/export/starexec/sandbox/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
We consider the system theBenchmark.

  Alphabet:

    0 : [] --> a 
    cons : [a * b] --> b 
    double : [b] --> b 
    inc : [b] --> b 
    map : [a -> a * b] --> b 
    nil : [] --> b 
    plus : [a] --> a -> a 
    s : [a] --> a 
    times : [a] --> a -> a 

  Rules:

    plus(0) x => x 
    plus(s(x)) y => s(plus(x) y) 
    times(0) x => 0 
    times(s(x)) y => plus(times(x) y) y 
    inc(x) => map(plus(s(0)), x) 
    double(x) => map(times(s(s(0))), x) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 

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

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  plus(0) X >? X 
  plus(s(X)) Y >? s(plus(X) Y) 
  times(0) X >? 0 
  times(s(X)) Y >? plus(times(X) Y) Y 
  inc(X) >? map(plus(s(0)), X) 
  double(X) >? map(times(s(s(0))), X) 
  map(F, nil) >? nil 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 

about to try horpo

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[nil]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, cons, double, inc, map, plus, s, times}, and the following precedence: double > inc > map > cons > times > @_{o -> o} > s > plus

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

  @_{o -> o}(plus(_|_), X) >= X 
  @_{o -> o}(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y)) 
  @_{o -> o}(times(_|_), X) >= _|_ 
  @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) 
  inc(X) >= map(plus(s(_|_)), X) 
  double(X) >= map(times(s(s(_|_))), X) 
  map(F, _|_) > _|_ 
  map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) 

With these choices, we have:

  1] @_{o -> o}(plus(_|_), X) >= X  because [2], by (Star) 
  2] @_{o -> o}*(plus(_|_), X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] @_{o -> o}(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y))  because [5], by (Star) 
  5] @_{o -> o}*(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y))  because @_{o -> o} > s and [6], by (Copy) 
  6] @_{o -> o}*(plus(s(X)), Y) >= @_{o -> o}(plus(X), Y)  because @_{o -> o} in Mul, [7] and [12], by (Stat) 
  7] plus(s(X)) > plus(X)  because [8], by definition 
  8] plus*(s(X)) >= plus(X)  because plus in Mul and [9], by (Stat) 
  9] s(X) > X  because [10], by definition 
  10] s*(X) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] Y >= Y  by (Meta) 

  13] @_{o -> o}(times(_|_), X) >= _|_  by (Bot) 

  14] @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [15], by (Star) 
  15] @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [16], by (Select) 
  16] times(s(X)) @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [17] 
  17] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because times > @_{o -> o}, [18] and [24], by (Copy) 
  18] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= plus(@_{o -> o}(times(X), Y))  because times > plus and [19], by (Copy) 
  19] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(times(X), Y)  because times > @_{o -> o}, [20] and [24], by (Copy) 
  20] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= times(X)  because times in Mul and [21], by (Stat) 
  21] s(X) > X  because [22], by definition 
  22] s*(X) >= X  because [23], by (Select) 
  23] X >= X  by (Meta) 
  24] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= Y  because [25], by (Select) 
  25] @_{o -> o}*(times(s(X)), Y) >= Y  because [26], by (Select) 
  26] Y >= Y  by (Meta) 

  27] inc(X) >= map(plus(s(_|_)), X)  because [28], by (Star) 
  28] inc*(X) >= map(plus(s(_|_)), X)  because inc > map, [29] and [32], by (Copy) 
  29] inc*(X) >= plus(s(_|_))  because inc > plus and [30], by (Copy) 
  30] inc*(X) >= s(_|_)  because inc > s and [31], by (Copy) 
  31] inc*(X) >= _|_  by (Bot) 
  32] inc*(X) >= X  because [33], by (Select) 
  33] X >= X  by (Meta) 

  34] double(X) >= map(times(s(s(_|_))), X)  because [35], by (Star) 
  35] double*(X) >= map(times(s(s(_|_))), X)  because double > map, [36] and [40], by (Copy) 
  36] double*(X) >= times(s(s(_|_)))  because double > times and [37], by (Copy) 
  37] double*(X) >= s(s(_|_))  because double > s and [38], by (Copy) 
  38] double*(X) >= s(_|_)  because double > s and [39], by (Copy) 
  39] double*(X) >= _|_  by (Bot) 
  40] double*(X) >= X  because [41], by (Select) 
  41] X >= X  by (Meta) 

  42] map(F, _|_) > _|_  because [43], by definition 
  43] map*(F, _|_) >= _|_  by (Bot) 

  44] map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because [45], by (Star) 
  45] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [46] and [53], by (Copy) 
  46] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [47] and [49], by (Copy) 
  47] map*(F, cons(X, Y)) >= F  because [48], by (Select) 
  48] F >= F  by (Meta) 
  49] map*(F, cons(X, Y)) >= X  because [50], by (Select) 
  50] cons(X, Y) >= X  because [51], by (Star) 
  51] cons*(X, Y) >= X  because [52], by (Select) 
  52] X >= X  by (Meta) 
  53] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [54] and [55], by (Stat) 
  54] F >= F  by (Meta) 
  55] cons(X, Y) > Y  because [56], by definition 
  56] cons*(X, Y) >= Y  because [57], by (Select) 
  57] Y >= Y  by (Meta) 

We can thus remove the following rules:

  map(F, nil) => nil 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  plus(0) X >? X 
  plus(s(X)) Y >? s(plus(X) Y) 
  times(0) X >? 0 
  times(s(X)) Y >? plus(times(X) Y) Y 
  inc(X) >? map(plus(s(0)), X) 
  double(X) >? map(times(s(s(0))), X) 
  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 

about to try horpo

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, cons, double, inc, map, plus, s, times}, and the following precedence: double > inc > times > plus > map > @_{o -> o} > cons > s

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

  @_{o -> o}(plus(_|_), X) >= X 
  @_{o -> o}(plus(s(X)), Y) > s(@_{o -> o}(plus(X), Y)) 
  @_{o -> o}(times(_|_), X) >= _|_ 
  @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) 
  inc(X) >= map(plus(s(_|_)), X) 
  double(X) >= map(times(s(s(_|_))), X) 
  map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y)) 

With these choices, we have:

  1] @_{o -> o}(plus(_|_), X) >= X  because [2], by (Star) 
  2] @_{o -> o}*(plus(_|_), X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] @_{o -> o}(plus(s(X)), Y) > s(@_{o -> o}(plus(X), Y))  because [5], by definition 
  5] @_{o -> o}*(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y))  because @_{o -> o} > s and [6], by (Copy) 
  6] @_{o -> o}*(plus(s(X)), Y) >= @_{o -> o}(plus(X), Y)  because @_{o -> o} in Mul, [7] and [12], by (Stat) 
  7] plus(s(X)) > plus(X)  because [8], by definition 
  8] plus*(s(X)) >= plus(X)  because plus in Mul and [9], by (Stat) 
  9] s(X) > X  because [10], by definition 
  10] s*(X) >= X  because [11], by (Select) 
  11] X >= X  by (Meta) 
  12] Y >= Y  by (Meta) 

  13] @_{o -> o}(times(_|_), X) >= _|_  by (Bot) 

  14] @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [15], by (Star) 
  15] @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [16], by (Select) 
  16] times(s(X)) @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [17] 
  17] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because times > @_{o -> o}, [18] and [27], by (Copy) 
  18] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= plus(@_{o -> o}(times(X), Y))  because times > plus and [19], by (Copy) 
  19] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(times(X), Y)  because [20], by (Select) 
  20] @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(times(X), Y)  because @_{o -> o} in Mul, [21] and [26], by (Stat) 
  21] times(s(X)) > times(X)  because [22], by definition 
  22] times*(s(X)) >= times(X)  because times in Mul and [23], by (Stat) 
  23] s(X) > X  because [24], by definition 
  24] s*(X) >= X  because [25], by (Select) 
  25] X >= X  by (Meta) 
  26] Y >= Y  by (Meta) 
  27] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= Y  because [28], by (Select) 
  28] @_{o -> o}*(times(s(X)), Y) >= Y  because [26], by (Select) 

  29] inc(X) >= map(plus(s(_|_)), X)  because [30], by (Star) 
  30] inc*(X) >= map(plus(s(_|_)), X)  because inc > map, [31] and [34], by (Copy) 
  31] inc*(X) >= plus(s(_|_))  because inc > plus and [32], by (Copy) 
  32] inc*(X) >= s(_|_)  because inc > s and [33], by (Copy) 
  33] inc*(X) >= _|_  by (Bot) 
  34] inc*(X) >= X  because [35], by (Select) 
  35] X >= X  by (Meta) 

  36] double(X) >= map(times(s(s(_|_))), X)  because [37], by (Star) 
  37] double*(X) >= map(times(s(s(_|_))), X)  because double > map, [38] and [42], by (Copy) 
  38] double*(X) >= times(s(s(_|_)))  because double > times and [39], by (Copy) 
  39] double*(X) >= s(s(_|_))  because double > s and [40], by (Copy) 
  40] double*(X) >= s(_|_)  because double > s and [41], by (Copy) 
  41] double*(X) >= _|_  by (Bot) 
  42] double*(X) >= X  because [43], by (Select) 
  43] X >= X  by (Meta) 

  44] map(F, cons(X, Y)) > cons(@_{o -> o}(F, X), map(F, Y))  because [45], by definition 
  45] map*(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y))  because map > cons, [46] and [53], by (Copy) 
  46] map*(F, cons(X, Y)) >= @_{o -> o}(F, X)  because map > @_{o -> o}, [47] and [49], by (Copy) 
  47] map*(F, cons(X, Y)) >= F  because [48], by (Select) 
  48] F >= F  by (Meta) 
  49] map*(F, cons(X, Y)) >= X  because [50], by (Select) 
  50] cons(X, Y) >= X  because [51], by (Star) 
  51] cons*(X, Y) >= X  because [52], by (Select) 
  52] X >= X  by (Meta) 
  53] map*(F, cons(X, Y)) >= map(F, Y)  because map in Mul, [54] and [55], by (Stat) 
  54] F >= F  by (Meta) 
  55] cons(X, Y) > Y  because [56], by definition 
  56] cons*(X, Y) >= Y  because [57], by (Select) 
  57] Y >= Y  by (Meta) 

We can thus remove the following rules:

  plus(s(X)) Y => s(plus(X) Y) 
  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  plus(0) X >? X 
  times(0) X >? 0 
  times(s(X)) Y >? plus(times(X) Y) Y 
  inc(X) >? map(plus(s(0)), X) 
  double(X) >? map(times(s(s(0))), X) 

about to try horpo

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, double, inc, map, plus, s, times}, and the following precedence: double > inc > s > times > @_{o -> o} > plus > map

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

  @_{o -> o}(plus(_|_), X) > X 
  @_{o -> o}(times(_|_), X) >= _|_ 
  @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) 
  inc(X) >= map(plus(s(_|_)), X) 
  double(X) >= map(times(s(s(_|_))), X) 

With these choices, we have:

  1] @_{o -> o}(plus(_|_), X) > X  because [2], by definition 
  2] @_{o -> o}*(plus(_|_), X) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] @_{o -> o}(times(_|_), X) >= _|_  by (Bot) 

