/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 
    eq : [a * a] --> c 
    false : [] --> c 
    fork : [b * a * b] --> b 
    if : [c * c * c] --> c 
    lt : [a * a] --> c 
    member : [a * b] --> c 
    null : [] --> b 
    s : [a] --> a 
    true : [] --> c 

  Rules:

    lt(s(x), s(y)) => lt(x, y) 
    lt(0, s(x)) => true 
    lt(x, 0) => false 
    eq(x, x) => true 
    eq(s(x), 0) => false 
    eq(0, s(x)) => false 
    member(x, null) => false 
    member(x, fork(y, z, u)) => if(lt(x, z), member(x, y), if(eq(x, z), true, member(x, u))) 

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]):

  lt(s(X), s(Y)) >? lt(X, Y) 
  lt(0, s(X)) >? true 
  lt(X, 0) >? false 
  eq(X, X) >? true 
  eq(s(X), 0) >? false 
  eq(0, s(X)) >? false 
  member(X, null) >? false 
  member(X, fork(Y, Z, U)) >? if(lt(X, Z), member(X, Y), if(eq(X, Z), true, member(X, U))) 

about to try horpo

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

Argument functions:

  [[false]] = _|_ 
  [[true]] = _|_ 

We choose Lex = {} and Mul = {0, eq, fork, if, lt, member, null, s}, and the following precedence: fork > null > s > eq = lt = member > 0 > if

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

  lt(s(X), s(Y)) >= lt(X, Y) 
  lt(0, s(X)) >= _|_ 
  lt(X, 0) >= _|_ 
  eq(X, X) > _|_ 
  eq(s(X), 0) >= _|_ 
  eq(0, s(X)) >= _|_ 
  member(X, null) >= _|_ 
  member(X, fork(Y, Z, U)) > if(lt(X, Z), member(X, Y), if(eq(X, Z), _|_, member(X, U))) 

With these choices, we have:

  1] lt(s(X), s(Y)) >= lt(X, