We consider the system AotoYamada_05__Ex5TermProof.

  Alphabet:

    0 : [] --> a 
    add : [a * a] --> a 
    fact : [] --> a -> a 
    mult : [] --> a -> a -> a 
    rec : [a -> a -> a * a] --> a -> a 
    s : [a] --> a 

  Rules:

    add(0, x) => x 
    add(s(x), y) => s(add(x, y)) 
    mult 0 x => 0 
    mult s(x) y => add(mult x y, y) 
    rec(f, x) 0 => x 
    rec(f, x) s(y) => f s(y) (rec(f, x) y) 
    fact => rec(mult, s(0)) 

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]).

In order to do so, we start by eta-expanding the system, which gives:

  add(0, X) => X 
  add(s(X), Y) => s(add(X, Y)) 
  mult(0, X) => 0 
  mult(s(X), Y) => add(mult(X, Y), Y) 
  rec(F, X, 0) => X 
  rec(F, X, s(Y)) => F s(Y) rec(F, X, Y) 
  fact(X) => rec(/\x./\y.mult(x, y), s(0), X) 

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

  Dependency Pairs P_0:

    0] add#(s(X), Y) =#> add#(X, Y)   
    1] mult#(s(X), Y) =#> add#(mult(X, Y), Y)   
    2] mult#(s(X), Y) =#> mult#(X, Y)   
    3] rec#(F, X, s(Y)) =#> rec#(F, X, Y)   
    4] fact#(X) =#> rec#(/\x./\y.mult(x, y), s(0), X)   
    5] fact#(X) =#> mult#(Y, Z)   

  Rules R_0:

    add(0, X) => X 
    add(s(X), Y) => s(add(X, Y)) 
    mult(0, X) => 0 
    mult(s(X), Y) => add(mult(X, Y), Y) 
    rec(F, X, 0) => X 
    rec(F, X, s(Y)) => F s(Y) rec(F, X, Y) 
    fact(X) => rec(/\x./\y.mult(x, y), s(0), X) 

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 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:

  add#(s(X), Y) >? add#(X, Y) 
  mult#(s(X), Y) >? add#(mult(X, Y), Y) 
  mult#(s(X), Y) >? mult#(X, Y) 
  rec#(F, X, s(Y)) >? rec#(F, X, Y) 
  fact#(X) >? rec#(/\x./\y.mult(x, y), s(0), X) 
  fact#(X) >? mult#(Y, Z) 
  add(0, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(0, X) >= 0 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, 0) >= X 
  rec(F, X, s(Y)) >= F s(Y) rec(F, X, Y) 
  fact(X) >= rec(/\x./\y.mult(x, y), s(0), X) 

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

  pi( fact(X) ) = #argfun-fact#(rec(/\x./\y.mult(x, y), s(0), 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-fact#(x_1)]] = x_1 
  [[add#(x_1, x_2)]] = _|_ 
  [[mult#(x_1, x_2)]] = _|_ 

We choose Lex = {} and Mul = {0, @_{o -> o -> o}, @_{o -> o}, add, fact, fact#, mult, rec, rec#, s}, and the following precedence: fact > rec > @_{o -> o -> o} > @_{o -> o} > fact# > mult > 0 > add > rec# > s

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

  _|_ >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 
  rec#(F, X, s(Y)) >= rec#(F, X, Y) 
  fact#(X) > rec#(/\x./\y.mult(x, y), s(0), X) 
  fact#(X) >= _|_ 
  add(0, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(0, X) >= 0 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, 0) >= X 
  rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y)) 
  rec(/\x./\y.mult(x, y), s(0), X) >= rec(/\x./\y.mult(x, y), s(0), X) 

With these choices, we have:

  1] _|_ >= _|_  by (Bot) 

  2] _|_ >= _|_  by (Bot) 

  3] _|_ >= _|_  by (Bot) 

  4] rec#(F, X, s(Y)) >= rec#(F, X, Y)  because rec# in Mul, [5], [6] and [7], by (Fun) 
  5] F >= F  by (Meta) 
  6] X >= X  by (Meta) 
  7] s(Y) >= Y  because [8], by (Star) 
  8] s*(Y) >= Y  because [9], by (Select) 
  9] Y >= Y  by (Meta) 

  10] fact#(X) > rec#(/\x./\y.mult(x, y), s(0), X)  because [11], by definition 
  11] fact#*(X) >= rec#(/\x./\y.mult(x, y), s(0), X)  because fact# > rec#, [12], [19] and [21], by (Copy) 
  12] fact#*(X) >= /\y./\z.mult(y, z)  because [13], by (F-Abs) 
  13] fact#*(X, x) >= /\z.mult(x, z)  because [14], by (F-Abs) 
  14] fact#*(X, x, y) >= mult(x, y)  because fact# > mult, [15] and [17], by (Copy) 
  15] fact#*(X, x, y) >= x  because [16], by (Select) 
  16] x >= x  by (Var) 
  17] fact#*(X, x, y) >= y  because [18], by (Select) 
  18] y >= y  by (Var) 
  19] fact#*(X) >= s(0)  because fact# > s and [20], by (Copy) 
  20] fact#*(X) >= 0  because fact# > 0, by (Copy) 
  21] fact#*(X) >= X  because [22], by (Select) 
  22] X >= X  by (Meta) 

  23] fact#(X) >= _|_  by (Bot) 

  24] add(0, X) >= X  because [25], by (Star) 
  25] add*(0, X) >= X  because [26], by (Select) 
  26] X >= X  by (Meta) 

  27] add(s(X), Y) >= s(add(X, Y))  because [28], by (Star) 
  28] add*(s(X), Y) >= s(add(X, Y))  because add > s and [29], by (Copy) 
  29] add*(s(X), Y) >= add(X, Y)  because add in Mul, [30] and [33], by (Stat) 
  30] s(X) > X  because [31], by definition 
  31] s*(X) >= X  because [32], by (Select) 
  32] X >= X  by (Meta) 
  33] Y >= Y  by (Meta) 

  34] mult(0, X) >= 0  because [35], by (Star) 
  35] mult*(0, X) >= 0  because mult > 0, by (Copy) 

  36] mult(s(X), Y) >= add(mult(X, Y), Y)  because [37], by (Star) 
  37] mult*(s(X), Y) >= add(mult(X, Y), Y)  because mult > add, [38] and [43], by (Copy) 
  38] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [39] and [42], by (Stat) 
  39] s(X) > X  because [40], by definition 
  40] s*(X) >= X  because [41], by (Select) 
  41] X >= X  by (Meta) 
  42] Y >= Y  by (Meta) 
  43] mult*(s(X), Y) >= Y  because [42], by (Select) 

  44] rec(F, X, 0) >= X  because [45], by (Star) 
  45] rec*(F, X, 0) >= X  because [46], by (Select) 
  46] X >= X  by (Meta) 

  47] rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because [48], by (Star) 
  48] rec*(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because rec > @_{o -> o}, [49] and [54], by (Copy) 
  49] rec*(F, X, s(Y)) >= @_{o -> o -> o}(F, s(Y))  because rec > @_{o -> o -> o}, [50] and [51], by (Copy) 
  50] rec*(F, X, s(Y)) >= F  because [5], by (Select) 
  51] rec*(F, X, s(Y)) >= s(Y)  because [52], by (Select) 
  52] s(Y) >= s(Y)  because s in Mul and [53], by (Fun) 
  53] Y >= Y  by (Meta) 
  54] rec*(F, X, s(Y)) >= rec(F, X, Y)  because rec in Mul, [5], [6] and [55], by (Stat) 
  55] s(Y) > Y  because [56], by definition 
  56] s*(Y) >= Y  because [53], by (Select) 

  57] rec(/\x./\y.mult(x, y), s(0), X) >= rec(/\x./\y.mult(x, y), s(0), X)  because rec in Mul, [58], [63] and [65], by (Fun) 
  58] /\x./\z.mult(x, z) >= /\x./\z.mult(x, z)  because [59], by (Abs) 
  59] /\z.mult(y, z) >= /\z.mult(y, z)  because [60], by (Abs) 
  60] mult(y, x) >= mult(y, x)  because mult in Mul, [61] and [62], by (Fun) 
  61] y >= y  by (Var) 
  62] x >= x  by (Var) 
  63] s(0) >= s(0)  because s in Mul and [64], by (Fun) 
  64] 0 >= 0  by (Fun) 
  65] 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_0, R_0, static, formative) by (P_1, R_0, static, formative), where P_1 consists of:

  add#(s(X), Y) =#> add#(X, Y)   
  mult#(s(X), Y) =#> add#(mult(X, Y), Y)   
  mult#(s(X), Y) =#> mult#(X, Y)   
  rec#(F, X, s(Y)) =#> rec#(F, X, Y)   
  fact#(X) =#> mult#(Y, Z)   

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:

  add#(s(X), Y) >? add#(X, Y) 
  mult#(s(X), Y) >? add#(mult(X, Y), Y) 
  mult#(s(X), Y) >? mult#(X, Y) 
  rec#(F, X, s(Y)) >? rec#(F, X, Y) 
  fact#(X) >? mult#(Y, Z) 
  add(0, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(0, X) >= 0 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, 0) >= X 
  rec(F, X, s(Y)) >= F s(Y) rec(F, X, Y) 
  fact(X) >= rec(/\x./\y.mult(x, y), s(0), X) 

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

  pi( fact(X) ) = #argfun-fact#(rec(/\x./\y.mult(x, y), s(0), 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:

  [[0]] = _|_ 
  [[add#(x_1, x_2)]] = _|_ 
  [[fact(x_1)]] = x_1 
  [[mult#(x_1, x_2)]] = _|_ 

We choose Lex = {} and Mul = {#argfun-fact#, @_{o -> o -> o}, @_{o -> o}, add, fact#, mult, rec, rec#, s}, and the following precedence: #argfun-fact# > mult > fact# > rec > @_{o -> o} > rec# > add > @_{o -> o -> o} > s

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

  _|_ >= _|_ 
  _|_ >= _|_ 
  _|_ >= _|_ 
  rec#(F, X, s(Y)) >= rec#(F, X, Y) 
  fact#(X) > _|_ 
  add(_|_, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, _|_) >= X 
  rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y)) 
  #argfun-fact#(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X) 

With these choices, we have:

  1] _|_ >= _|_  by (Bot) 

  2] _|_ >= _|_  by (Bot) 

  3] _|_ >= _|_  by (Bot) 

  4] rec#(F, X, s(Y)) >= rec#(F, X, Y)  because [5], by (Star) 
  5] rec#*(F, X, s(Y)) >= rec#(F, X, Y)  because rec# in Mul, [6], [7] and [8], by (Stat) 
  6] F >= F  by (Meta) 
  7] X >= X  by (Meta) 
  8] s(Y) > Y  because [9], by definition 
  9] s*(Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 

  11] fact#(X) > _|_  because [12], by definition 
  12] fact#*(X) >= _|_  by (Bot) 

  13] add(_|_, X) >= X  because [14], by (Star) 
  14] add*(_|_, X) >= X  because [15], by (Select) 
  15] X >= X  by (Meta) 

  16] add(s(X), Y) >= s(add(X, Y))  because [17], by (Star) 
  17] add*(s(X), Y) >= s(add(X, Y))  because add > s and [18], by (Copy) 
  18] add*(s(X), Y) >= add(X, Y)  because add in Mul, [19] and [22], by (Stat) 
  19] s(X) > X  because [20], by definition 
  20] s*(X) >= X  because [21], by (Select) 
  21] X >= X  by (Meta) 
  22] Y >= Y  by (Meta) 

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

  24] mult(s(X), Y) >= add(mult(X, Y), Y)  because [25], by (Star) 
  25] mult*(s(X), Y) >= add(mult(X, Y), Y)  because mult > add, [26] and [31], by (Copy) 
  26] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [27] and [30], by (Stat) 
  27] s(X) > X  because [28], by definition 
  28] s*(X) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 
  30] Y >= Y  by (Meta) 
  31] mult*(s(X), Y) >= Y  because [30], by (Select) 

  32] rec(F, X, _|_) >= X  because [33], by (Star) 
  33] rec*(F, X, _|_) >= X  because [34], by (Select) 
  34] X >= X  by (Meta) 