  5] @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [6], by (Star) 
  6] @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [7], by (Select) 
  7] times(s(X)) @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [8] 
  8] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because times > @_{o -> o}, [9] and [15], by (Copy) 
  9] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= plus(@_{o -> o}(times(X), Y))  because times > plus and [10], by (Copy) 
  10] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(times(X), Y)  because times > @_{o -> o}, [11] and [15], by (Copy) 
  11] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= times(X)  because times in Mul and [12], by (Stat) 
  12] s(X) > X  because [13], by definition 
  13] s*(X) >= X  because [14], by (Select) 
  14] X >= X  by (Meta) 
  15] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= Y  because [16], by (Select) 
  16] @_{o -> o}*(times(s(X)), Y) >= Y  because [17], by (Select) 
  17] Y >= Y  by (Meta) 

  18] inc(X) >= map(plus(s(_|_)), X)  because [19], by (Star) 
  19] inc*(X) >= map(plus(s(_|_)), X)  because inc > map, [20] and [23], by (Copy) 
  20] inc*(X) >= plus(s(_|_))  because inc > plus and [21], by (Copy) 
  21] inc*(X) >= s(_|_)  because inc > s and [22], by (Copy) 
  22] inc*(X) >= _|_  by (Bot) 
  23] inc*(X) >= X  because [24], by (Select) 
  24] X >= X  by (Meta) 

  25] double(X) >= map(times(s(s(_|_))), X)  because [26], by (Star) 
  26] double*(X) >= map(times(s(s(_|_))), X)  because double > map, [27] and [31], by (Copy) 
  27] double*(X) >= times(s(s(_|_)))  because double > times and [28], by (Copy) 
  28] double*(X) >= s(s(_|_))  because double > s and [29], by (Copy) 
  29] double*(X) >= s(_|_)  because double > s and [30], by (Copy) 
  30] double*(X) >= _|_  by (Bot) 
  31] double*(X) >= X  because [32], by (Select) 
  32] X >= X  by (Meta) 

We can thus remove the following rules:

  plus(0) X => X 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  times(0) X >? 0 
  times(s(X)) Y >? plus(times(X) Y) Y 
  inc(X) >? map(plus(s(0)), X) 
  double(X) >? map(times(s(s(0))), X) 

about to try horpo

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, double, inc, map, plus, s, times}, and the following precedence: double > inc > map > s > times > @_{o -> o} > plus

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

  @_{o -> o}(times(_|_), X) > _|_ 
  @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) 
  inc(X) >= map(plus(s(_|_)), X) 
  double(X) >= map(times(s(s(_|_))), X) 

With these choices, we have:

  1] @_{o -> o}(times(_|_), X) > _|_  because [2], by definition 
  2] @_{o -> o}*(times(_|_), X) >= _|_  by (Bot) 

  3] @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [4], by (Star) 
  4] @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [5], by (Select) 
  5] times(s(X)) @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [6] 
  6] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because times > @_{o -> o}, [7] and [13], by (Copy) 
  7] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= plus(@_{o -> o}(times(X), Y))  because times > plus and [8], by (Copy) 
  8] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(times(X), Y)  because times > @_{o -> o}, [9] and [13], by (Copy) 
  9] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= times(X)  because times in Mul and [10], by (Stat) 
  10] s(X) > X  because [11], by definition 
  11] s*(X) >= X  because [12], by (Select) 
  12] X >= X  by (Meta) 
  13] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= Y  because [14], by (Select) 
  14] @_{o -> o}*(times(s(X)), Y) >= Y  because [15], by (Select) 
  15] Y >= Y  by (Meta) 

  16] inc(X) >= map(plus(s(_|_)), X)  because [17], by (Star) 
  17] inc*(X) >= map(plus(s(_|_)), X)  because inc > map, [18] and [21], by (Copy) 
  18] inc*(X) >= plus(s(_|_))  because inc > plus and [19], by (Copy) 
  19] inc*(X) >= s(_|_)  because inc > s and [20], by (Copy) 
  20] inc*(X) >= _|_  by (Bot) 
  21] inc*(X) >= X  because [22], by (Select) 
  22] X >= X  by (Meta) 

