedence:
 || trivial
 || 
 || 
 || Status:
 || max_2: multiset status
 || 0: multiset status
 || 
 || 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
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (4)
 || Obligation:
 || Q restricted rewrite system:
 || R is empty.
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (5) RisEmptyProof (EQUIVALENT)
 || The TRS R is empty. Hence, termination is trivially proven.
 || ----------------------------------------
 || 
 || (6)
 || 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_3, x_4, x_1) 
  [[insert#(x_1, x_2, x_3, x_4)]] = insert#(x_2, x_3, x_4, 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-descending!6220sort# > #argfun-descending!6220sort## > #argfun-ascending!6220sort# > #argfun-ascending!6220sort## > min- > sort# > insert# > sort > insert > @_{o -> o -> o} > cons > @_{o -> o} > max = max-

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

  23] insert#(X, cons(Y, Z), F, G) > @_{o -> o}(@_{o -> o -> o}(G, X), Y)  because [24], by definition 
  24] insert#*(X, cons(Y, Z), F, G) >= @_{o -> o}(@_{o -> o -> o}(G, X), Y)  because insert# > @_{o -> o}, [18] and [8], by (Copy) 

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

  40] sort#(cons(X, Y), F, G) >= sort#(Y, F, G)  because sort# in Mul, [41], [42] and [43], by (Fun) 
  41] cons(X, Y) >= Y  because [34], by (Star) 
  42] F >= F  by (Meta) 
  43] G >= G  by (Meta) 

  44] #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 [45], by (Star) 
  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 #argfun-ascending!6220sort## > sort#, [46], [50] and [57], by (Copy) 
  46] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= X  because [47], by (Select) 
  47] sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) >= X  because [48], by (Star) 
  48] sort#*(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) >= X  because [49], by (Select) 
  49] X >= X  by (Meta) 
  50] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= /\x./\y.min-(x, y)  because [51], by (F-Abs) 
  51] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v) >= /\x.min-(v, x)  because [52], by (F-Abs) 
  52] #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-, [53] and [55], by (Copy) 
  53] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= v  because [54], by (Select) 
  54] v >= v  by (Var) 
  55] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= w  because [56], by (Select) 
  56] w >= w  by (Var) 
  57] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= /\x./\y.max-(x, y)  because [58], by (F-Abs) 
  58] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x') >= /\x.max-(x', x)  because [59], by (F-Abs) 
  59] #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-, [60] and [62], by (Copy) 
  60] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= x'  because [61], by (Select) 
  61] x' >= x'  by (Var) 
  62] #argfun-ascending!6220sort##*(sort#(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= y'  because [63], by (Select) 
  63] y' >= y'  by (Var) 

  64] #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 [65], by (Star) 
  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 #argfun-descending!6220sort## > sort#, [66], [70] and [77], by (Copy) 
  66] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= X  because [67], by (Select) 
  67] sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) >= X  because [68], by (Star) 
  68] sort#*(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) >= X  because [69], by (Select) 
  69] X >= X  by (Meta) 
  70] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= /\x./\y.max-(x, y)  because [71], by (F-Abs) 
  71] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v) >= /\x.max-(v, x)  because [72], by (F-Abs) 
  72] #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-, [73] and [75], by (Copy) 
  73] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= v  because [74], by (Select) 
  74] v >= v  by (Var) 
  75] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= w  because [76], by (Select) 
  76] w >= w  by (Var) 
  77] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= /\x./\y.min-(x, y)  because [78], by (F-Abs) 
  78] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x') >= /\x.min-(x', x)  because [79], by (F-Abs) 
  79] #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-, [80] and [82], by (Copy) 
  80] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= x'  because [81], by (Select) 
  81] x' >= x'  by (Var) 
  82] #argfun-descending!6220sort##*(sort#(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= y'  because [83], by (Select) 
  83] y' >= y'  by (Var) 