  35] rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because [36], by (Star) 
  36] rec*(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because rec > @_{o -> o}, [37] and [42], by (Copy) 
  37] rec*(F, X, s(Y)) >= @_{o -> o -> o}(F, s(Y))  because rec > @_{o -> o -> o}, [38] and [39], by (Copy) 
  38] rec*(F, X, s(Y)) >= F  because [6], by (Select) 
  39] rec*(F, X, s(Y)) >= s(Y)  because [40], by (Select) 
  40] s(Y) >= s(Y)  because s in Mul and [41], by (Fun) 
  41] Y >= Y  by (Meta) 
  42] rec*(F, X, s(Y)) >= rec(F, X, Y)  because rec in Mul, [6], [7] and [8], by (Stat) 

  43] #argfun-fact#(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X)  because [44], by (Star) 
  44] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X)  because #argfun-fact# > rec, [45], [52] and [54], by (Copy) 
  45] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= /\x./\y.mult(x, y)  because [46], by (F-Abs) 
  46] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z) >= /\x.mult(z, x)  because [47], by (F-Abs) 
  47] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= mult(z, u)  because #argfun-fact# > mult, [48] and [50], by (Copy) 
  48] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= z  because [49], by (Select) 
  49] z >= z  by (Var) 
  50] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= u  because [51], by (Select) 
  51] u >= u  by (Var) 
  52] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= s(_|_)  because #argfun-fact# > s and [53], by (Copy) 
  53] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= _|_  by (Bot) 
  54] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= X  because [55], by (Select) 
  55] rec(/\x./\y.mult(x, y), s(_|_), X) >= X  because [56], by (Star) 
  56] rec*(/\x./\y.mult(x, y), s(_|_), X) >= X  because [57], by (Select) 
  57] 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_0, static, formative) by (P_2, R_0, static, formative), where P_2 consists of:

  add#(s(X), Y) =#> add#(X, Y)   
  mult#(s(X), Y) =#> add#(mult(X, Y), Y)   
  mult#(s(X), Y) =#> mult#(X, Y)   
  rec#(F, X, s(Y)) =#> rec#(F, X, Y)   

Thus, the original system is terminating if (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:

  add#(s(X), Y) >? add#(X, Y) 
  mult#(s(X), Y) >? add#(mult(X, Y), Y) 
  mult#(s(X), Y) >? mult#(X, Y) 
  rec#(F, X, s(Y)) >? rec#(F, X, Y) 
  add(0, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(0, X) >= 0 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, 0) >= X 
  rec(F, X, s(Y)) >= F s(Y) rec(F, X, Y) 
  fact(X) >= rec(/\x./\y.mult(x, y), s(0), X) 

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

  pi( fact(X) ) = #argfun-fact#(rec(/\x./\y.mult(x, y), s(0), 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:

  [[0]] = _|_ 
  [[add#(x_1, x_2)]] = add#(x_2) 
  [[fact(x_1)]] = x_1 

We choose Lex = {} and Mul = {#argfun-fact#, @_{o -> o -> o}, @_{o -> o}, add, add#, mult, mult#, rec, rec#, s}, and the following precedence: #argfun-fact# > mult > rec > add > @_{o -> o} > @_{o -> o -> o} > rec# > s > add# = mult#

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

  add#(X) >= add#(X) 
  mult#(s(X), Y) > add#(Y) 
  mult#(s(X), Y) > mult#(X, Y) 
  rec#(F, X, s(Y)) >= rec#(F, X, Y) 
  add(_|_, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, _|_) >= X 
  rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y)) 
  #argfun-fact#(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X) 

With these choices, we have:

  1] add#(X) >= add#(X)  because add# in Mul and [2], by (Fun) 
  2] X >= X  by (Meta) 

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

  6] mult#(s(X), Y) > mult#(X, Y)  because [7], by definition 
  7] mult#*(s(X), Y) >= mult#(X, Y)  because mult# in Mul, [8] and [5], by (Stat) 
  8] s(X) > X  because [9], by definition 
  9] s*(X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 

