We consider the system 06plusmult.

  Alphabet:

    mult : [N * N] --> N 
    plus : [N * N] --> N 
    s : [N] --> N 
    z : [] --> N 

  Rules:

    plus(z, x) => x 
    plus(s(x), y) => plus(x, s(y)) 
    plus(plus(x, y), u) => plus(x, plus(y, u)) 
    mult(z, x) => z 
    mult(s(x), y) => plus(mult(x, y), y) 
    mult(plus(x, y), u) => plus(mult(x, u), mult(y, u)) 

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

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).

We thus obtain the following dependency pair problem (P_0, R_0, static, formative):

  Dependency Pairs P_0:

    0] plus#(s(X), Y) =#> plus#(X, s(Y))   
    1] plus#(plus(X, Y), Z) =#> plus#(X, plus(Y, Z))   
    2] plus#(plus(X, Y), Z) =#> plus#(Y, Z)   
    3] mult#(s(X), Y) =#> plus#(mult(X, Y), Y)   
    4] mult#(s(X), Y) =#> mult#(X, Y)   
    5] mult#(plus(X, Y), Z) =#> plus#(mult(X, Z), mult(Y, Z))   
    6] mult#(plus(X, Y), Z) =#> mult#(X, Z)   
    7] mult#(plus(X, Y), Z) =#> mult#(Y, Z)   

  Rules R_0:

    plus(z, X) => X 
    plus(s(X), Y) => plus(X, s(Y)) 
    plus(plus(X, Y), Z) => plus(X, plus(Y, Z)) 
    mult(z, X) => z 
    mult(s(X), Y) => plus(mult(X, Y), Y) 
    mult(plus(X, Y), Z) => plus(mult(X, Z), mult(Y, Z)) 

Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.

We consider the dependency pair problem (P_0, R_0, static, 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 
    * 1 : 0, 1, 2 
    * 2 : 0, 1, 2 
    * 3 : 0, 1, 2 
    * 4 : 3, 4, 5, 6, 7 
    * 5 : 0, 1, 2 
    * 6 : 3, 4, 5, 6, 7 
    * 7 : 3, 4, 5, 6, 7 

This graph has the following strongly connected components:

  P_1:

    plus#(s(X), Y) =#> plus#(X, s(Y))   
    plus#(plus(X, Y), Z) =#> plus#(X, plus(Y, Z))   
    plus#(plus(X, Y), Z) =#> plus#(Y, Z)   

  P_2:

    mult#(s(X), Y) =#> mult#(X, Y)   
    mult#(plus(X, Y), Z) =#> mult#(X, Z)   
    mult#(plus(X, Y), Z) =#> mult#(Y, Z)   

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) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.

We consider the dependency pair problem (P_2, R_0, static, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  mult#(s(X), Y) >? mult#(X, Y) 
  mult#(plus(X, Y), Z) >? mult#(X, Z) 
  mult#(plus(X, Y), Z) >? mult#(Y, Z) 
  plus(z, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(z, X) >= z 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

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:

  [[z]] = _|_ 

We choose Lex = {plus} and Mul = {mult, mult#, s}, and the following precedence: mult# > mult > plus > s

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

  mult#(s(X), Y) >= mult#(X, Y) 
  mult#(plus(X, Y), Z) >= mult#(X, Z) 
  mult#(plus(X, Y), Z) > mult#(Y, Z) 
  plus(_|_, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

With these choices, we have:

  1] mult#(s(X), Y) >= mult#(X, Y)  because mult# in Mul, [2] and [5], by (Fun) 
  2] s(X) >= X  because [3], by (Star) 
  3] s*(X) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 
  5] Y >= Y  by (Meta) 

  6] mult#(plus(X, Y), Z) >= mult#(X, Z)  because mult# in Mul, [7] and [10], by (Fun) 
  7] plus(X, Y) >= X  because [8], by (Star) 
  8] plus*(X, Y) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] Z >= Z  by (Meta) 

  11] mult#(plus(X, Y), Z) > mult#(Y, Z)  because [12], by definition 
  12] mult#*(plus(X, Y), Z) >= mult#(Y, Z)  because mult# in Mul, [13] and [10], by (Stat) 
  13] plus(X, Y) > Y  because [14], by definition 
  14] plus*(X, Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 

  16] plus(_|_, X) >= X  because [17], by (Star) 
  17] plus*(_|_, X) >= X  because [18], by (Select) 
  18] X >= X  by (Meta) 

  19] plus(s(X), Y) >= plus(X, s(Y))  because [20], by (Star) 
  20] plus*(s(X), Y) >= plus(X, s(Y))  because [21], [24] and [26], by (Stat) 
  21] s(X) > X  because [22], by definition 
  22] s*(X) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] plus*(s(X), Y) >= X  because [25], by (Select) 
  25] s(X) >= X  because [22], by (Star) 
  26] plus*(s(X), Y) >= s(Y)  because plus > s and [27], by (Copy) 
  27] plus*(s(X), Y) >= Y  because [28], by (Select) 
  28] Y >= Y  by (Meta) 

  29] plus(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [30], by (Star) 
  30] plus*(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [31], [34] and [36], by (Stat) 
  31] plus(X, Y) > X  because [32], by definition 
  32] plus*(X, Y) >= X  because [33], by (Select) 
  33] X >= X  by (Meta) 
  34] plus*(plus(X, Y), Z) >= X  because [35], by (Select) 
  35] plus(X, Y) >= X  because [32], by (Star) 
  36] plus*(plus(X, Y), Z) >= plus(Y, Z)  because [37], [40] and [42], by (Stat) 
  37] plus(X, Y) > Y  because [38], by definition 
  38] plus*(X, Y) >= Y  because [39], by (Select) 
  39] Y >= Y  by (Meta) 
  40] plus*(plus(X, Y), Z) >= Y  because [41], by (Select) 
  41] plus(X, Y) >= Y  because [38], by (Star) 
  42] plus*(plus(X, Y), Z) >= Z  because [43], by (Select) 
  43] Z >= Z  by (Meta) 

  44] mult(_|_, X) >= _|_  by (Bot) 

  45] mult(s(X), Y) >= plus(mult(X, Y), Y)  because [46], by (Star) 
  46] mult*(s(X), Y) >= plus(mult(X, Y), Y)  because mult > plus, [47] and [50], by (Copy) 
  47] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [48] and [5], by (Stat) 
  48] s(X) > X  because [49], by definition 
  49] s*(X) >= X  because [4], by (Select) 
  50] mult*(s(X), Y) >= Y  because [5], by (Select) 

  51] mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because [52], by (Star) 
  52] mult*(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because mult > plus, [53] and [56], by (Copy) 
  53] mult*(plus(X, Y), Z) >= mult(X, Z)  because mult in Mul, [54] and [10], by (Stat) 
  54] plus(X, Y) > X  because [55], by definition 
  55] plus*(X, Y) >= X  because [9], by (Select) 
  56] mult*(plus(X, Y), Z) >= mult(Y, Z)  because mult in Mul, [13] and [10], by (Stat) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_2, R_0, static, formative) by (P_3, R_0, static, formative), where P_3 consists of:

  mult#(s(X), Y) =#> mult#(X, Y)   
  mult#(plus(X, Y), Z) =#> mult#(X, Z)   

Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_3, R_0, static, formative) is finite.

We consider the dependency pair problem (P_3, R_0, static, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  mult#(s(X), Y) >? mult#(X, Y) 
  mult#(plus(X, Y), Z) >? mult#(X, Z) 
  plus(z, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(z, X) >= z 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

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:

  [[z]] = _|_ 

We choose Lex = {plus} and Mul = {mult, mult#, s}, and the following precedence: mult > mult# > plus > s

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

  mult#(s(X), Y) > mult#(X, Y) 
  mult#(plus(X, Y), Z) >= mult#(X, Z) 
  plus(_|_, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

With these choices, we have:

