We consider the system Applicative_05__Ex7Sorting.

  Alphabet:

    0 : [] --> a 
    asort : [b] --> b 
    cons : [a * b] --> b 
    dsort : [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:

    sort(f, g, nil) => nil 
    sort(f, g, cons(x, y)) => insert(f, g, sort(f, g, y), x) 
    insert(f, g, nil, x) => cons(x, nil) 
    insert(f, g, cons(x, y), z) => cons(f x z, insert(f, g, y, g x z)) 
    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 
    asort(x) => sort(min, max, x) 
    dsort(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] sort#(F, G, cons(X, Y)) =#> insert#(F, G, sort(F, G, Y), X)   
    1] sort#(F, G, cons(X, Y)) =#> sort#(F, G, Y)   
    2] insert#(F, G, cons(X, Y), Z) =#> F(X, Z)   
    3] insert#(F, G, cons(X, Y), Z) =#> insert#(F, G, Y, G X Z)   
    4] insert#(F, G, cons(X, Y), Z) =#> G(X, Z)   
    5] asort#(X) =#> sort#(min, max, X)   
    6] asort#(X) =#> min#   
    7] asort#(X) =#> max#   
    8] dsort#(X) =#> sort#(max, min, X)   
    9] dsort#(X) =#> max#   
    10] dsort#(X) =#> min#   

  Rules R_0:

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

This graph has the following strongly connected components:

  P_1:

    sort#(F, G, cons(X, Y)) =#> insert#(F, G, sort(F, G, Y), X)   
    sort#(F, G, cons(X, Y)) =#> sort#(F, G, Y)   
    insert#(F, G, cons(X, Y), Z) =#> F(X, Z)   
    insert#(F, G, cons(X, Y), Z) =#> insert#(F, G, Y, G X Z)   
    insert#(F, G, cons(X, Y), Z) =#> G(X, Z)   
    asort#(X) =#> sort#(min, max, X)   
    dsort#(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 ::=

  sort(F, G, nil) => nil 
  sort(F, G, cons(X, Y)) => insert(F, G, sort(F, G, Y), X) 
  insert(F, G, nil, X) => cons(X, nil) 
  insert(F, G, cons(X, Y), Z) => cons(F X Z, insert(F, G, Y, G X Z)) 
  asort(X) => sort(min, max, X) 
  dsort(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:

  sort#(F, G, cons(X, Y)) >? insert#(F, G, sort(F, G, Y), X) 
  sort#(F, G, cons(X, Y)) >? sort#(F, G, Y) 
  insert#(F, G, cons(X, Y), Z) >? F(X, Z) 
  insert#(F, G, cons(X, Y), Z) >? insert#(F, G, Y, G X Z) 
  insert#(F, G, cons(X, Y), Z) >? G(X, Z) 
  asort#(X) >? sort#(min, max, X) 
  dsort#(X) >? sort#(max, min, X) 
  sort(F, G, nil) >= nil 
  sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X) 
  insert(F, G, nil, X) >= cons(X, nil) 
  insert(F, G, cons(X, Y), Z) >= cons(F X Z, insert(F, G, Y, G X Z)) 
  asort(X) >= sort(min, max, X) 
  dsort(X) >= sort(max, min, X) 

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

  pi( asort(X) ) = #argfun-asort#(sort(min, max, X)) 
  pi( asort#(X) ) = #argfun-asort##(sort#(min, max, X)) 
  pi( dsort(X) ) = #argfun-dsort#(sort(max, min, X)) 
  pi( dsort#(X) ) = #argfun-dsort##(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-dsort#(x_1)]] = x_1 
  [[#argfun-dsort##(x_1)]] = x_1 
  [[asort(x_1)]] = x_1 
  [[asort#(x_1)]] = x_1 
  [[dsort(x_1)]] = x_1 
  [[insert(x_1, x_2, x_3, x_4)]] = insert(x_3, x_2, x_1, x_4) 
  [[insert#(x_1, x_2, x_3, x_4)]] = insert#(x_3, x_1, x_2, x_4) 
  [[nil]] = _|_ 

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

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

With these choices, we have:

  1] sort#(F, G, cons(X, Y)) >= insert#(F, G, sort(F, G, Y), X)  because [2], by (Star) 
  2] sort#*(F, G, cons(X, Y)) >= insert#(F, G, sort(F, G, Y), X)  because sort# > insert#, [3], [5], [7] and [13], by (Copy) 
  3] sort#*(F, G, cons(X, Y)) >= F  because [4], by (Select) 
  4] F >= F  by (Meta) 
  5] sort#*(F, G, cons(X, Y)) >= G  because [6], by (Select) 
  6] G >= G  by (Meta) 
  7] sort#*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort# = sort, sort# in Mul, [8], [9] and [10], by (Stat) 
  8] F >= F  by (Meta) 
  9] G >= G  by (Meta) 
  10] cons(X, Y) > Y  because [11], by definition 
  11] cons*(X, Y) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 
  13] sort#*(F, G, cons(X, Y)) >= X  because [14], by (Select) 
  14] cons(X, Y) >= X  because [15], by (Star) 
  15] cons*(X, Y) >= X  because [16], by (Select) 
  16] X >= X  by (Meta) 

  17] sort#(F, G, cons(X, Y)) >= sort#(F, G, Y)  because [18], by (Star) 
  18] sort#*(F, G, cons(X, Y)) >= sort#(F, G, Y)  because sort# in Mul, [8], [9] and [10], by (Stat) 

  19] insert#(F, G, cons(X, Y), Z) > @_{o -> o}(@_{o -> o -> o}(F, X), Z)  because [20], by definition 
  20] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, X), Z)  because insert# > @_{o -> o}, [21] and [28], by (Copy) 
  21] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, X)  because insert# > @_{o -> o -> o}, [22] and [24], by (Copy) 
  22] insert#*(F, G, cons(X, Y), Z) >= F  because [23], by (Select) 
  23] F >= F  by (Meta) 
  24] insert#*(F, G, cons(X, Y), Z) >= X  because [25], by (Select) 
  25] cons(X, Y) >= X  because [26], by (Star) 
  26] cons*(X, Y) >= X  because [27], by (Select) 
  27] X >= X  by (Meta) 
  28] insert#*(F, G, cons(X, Y), Z) >= Z  because [29], by (Select) 
  29] Z >= Z  by (Meta) 

  30] insert#(F, G, cons(X, Y), Z) > insert#(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z))  because [31], by definition 
  31] insert#*(F, G, cons(X, Y), Z) >= insert#(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z))  because [32], [22], [35], [37] and [39], by (Stat) 
  32] cons(X, Y) > Y  because [33], by definition 
  33] cons*(X, Y) >= Y  because [34], by (Select) 
  34] Y >= Y  by (Meta) 
  35] insert#*(F, G, cons(X, Y), Z) >= G  because [36], by (Select) 
  36] G >= G  by (Meta) 
  37] insert#*(F, G, cons(X, Y), Z) >= Y  because [38], by (Select) 
  38] cons(X, Y) >= Y  because [33], by (Star) 
  39] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, X), Z)  because insert# > @_{o -> o}, [40] and [28], by (Copy) 
  40] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, X)  because insert# > @_{o -> o -> o}, [35] and [24], by (Copy) 

  41] insert#(F, G, cons(X, Y), Z) > @_{o -> o}(@_{o -> o -> o}(G, X), Z)  because [42], by definition 
  42] insert#*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, X), Z)  because insert# > @_{o -> o}, [40] and [28], by (Copy) 

  43] #argfun-asort##(sort#(min, max, X)) > sort#(min, max, X)  because [44], by definition 
  44] #argfun-asort##*(sort#(min, max, X)) >= sort#(min, max, X)  because #argfun-asort## > sort#, [45], [46] and [47], by (Copy) 
  45] #argfun-asort##*(sort#(min, max, X)) >= min  because #argfun-asort## > min, by (Copy) 
  46] #argfun-asort##*(sort#(min, max, X)) >= max  because #argfun-asort## > max, by (Copy) 
  47] #argfun-asort##*(sort#(min, max, X)) >= X  because [48], by (Select) 
  48] sort#(min, max, X) >= X  because [49], by (Star) 
  49] sort#*(min, max, X) >= X  because [50], by (Select) 
  50] X >= X  by (Meta) 

  51] sort#(max, min, X) >= sort#(max, min, X)  because sort# in Mul, [52], [53] and [54], by (Fun) 
  52] max >= max  by (Fun) 
  53] min >= min  by (Fun) 
  54] X >= X  by (Meta) 

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

  56] sort(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because [57], by (Star) 
  57] sort*(F, G, cons(X, Y)) >= insert(F, G, sort(F, G, Y), X)  because sort > insert, [58], [59], [60] and [61], by (Copy) 
  58] sort*(F, G, cons(X, Y)) >= F  because [8], by (Select) 
  59] sort*(F, G, cons(X, Y)) >= G  because [9], by (Select) 
  60] sort*(F, G, cons(X, Y)) >= sort(F, G, Y)  because sort in Mul, [8], [9] and [10], by (Stat) 
  61] sort*(F, G, cons(X, Y)) >= X  because [14], by (Select) 

  62] insert(F, G, _|_, X) >= cons(X, _|_)  because [63], by (Star) 
  63] insert*(F, G, _|_, X) >= cons(X, _|_)  because insert > cons, [64] and [66], by (Copy) 
  64] insert*(F, G, _|_, X) >= X  because [65], by (Select) 
  65] X >= X  by (Meta) 
  66] insert*(F, G, _|_, X) >= _|_  by (Bot) 

  67] insert(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, X), Z), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z)))  because [68], by (Star) 
  68] insert*(F, G, cons(X, Y), Z) >= cons(@_{o -> o}(@_{o -> o -> o}(F, X), Z), insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z)))  because insert > cons, [69] and [74], by (Copy) 
  69] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(F, X), Z)  because insert > @_{o -> o}, [70] and [73], by (Copy) 
  70] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(F, X)  because insert > @_{o -> o -> o}, [71] and [72], by (Copy) 
  71] insert*(F, G, cons(X, Y), Z) >= F  because [23], by (Select) 
  72] insert*(F, G, cons(X, Y), Z) >= X  because [25], by (Select) 
  73] insert*(F, G, cons(X, Y), Z) >= Z  because [29], by (Select) 
  74] insert*(F, G, cons(X, Y), Z) >= insert(F, G, Y, @_{o -> o}(@_{o -> o -> o}(G, X), Z))  because [32], [71], [75], [76] and [77], by (Stat) 
  75] insert*(F, G, cons(X, Y), Z) >= G  because [36], by (Select) 
  76] insert*(F, G, cons(X, Y), Z) >= Y  because [38], by (Select) 
  77] insert*(F, G, cons(X, Y), Z) >= @_{o -> o}(@_{o -> o -> o}(G, X), Z)  because insert > @_{o -> o}, [78] and [73], by (Copy) 
  78] insert*(F, G, cons(X, Y), Z) >= @_{o -> o -> o}(G, X)  because insert > @_{o -> o -> o}, [75] and [72], by (Copy) 

  79] #argfun-asort#(sort(min, max, X)) >= sort(min, max, X)  because [80], by (Star) 
  80] #argfun-asort#*(sort(min, max, X)) >= sort(min, max, X)  because #argfun-asort# > sort, [81], [82] and [83], by (Copy) 
  81] #argfun-asort#*(sort(min, max, X)) >= min  because #argfun-asort# > min, by (Copy) 
  82] #argfun-asort#*(sort(min, max, X)) >= max  because #argfun-asort# > max, by (Copy) 
  83] #argfun-asort#*(sort(min, max, X)) >= X  because [84], by (Select) 
  84] sort(min, max, X) >= X  because [85], by (Star) 
  85] sort*(min, max, X) >= X  because [50], by (Select) 

  86] sort(max, min, X) >= sort(max, min, X)  because sort in Mul, [52], [53] and [54], by (Fun) 

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:

  sort#(F, G, cons(X, Y)) =#> insert#(F, G, sort(F, G, Y), X)   
  sort#(F, G, cons(X, Y)) =#> sort#(F, G, Y)   
  dsort#(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 place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  
    * 1 : 0, 1 
    * 2 : 0, 1 

This graph has the following strongly connected components:

  P_3:

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

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).

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(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_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.