  23] double(X) >= map(times(s(s(_|_))), X)  because [24], by (Star) 
  24] double*(X) >= map(times(s(s(_|_))), X)  because double > map, [25] and [29], by (Copy) 
  25] double*(X) >= times(s(s(_|_)))  because double > times and [26], by (Copy) 
  26] double*(X) >= s(s(_|_))  because double > s and [27], by (Copy) 
  27] double*(X) >= s(_|_)  because double > s and [28], by (Copy) 
  28] double*(X) >= _|_  by (Bot) 
  29] double*(X) >= X  because [30], by (Select) 
  30] X >= X  by (Meta) 

We can thus remove the following rules:

  times(0) X => 0 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  times(s(X)) Y >? plus(times(X) Y) Y 
  inc(X) >? map(plus(s(0)), X) 
  double(X) >? map(times(s(s(0))), X) 

about to try horpo

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 

We choose Lex = {} and Mul = {@_{o -> o}, double, inc, map, plus, s, times}, and the following precedence: inc > double > map > s > times > @_{o -> o} > plus

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

  @_{o -> o}(times(s(X)), Y) > @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) 
  inc(X) >= map(plus(s(_|_)), X) 
  double(X) >= map(times(s(s(_|_))), X) 

With these choices, we have:

  1] @_{o -> o}(times(s(X)), Y) > @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [2], by definition 
  2] @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [3], by (Select) 
  3] times(s(X)) @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because [4] 
  4] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y)  because times > @_{o -> o}, [5] and [11], by (Copy) 
  5] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= plus(@_{o -> o}(times(X), Y))  because times > plus and [6], by (Copy) 
  6] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(times(X), Y)  because times > @_{o -> o}, [7] and [11], by (Copy) 
  7] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= times(X)  because times in Mul and [8], by (Stat) 
  8] s(X) > X  because [9], by definition 
  9] s*(X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= Y  because [12], by (Select) 
  12] @_{o -> o}*(times(s(X)), Y) >= Y  because [13], by (Select) 
  13] Y >= Y  by (Meta) 

  14] inc(X) >= map(plus(s(_|_)), X)  because [15], by (Star) 
  15] inc*(X) >= map(plus(s(_|_)), X)  because inc > map, [16] and [19], by (Copy) 
  16] inc*(X) >= plus(s(_|_))  because inc > plus and [17], by (Copy) 
  17] inc*(X) >= s(_|_)  because inc > s and [18], by (Copy) 
  18] inc*(X) >= _|_  by (Bot) 
  19] inc*(X) >= X  because [20], by (Select) 
  20] X >= X  by (Meta) 

  21] double(X) >= map(times(s(s(_|_))), X)  because [22], by (Star) 
  22] double*(X) >= map(times(s(s(_|_))), X)  because double > map, [23] and [27], by (Copy) 
  23] double*(X) >= times(s(s(_|_)))  because double > times and [24], by (Copy) 
  24] double*(X) >= s(s(_|_))  because double > s and [25], by (Copy) 
  25] double*(X) >= s(_|_)  because double > s and [26], by (Copy) 
  26] double*(X) >= _|_  by (Bot) 
  27] double*(X) >= X  because [28], by (Select) 
  28] X >= X  by (Meta) 

We can thus remove the following rules:

  times(s(X)) Y => plus(times(X) Y) Y 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  inc(X) >? map(plus(s(0)), X) 
  double(X) >? map(times(s(s(0))), X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  double = \y0.3 + 3y0 
  inc = \y0.3 + 3y0 
  map = \G0y1.y1 + G0(0) 
  plus = \y0y1.y0 
  s = \y0.y0 
  times = \y0y1.y0 

Using this interpretation, the requirements translate to:

  [[inc(_x0)]] = 3 + 3x0 > x0 = [[map(plus(s(0)), _x0)]] 
  [[double(_x0)]] = 3 + 3x0 > x0 = [[map(times(s(s(0))), _x0)]] 

We can thus remove the following rules:

  inc(X) => map(plus(s(0)), X) 
  double(X) => map(times(s(s(0))), X) 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

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