  1] mult#(s(X), Y) > mult#(X, Y)  because [2], by definition 
  2] mult#*(s(X), Y) >= mult#(X, Y)  because mult# in Mul, [3] and [6], by (Stat) 
  3] s(X) > X  because [4], by definition 
  4] s*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 
  6] Y >= Y  by (Meta) 

  7] mult#(plus(X, Y), Z) >= mult#(X, Z)  because mult# in Mul, [8] and [11], by (Fun) 
  8] plus(X, Y) >= X  because [9], by (Star) 
  9] plus*(X, Y) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] Z >= Z  by (Meta) 

  12] plus(_|_, X) >= X  because [13], by (Star) 
  13] plus*(_|_, X) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 

  15] plus(s(X), Y) >= plus(X, s(Y))  because [16], by (Star) 
  16] plus*(s(X), Y) >= plus(X, s(Y))  because [17], [20] and [22], by (Stat) 
  17] s(X) > X  because [18], by definition 
  18] s*(X) >= X  because [19], by (Select) 
  19] X >= X  by (Meta) 
  20] plus*(s(X), Y) >= X  because [21], by (Select) 
  21] s(X) >= X  because [18], by (Star) 
  22] plus*(s(X), Y) >= s(Y)  because plus > s and [23], by (Copy) 
  23] plus*(s(X), Y) >= Y  because [24], by (Select) 
  24] Y >= Y  by (Meta) 

  25] plus(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [26], by (Star) 
  26] plus*(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [27], [30] and [32], by (Stat) 
  27] plus(X, Y) > X  because [28], by definition 
  28] plus*(X, Y) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 
  30] plus*(plus(X, Y), Z) >= X  because [31], by (Select) 
  31] plus(X, Y) >= X  because [28], by (Star) 
  32] plus*(plus(X, Y), Z) >= plus(Y, Z)  because [33], [36] and [38], by (Stat) 
  33] plus(X, Y) > Y  because [34], by definition 
  34] plus*(X, Y) >= Y  because [35], by (Select) 
  35] Y >= Y  by (Meta) 
  36] plus*(plus(X, Y), Z) >= Y  because [37], by (Select) 
  37] plus(X, Y) >= Y  because [34], by (Star) 
  38] plus*(plus(X, Y), Z) >= Z  because [39], by (Select) 
  39] Z >= Z  by (Meta) 

  40] mult(_|_, X) >= _|_  by (Bot) 

  41] mult(s(X), Y) >= plus(mult(X, Y), Y)  because [42], by (Star) 
  42] mult*(s(X), Y) >= plus(mult(X, Y), Y)  because mult > plus, [43] and [44], by (Copy) 
  43] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [3] and [6], by (Stat) 
  44] mult*(s(X), Y) >= Y  because [6], by (Select) 

  45] mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because [46], by (Star) 
  46] mult*(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because mult > plus, [47] and [50], by (Copy) 
  47] mult*(plus(X, Y), Z) >= mult(X, Z)  because mult in Mul, [48] and [11], by (Stat) 
  48] plus(X, Y) > X  because [49], by definition 
  49] plus*(X, Y) >= X  because [10], by (Select) 
  50] mult*(plus(X, Y), Z) >= mult(Y, Z)  because mult in Mul, [51] and [11], by (Stat) 
  51] plus(X, Y) > Y  because [52], by definition 
  52] plus*(X, Y) >= Y  because [53], by (Select) 
  53] Y >= Y  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_0, static, formative) by (P_4, R_0, static, formative), where P_4 consists of:

  mult#(plus(X, Y), Z) =#> mult#(X, Z)   

Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_4, R_0, static, formative) is finite.

We consider the dependency pair problem (P_4, R_0, static, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  mult#(plus(X, Y), Z) >? mult#(X, Z) 
  plus(z, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(z, X) >= z 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

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:

  [[mult#(x_1, x_2)]] = x_1 
  [[z]] = _|_ 

We choose Lex = {plus} and Mul = {mult, s}, and the following precedence: mult > plus > s

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

  plus(X, Y) > X 
  plus(_|_, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

With these choices, we have:

  1] plus(X, Y) > X  because [2], by definition 
  2] plus*(X, Y) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] plus(_|_, X) >= X  because [5], by (Star) 
  5] plus*(_|_, X) >= X  because [6], by (Select) 
  6] X >= X  by (Meta) 

  7] plus(s(X), Y) >= plus(X, s(Y))  because [8], by (Star) 
  8] plus*(s(X), Y) >= plus(X, s(Y))  because [9], [12] and [14], by (Stat) 
  9] s(X) > X  because [10], by definition 
  10] s*(X) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] plus*(s(X), Y) >= X  because [13], by (Select) 
  13] s(X) >= X  because [10], by (Star) 
  14] plus*(s(X), Y) >= s(Y)  because plus > s and [15], by (Copy) 
  15] plus*(s(X), Y) >= Y  because [16], by (Select) 
  16] Y >= Y  by (Meta) 

  17] plus(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [18], by (Star) 
  18] plus*(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [19], [22] and [24], by (Stat) 
  19] plus(X, Y) > X  because [20], by definition 
  20] plus*(X, Y) >= X  because [21], by (Select) 
  21] X >= X  by (Meta) 
  22] plus*(plus(X, Y), Z) >= X  because [23], by (Select) 
  23] plus(X, Y) >= X  because [20], by (Star) 
  24] plus*(plus(X, Y), Z) >= plus(Y, Z)  because [25], [28] and [30], by (Stat) 
  25] plus(X, Y) > Y  because [26], by definition 
  26] plus*(X, Y) >= Y  because [27], by (Select) 
  27] Y >= Y  by (Meta) 
  28] plus*(plus(X, Y), Z) >= Y  because [29], by (Select) 
  29] plus(X, Y) >= Y  because [26], by (Star) 
  30] plus*(plus(X, Y), Z) >= Z  because [31], by (Select) 
  31] Z >= Z  by (Meta) 

  32] mult(_|_, X) >= _|_  by (Bot) 

  33] mult(s(X), Y) >= plus(mult(X, Y), Y)  because [34], by (Star) 
  34] mult*(s(X), Y) >= plus(mult(X, Y), Y)  because mult > plus, [35] and [40], by (Copy) 
  35] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [36] and [39], by (Stat) 
  36] s(X) > X  because [37], by definition 
  37] s*(X) >= X  because [38], by (Select) 
  38] X >= X  by (Meta) 
  39] Y >= Y  by (Meta) 
  40] mult*(s(X), Y) >= Y  because [39], by (Select) 

  41] mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because [42], by (Star) 
  42] mult*(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because mult > plus, [43] and [46], by (Copy) 
  43] mult*(plus(X, Y), Z) >= mult(X, Z)  because mult in Mul, [44] and [45], by (Stat) 
  44] plus(X, Y) > X  because [2], by definition 
  45] Z >= Z  by (Meta) 
  46] mult*(plus(X, Y), Z) >= mult(Y, Z)  because mult in Mul, [47] and [45], by (Stat) 
  47] plus(X, Y) > Y  because [48], by definition 
  48] plus*(X, Y) >= Y  because [49], by (Select) 
  49] Y >= Y  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_4, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.

We consider the dependency pair problem (P_1, R_0, static, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  plus#(s(X), Y) >? plus#(X, s(Y)) 
  plus#(plus(X, Y), Z) >? plus#(X, plus(Y, Z)) 
  plus#(plus(X, Y), Z) >? plus#(Y, Z) 
  plus(z, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(z, X) >= z 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

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:

  [[plus#(x_1, x_2)]] = plus#(x_1) 
  [[z]] = _|_ 

We choose Lex = {plus} and Mul = {mult, plus#, s}, and the following precedence: mult > plus > plus# > s

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

  plus#(s(X)) >= plus#(X) 
  plus#(plus(X, Y)) > plus#(X) 
  plus#(plus(X, Y)) >= plus#(Y) 
  plus(_|_, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

With these choices, we have:

  1] plus#(s(X)) >= plus#(X)  because plus# in Mul and [2], by (Fun) 
  2] s(X) >= X  because [3], by (Star) 
  3] s*(X) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 

  5] plus#(plus(X, Y)) > plus#(X)  because [6], by definition 
  6] plus#*(plus(X, Y)) >= plus#(X)  because [7], by (Select) 
  7] plus(X, Y) >= plus#(X)  because [8], by (Star) 
  8] plus*(X, Y) >= plus#(X)  because plus > plus# and [9], by (Copy) 
  9] plus*(X, Y) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 

  11] plus#(plus(X, Y)) >= plus#(Y)  because plus# in Mul and [12], by (Fun) 
  12] plus(X, Y) >= Y  because [13], by (Star) 
  13] plus*(X, Y) >= Y  because [14], by (Select) 
  14] Y >= Y  by (Meta) 

  15] plus(_|_, X) >= X  because [16], by (Star) 
  16] plus*(_|_, X) >= X  because [17], by (Select) 
  17] X >= X  by (Meta) 

  18] plus(s(X), Y) >= plus(X, s(Y))  because [19], by (Star) 
  19] plus*(s(X), Y) >= plus(X, s(Y))  because [20], [22] and [23], by (Stat) 
  20] s(X) > X  because [21], by definition 
  21] s*(X) >= X  because [4], by (Select) 
  22] plus*(s(X), Y) >= X  because [2], by (Select) 
  23] plus*(s(X), Y) >= s(Y)  because plus > s and [24], by (Copy) 
  24] plus*(s(X), Y) >= Y  because [25], by (Select) 
  25] Y >= Y  by (Meta) 

  26] plus(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [27], by (Star) 
  27] plus*(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [28], [29] and [31], by (Stat) 
  28] plus(X, Y) > X  because [9], by definition 
  29] plus*(plus(X, Y), Z) >= X  because [30], by (Select) 
  30] plus(X, Y) >= X  because [9], by (Star) 
  31] plus*(plus(X, Y), Z) >= plus(Y, Z)  because [32], [34] and [35], by (Stat) 
  32] plus(X, Y) > Y  because [33], by definition 
  33] plus*(X, Y) >= Y  because [14], by (Select) 
  34] plus*(plus(X, Y), Z) >= Y  because [12], by (Select) 
  35] plus*(plus(X, Y), Z) >= Z  because [36], by (Select) 
  36] Z >= Z  by (Meta) 

  37] mult(_|_, X) >= _|_  by (Bot) 

  38] mult(s(X), Y) >= plus(mult(X, Y), Y)  because [39], by (Star) 
  39] mult*(s(X), Y) >= plus(mult(X, Y), Y)  because mult > plus, [40] and [45], by (Copy) 
  40] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [41] and [44], by (Stat) 
  41] s(X) > X  because [42], by definition 
  42] s*(X) >= X  because [43], by (Select) 
  43] X >= X  by (Meta) 
  44] Y >= Y  by (Meta) 
  45] mult*(s(X), Y) >= Y  because [44], by (Select) 

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

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, static, formative) by (P_5, R_0, static, formative), where P_5 consists of:

  plus#(s(X), Y) =#> plus#(X, s(Y))   
  plus#(plus(X, Y), Z) =#> plus#(Y, Z)   

Thus, the original system is terminating if (P_5, R_0, static, formative) is finite.

We consider the dependency pair problem (P_5, R_0, static, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  plus#(s(X), Y) >? plus#(X, s(Y)) 
  plus#(plus(X, Y), Z) >? plus#(Y, Z) 
  plus(z, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(z, X) >= z 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

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:

  [[plus#(x_1, x_2)]] = plus#(x_1) 
  [[z]] = _|_ 

We choose Lex = {plus} and Mul = {mult, plus#, s}, and the following precedence: mult > plus > s > plus#

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

  plus#(s(X)) > plus#(X) 
  plus#(plus(X, Y)) >= plus#(Y) 
  plus(_|_, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

With these choices, we have:

  1] plus#(s(X)) > plus#(X)  because [2], by definition 
  2] plus#*(s(X)) >= plus#(X)  because plus# in Mul and [3], by (Stat) 
  3] s(X) > X  because [4], by definition 
  4] s*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] plus#(plus(X, Y)) >= plus#(Y)  because plus# in Mul and [7], by (Fun) 
  7] plus(X, Y) >= Y  because [8], by (Star) 
  8] plus*(X, Y) >= Y  because [9], by (Select) 
  9] Y >= Y  by (Meta) 

  10] plus(_|_, X) >= X  because [11], by (Star) 
  11] plus*(_|_, X) >= X  because [12], by (Select) 
  12] X >= X  by (Meta) 

  13] plus(s(X), Y) >= plus(X, s(Y))  because [14], by (Star) 
  14] plus*(s(X), Y) >= plus(X, s(Y))  because [3], [15] and [17], by (Stat) 
  15] plus*(s(X), Y) >= X  because [16], by (Select) 
  16] s(X) >= X  because [4], by (Star) 
  17] plus*(s(X), Y) >= s(Y)  because plus > s and [18], by (Copy) 
  18] plus*(s(X), Y) >= Y  because [19], by (Select) 
  19] Y >= Y  by (Meta) 

  20] plus(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [21], by (Star) 
  21] plus*(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [22], [25] and [27], by (Stat) 
  22] plus(X, Y) > X  because [23], by definition 
  23] plus*(X, Y) >= X  because [24], by (Select) 
  24] X >= X  by (Meta) 
  25] plus*(plus(X, Y), Z) >= X  because [26], by (Select) 
  26] plus(X, Y) >= X  because [23], by (Star) 
  27] plus*(plus(X, Y), Z) >= plus(Y, Z)  because [28], [30] and [31], by (Stat) 
  28] plus(X, Y) > Y  because [29], by definition 
  29] plus*(X, Y) >= Y  because [9], by (Select) 
  30] plus*(plus(X, Y), Z) >= Y  because [7], by (Select) 
  31] plus*(plus(X, Y), Z) >= Z  because [32], by (Select) 
  32] Z >= Z  by (Meta) 

  33] mult(_|_, X) >= _|_  by (Bot) 

  34] mult(s(X), Y) >= plus(mult(X, Y), Y)  because [35], by (Star) 
  35] mult*(s(X), Y) >= plus(mult(X, Y), Y)  because mult > plus, [36] and [41], by (Copy) 
  36] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [37] and [40], by (Stat) 
  37] s(X) > X  because [38], by definition 
  38] s*(X) >= X  because [39], by (Select) 
  39] X >= X  by (Meta) 
  40] Y >= Y  by (Meta) 
  41] mult*(s(X), Y) >= Y  because [40], by (Select) 

  42] mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because [43], by (Star) 
  43] mult*(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because mult > plus, [44] and [49], by (Copy) 
  44] mult*(plus(X, Y), Z) >= mult(X, Z)  because mult in Mul, [45] and [48], by (Stat) 
  45] plus(X, Y) > X  because [46], by definition 
  46] plus*(X, Y) >= X  because [47], by (Select) 
  47] X >= X  by (Meta) 
  48] Z >= Z  by (Meta) 
  49] mult*(plus(X, Y), Z) >= mult(Y, Z)  because mult in Mul, [50] and [48], by (Stat) 
  50] plus(X, Y) > Y  because [51], by definition 
  51] plus*(X, Y) >= Y  because [52], by (Select) 
  52] Y >= Y  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_0, static, formative) by (P_6, R_0, static, formative), where P_6 consists of:

  plus#(plus(X, Y), Z) =#> plus#(Y, Z)   

Thus, the original system is terminating if (P_6, R_0, static, formative) is finite.