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

  89] 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 [90], by (Star) 
  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 insert > cons, [91] and [96], by (Copy) 
  91] insert*(X, cons(Y, Z), F, G) >= @_{o -> o}(@_{o -> o -> o}(F, X), Y)  because insert > @_{o -> o}, [92] and [95], by (Copy) 
  92] insert*(X, cons(Y, Z), F, G) >= @_{o -> o -> o}(F, X)  because insert > @_{o -> o -> o}, [93] and [94], by (Copy) 
  93] insert*(X, cons(Y, Z), F, G) >= F  because [5], by (Select) 
  94] insert*(X, cons(Y, Z), F, G) >= X  because [7], by (Select) 
  95] insert*(X, cons(Y, Z), F, G) >= Y  because [9], by (Select) 
  96] insert*(X, cons(Y, Z), F, G) >= insert(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G)  because [14], [97], [100], [93] and [99], by (Stat) 
  97] insert*(X, cons(Y, Z), F, G) >= @_{o -> o}(@_{o -> o -> o}(G, X), Y)  because insert > @_{o -> o}, [98] and [95], by (Copy) 
  98] insert*(X, cons(Y, Z), F, G) >= @_{o -> o -> o}(G, X)  because insert > @_{o -> o -> o}, [99] and [94], by (Copy) 
  99] insert*(X, cons(Y, Z), F, G) >= G  because [20], by (Select) 
  100] insert*(X, cons(Y, Z), F, G) >= Z  because [22], by (Select) 

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

  102] sort(cons(X, Y), F, G) >= insert(X, sort(Y, F, G), F, G)  because [103], by (Star) 
  103] sort*(cons(X, Y), F, G) >= insert(X, sort(Y, F, G), F, G)  because sort > insert, [104], [105], [108] and [109], by (Copy) 
  104] sort*(cons(X, Y), F, G) >= X  because [28], by (Select) 
  105] sort*(cons(X, Y), F, G) >= sort(Y, F, G)  because sort in Mul, [106], [42] and [43], by (Stat) 
  106] cons(X, Y) > Y  because [107], by definition 
  107] cons*(X, Y) >= Y  because [35], by (Select) 
  108] sort*(cons(X, Y), F, G) >= F  because [42], by (Select) 
  109] sort*(cons(X, Y), F, G) >= G  because [43], by (Select) 

  110] #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 [111], by (Star) 
  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 #argfun-ascending!6220sort# > sort, [112], [115] and [122], by (Copy) 
  112] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= X  because [113], by (Select) 
  113] sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) >= X  because [114], by (Star) 
  114] sort*(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)) >= X  because [49], by (Select) 
  115] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= /\x./\y.min-(x, y)  because [116], by (F-Abs) 
  116] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v) >= /\x.min-(v, x)  because [117], by (F-Abs) 
  117] #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-, [118] and [120], by (Copy) 
  118] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= v  because [119], by (Select) 
  119] v >= v  by (Var) 
  120] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), v, w) >= w  because [121], by (Select) 
  121] w >= w  by (Var) 
  122] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u))) >= /\x./\y.max-(x, y)  because [123], by (F-Abs) 
  123] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x') >= /\x.max-(x', x)  because [124], by (F-Abs) 
  124] #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-, [125] and [127], by (Copy) 
  125] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= x'  because [126], by (Select) 
  126] x' >= x'  by (Var) 
  127] #argfun-ascending!6220sort#*(sort(X, /\x./\y.min-(x, y), /\z./\u.max-(z, u)), x', y') >= y'  because [128], by (Select) 
  128] y' >= y'  by (Var) 

  129] #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 [130], by (Star) 
  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 #argfun-descending!6220sort# > sort, [131], [134] and [141], by (Copy) 
  131] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= X  because [132], by (Select) 
  132] sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) >= X  because [133], by (Star) 
  133] sort*(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)) >= X  because [69], by (Select) 
  134] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= /\x./\y.max-(x, y)  because [135], by (F-Abs) 
  135] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v) >= /\x.max-(v, x)  because [136], by (F-Abs) 
  136] #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-, [137] and [139], by (Copy) 
  137] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= v  because [138], by (Select) 
  138] v >= v  by (Var) 
  139] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), v, w) >= w  because [140], by (Select) 
  140] w >= w  by (Var) 
  141] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u))) >= /\x./\y.min-(x, y)  because [142], by (F-Abs) 
  142] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x') >= /\x.min-(x', x)  because [143], by (F-Abs) 
  143] #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-, [144] and [146], by (Copy) 
  144] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= x'  because [145], by (Select) 
  145] x' >= x'  by (Var) 
  146] #argfun-descending!6220sort#*(sort(X, /\x./\y.max-(x, y), /\z./\u.min-(z, u)), x', y') >= y'  because [147], by (Select) 
  147] y' >= y'  by (Var) 

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

  151] min-(X, Y) >= X  because [152], by (Star) 
  152] min-*(X, Y) >= X  because [153], by (Select) 
  153] 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)   
  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_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 
    * 4 : 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 [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.

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 [FuhKop19, Thm. 61], 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 +++

[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.