  11] rec#(F, X, s(Y)) >= rec#(F, X, Y)  because [12], by (Star) 
  12] rec#*(F, X, s(Y)) >= rec#(F, X, Y)  because rec# in Mul, [13], [14] and [15], by (Stat) 
  13] F >= F  by (Meta) 
  14] X >= X  by (Meta) 
  15] s(Y) > Y  because [16], by definition 
  16] s*(Y) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 

  18] add(_|_, X) >= X  because [19], by (Star) 
  19] add*(_|_, X) >= X  because [20], by (Select) 
  20] X >= X  by (Meta) 

  21] add(s(X), Y) >= s(add(X, Y))  because [22], by (Star) 
  22] add*(s(X), Y) >= s(add(X, Y))  because add > s and [23], by (Copy) 
  23] add*(s(X), Y) >= add(X, Y)  because add in Mul, [24] and [2], by (Stat) 
  24] s(X) > X  because [25], by definition 
  25] s*(X) >= X  because [26], by (Select) 
  26] X >= X  by (Meta) 

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

  28] mult(s(X), Y) >= add(mult(X, Y), Y)  because [29], by (Star) 
  29] mult*(s(X), Y) >= add(mult(X, Y), Y)  because mult > add, [30] and [31], by (Copy) 
  30] mult*(s(X), Y) >= mult(X, Y)  because mult in Mul, [8] and [5], by (Stat) 
  31] mult*(s(X), Y) >= Y  because [5], by (Select) 

  32] rec(F, X, _|_) >= X  because [33], by (Star) 
  33] rec*(F, X, _|_) >= X  because [34], by (Select) 
  34] X >= X  by (Meta) 

  35] rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because [36], by (Star) 
  36] rec*(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because rec > @_{o -> o}, [37] and [42], by (Copy) 
  37] rec*(F, X, s(Y)) >= @_{o -> o -> o}(F, s(Y))  because rec > @_{o -> o -> o}, [38] and [39], by (Copy) 
  38] rec*(F, X, s(Y)) >= F  because [13], by (Select) 
  39] rec*(F, X, s(Y)) >= s(Y)  because [40], by (Select) 
  40] s(Y) >= s(Y)  because s in Mul and [41], by (Fun) 
  41] Y >= Y  by (Meta) 
  42] rec*(F, X, s(Y)) >= rec(F, X, Y)  because rec in Mul, [13], [14] and [15], by (Stat) 

  43] #argfun-fact#(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X)  because [44], by (Star) 
  44] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X)  because #argfun-fact# > rec, [45], [52] and [54], by (Copy) 
  45] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= /\x./\y.mult(x, y)  because [46], by (F-Abs) 
  46] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z) >= /\x.mult(z, x)  because [47], by (F-Abs) 
  47] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= mult(z, u)  because #argfun-fact# > mult, [48] and [50], by (Copy) 
  48] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= z  because [49], by (Select) 
  49] z >= z  by (Var) 
  50] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= u  because [51], by (Select) 
  51] u >= u  by (Var) 
  52] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= s(_|_)  because #argfun-fact# > s and [53], by (Copy) 
  53] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= _|_  by (Bot) 
  54] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= X  because [55], by (Select) 
  55] rec(/\x./\y.mult(x, y), s(_|_), X) >= X  because [56], by (Star) 
  56] rec*(/\x./\y.mult(x, y), s(_|_), X) >= X  because [57], by (Select) 
  57] X >= X  by (Meta) 

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

  add#(s(X), Y) =#> add#(X, Y)   
  rec#(F, X, s(Y)) =#> rec#(F, X, Y)   

Thus, the original system is terminating if (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:

  add#(s(X), Y) >? add#(X, Y) 
  rec#(F, X, s(Y)) >? rec#(F, X, Y) 
  add(0, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(0, X) >= 0 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, 0) >= X 
  rec(F, X, s(Y)) >= F s(Y) rec(F, X, Y) 
  fact(X) >= rec(/\x./\y.mult(x, y), s(0), X) 

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

  pi( fact(X) ) = #argfun-fact#(rec(/\x./\y.mult(x, y), s(0), 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-fact#(x_1)]] = x_1 
  [[add#(x_1, x_2)]] = x_1 
  [[fact(x_1)]] = x_1 

We choose Lex = {} and Mul = {0, @_{o -> o -> o}, @_{o -> o}, add, mult, rec, rec#, s}, and the following precedence: mult > 0 > add > rec > @_{o -> o -> o} > @_{o -> o} > rec# > s

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

  s(X) > X 
  rec#(F, X, s(Y)) >= rec#(F, X, Y) 
  add(0, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(0, X) >= 0 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, 0) >= X 
  rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y)) 
  rec(/\x./\y.mult(x, y), s(0), X) >= rec(/\x./\y.mult(x, y), s(0), X) 

With these choices, we have:

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

  4] rec#(F, X, s(Y)) >= rec#(F, X, Y)  because [5], by (Star) 
  5] rec#*(F, X, s(Y)) >= rec#(F, X, Y)  because rec# in Mul, [6], [7] and [8], by (Stat) 
  6] F >= F  by (Meta) 
  7] X >= X  by (Meta) 
  8] s(Y) > Y  because [9], by definition 
  9] s*(Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 

  11] add(0, X) >= X  because [12], by (Star) 
  12] add*(0, X) >= X  because [13], by (Select) 
  13] X >= X  by (Meta) 

  14] add(s(X), Y) >= s(add(X, Y))  because [15], by (Star) 
  15] add*(s(X), Y) >= s(add(X, Y))  because add > s and [16], by (Copy) 
  16] add*(s(X), Y) >= add(X, Y)  because add in Mul, [17] and [18], by (Stat) 
  17] s(X) > X  because [2], by definition 
  18] Y >= Y  by (Meta) 

  19] mult(0, X) >= 0  because [20], by (Star) 
  20] mult*(0, X) >= 0  because mult > 0, by (Copy) 

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

  29] rec(F, X, 0) >= X  because [30], by (Star) 
  30] rec*(F, X, 0) >= X  because [31], by (Select) 
  31] X >= X  by (Meta) 

  32] rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because [33], by (Star) 
  33] rec*(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because rec > @_{o -> o}, [34] and [39], by (Copy) 
  34] rec*(F, X, s(Y)) >= @_{o -> o -> o}(F, s(Y))  because rec > @_{o -> o -> o}, [35] and [36], by (Copy) 
  35] rec*(F, X, s(Y)) >= F  because [6], by (Select) 
  36] rec*(F, X, s(Y)) >= s(Y)  because [37], by (Select) 
  37] s(Y) >= s(Y)  because s in Mul and [38], by (Fun) 
  38] Y >= Y  by (Meta) 
  39] rec*(F, X, s(Y)) >= rec(F, X, Y)  because rec in Mul, [6], [7] and [8], by (Stat) 

  40] rec(/\x./\y.mult(x, y), s(0), X) >= rec(/\x./\y.mult(x, y), s(0), X)  because rec in Mul, [41], [46] and [48], by (Fun) 
  41] /\x./\z.mult(x, z) >= /\x./\z.mult(x, z)  because [42], by (Abs) 
  42] /\z.mult(y, z) >= /\z.mult(y, z)  because [43], by (Abs) 
  43] mult(y, x) >= mult(y, x)  because mult in Mul, [44] and [45], by (Fun) 
  44] y >= y  by (Var) 
  45] x >= x  by (Var) 
  46] s(0) >= s(0)  because s in Mul and [47], by (Fun) 
  47] 0 >= 0  by (Fun) 
  48] 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_3, R_0, static, formative) by (P_4, R_0, static, formative), where P_4 consists of:

  rec#(F, X, s(Y)) =#> rec#(F, X, Y)   

Thus, the original system is terminating if (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:

  rec#(F, X, s(Y)) >? rec#(F, X, Y) 
  add(0, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(0, X) >= 0 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, 0) >= X 
  rec(F, X, s(Y)) >= F s(Y) rec(F, X, Y) 
  fact(X) >= rec(/\x./\y.mult(x, y), s(0), X) 

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

  pi( fact(X) ) = #argfun-fact#(rec(/\x./\y.mult(x, y), s(0), 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:

  [[0]] = _|_ 
  [[fact(x_1)]] = x_1 

We choose Lex = {} and Mul = {#argfun-fact#, @_{o -> o -> o}, @_{o -> o}, add, mult, rec, rec#, s}, and the following precedence: #argfun-fact# > mult > add > rec > @_{o -> o -> o} > @_{o -> o} > rec# > s

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

  rec#(F, X, s(Y)) > rec#(F, X, Y) 
  add(_|_, X) >= X 
  add(s(X), Y) >= s(add(X, Y)) 
  mult(_|_, X) >= _|_ 
  mult(s(X), Y) >= add(mult(X, Y), Y) 
  rec(F, X, _|_) >= X 
  rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y)) 
  #argfun-fact#(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X) 

With these choices, we have:

  1] rec#(F, X, s(Y)) > rec#(F, X, Y)  because [2], by definition 
  2] rec#*(F, X, s(Y)) >= rec#(F, X, Y)  because rec# in Mul, [3], [4] and [5], by (Stat) 
  3] F >= F  by (Meta) 
  4] X >= X  by (Meta) 
  5] s(Y) > Y  because [6], by definition 
  6] s*(Y) >= Y  because [7], by (Select) 
  7] Y >= Y  by (Meta) 

  8] add(_|_, X) >= X  because [9], by (Star) 
  9] add*(_|_, X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 

  11] add(s(X), Y) >= s(add(X, Y))  because [12], by (Star) 
  12] add*(s(X), Y) >= s(add(X, Y))  because add > s and [13], by (Copy) 
  13] add*(s(X), Y) >= add(X, Y)  because add in Mul, [14] and [17], by (Stat) 
  14] s(X) > X  because [15], by definition 
  15] s*(X) >= X  because [16], by (Select) 
  16] X >= X  by (Meta) 
  17] Y >= Y  by (Meta) 

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

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

  27] rec(F, X, _|_) >= X  because [28], by (Star) 
  28] rec*(F, X, _|_) >= X  because [29], by (Select) 
  29] X >= X  by (Meta) 

  30] rec(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because [31], by (Star) 
  31] rec*(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because rec > @_{o -> o}, [32] and [37], by (Copy) 
  32] rec*(F, X, s(Y)) >= @_{o -> o -> o}(F, s(Y))  because rec > @_{o -> o -> o}, [33] and [34], by (Copy) 
  33] rec*(F, X, s(Y)) >= F  because [3], by (Select) 
  34] rec*(F, X, s(Y)) >= s(Y)  because rec > s and [35], by (Copy) 
  35] rec*(F, X, s(Y)) >= Y  because [36], by (Select) 
  36] s(Y) >= Y  because [6], by (Star) 
  37] rec*(F, X, s(Y)) >= rec(F, X, Y)  because rec in Mul, [3], [4] and [5], by (Stat) 

  38] #argfun-fact#(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X)  because [39], by (Star) 
  39] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= rec(/\x./\y.mult(x, y), s(_|_), X)  because #argfun-fact# > rec, [40], [47] and [51], by (Copy) 
  40] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= /\x./\y.mult(x, y)  because [41], by (F-Abs) 
  41] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z) >= /\x.mult(z, x)  because [42], by (F-Abs) 
  42] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= mult(z, u)  because #argfun-fact# > mult, [43] and [45], by (Copy) 
  43] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= z  because [44], by (Select) 
  44] z >= z  by (Var) 
  45] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X), z, u) >= u  because [46], by (Select) 
  46] u >= u  by (Var) 
  47] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= s(_|_)  because [48], by (Select) 
  48] rec(/\x./\y.mult(x, y), s(_|_), X) >= s(_|_)  because [49], by (Star) 
  49] rec*(/\x./\y.mult(x, y), s(_|_), X) >= s(_|_)  because rec > s and [50], by (Copy) 
  50] rec*(/\x./\y.mult(x, y), s(_|_), X) >= _|_  by (Bot) 
  51] #argfun-fact#*(rec(/\x./\y.mult(x, y), s(_|_), X)) >= X  because [52], by (Select) 
  52] rec(/\x./\y.mult(x, y), s(_|_), X) >= X  because [53], by (Star) 
  53] rec*(/\x./\y.mult(x, y), s(_|_), X) >= X  because [54], by (Select) 
  54] X >= X  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.

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.