We consider the dependency pair problem (P_6, R_0, static, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  plus#(plus(X, Y), Z) >? plus#(Y, Z) 
  plus(z, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(z, X) >= z 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

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:

  [[z]] = _|_ 

We choose Lex = {plus} and Mul = {mult, plus#, s}, and the following precedence: mult > plus > plus# > s

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

  plus#(plus(X, Y), Z) > plus#(Y, Z) 
  plus(_|_, X) >= X 
  plus(s(X), Y) >= plus(X, s(Y)) 
  plus(plus(X, Y), Z) >= plus(X, plus(Y, Z)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= plus(mult(X, Y), Y) 
  mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z)) 

With these choices, we have:

  1] plus#(plus(X, Y), Z) > plus#(Y, Z)  because [2], by definition 
  2] plus#*(plus(X, Y), Z) >= plus#(Y, Z)  because plus# in Mul, [3] and [6], by (Stat) 
  3] plus(X, Y) > Y  because [4], by definition 
  4] plus*(X, Y) >= Y  because [5], by (Select) 
  5] Y >= Y  by (Meta) 
  6] Z >= Z  by (Meta) 

  7] plus(_|_, X) >= X  because [8], by (Star) 
  8] plus*(_|_, X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 

  10] plus(s(X), Y) >= plus(X, s(Y))  because [11], by (Star) 
  11] plus*(s(X), Y) >= plus(X, s(Y))  because [12], [15] and [17], by (Stat) 
  12] s(X) > X  because [13], by definition 
  13] s*(X) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 
  15] plus*(s(X), Y) >= X  because [16], by (Select) 
  16] s(X) >= X  because [13], by (Star) 
  17] plus*(s(X), Y) >= s(Y)  because plus > s and [18], by (Copy) 
  18] plus*(s(X), Y) >= Y  because [19], by (Select) 
  19] Y >= Y  by (Meta) 

  20] plus(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [21], by (Star) 
  21] plus*(plus(X, Y), Z) >= plus(X, plus(Y, Z))  because [22], [25] and [27], by (Stat) 
  22] plus(X, Y) > X  because [23], by definition 
  23] plus*(X, Y) >= X  because [24], by (Select) 
  24] X >= X  by (Meta) 
  25] plus*(plus(X, Y), Z) >= X  because [26], by (Select) 
  26] plus(X, Y) >= X  because [23], by (Star) 
  27] plus*(plus(X, Y), Z) >= plus(Y, Z)  because [3], [28] and [30], by (Stat) 
  28] plus*(plus(X, Y), Z) >= Y  because [29], by (Select) 
  29] plus(X, Y) >= Y  because [4], by (Star) 
  30] plus*(plus(X, Y), Z) >= Z  because [6], by (Select) 

  31] mult(_|_, X) >= _|_  by (Bot) 

  32] mult(s(X), Y) >= plus(mult(X, Y), Y)  because [33], by (Star) 
  33] mult*(s(X), Y) >= plus(mult(X, Y), Y)  because mult > plus, [34] and [39], by (Copy) 
  34] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [35] and [38], by (Stat) 
  35] s(X) > X  because [36], by definition 
  36] s*(X) >= X  because [37], by (Select) 
  37] X >= X  by (Meta) 
  38] Y >= Y  by (Meta) 
  39] mult*(s(X), Y) >= Y  because [38], by (Select) 

  40] mult(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because [41], by (Star) 
  41] mult*(plus(X, Y), Z) >= plus(mult(X, Z), mult(Y, Z))  because mult > plus, [42] and [47], by (Copy) 
  42] mult*(plus(X, Y), Z) >= mult(X, Z)  because mult in Mul, [43] and [46], by (Stat) 
  43] plus(X, Y) > X  because [44], by definition 
  44] plus*(X, Y) >= X  because [45], by (Select) 
  45] X >= X  by (Meta) 
  46] Z >= Z  by (Meta) 
  47] mult*(plus(X, Y), Z) >= mult(Y, Z)  because mult in Mul, [48] and [46], by (Stat) 
  48] plus(X, Y) > Y  because [49], by definition 
  49] plus*(X, Y) >= Y  because [50], by (Select) 
  50] Y >= Y  by (Meta) 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_6, R_0) by ({}, R_0).  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.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.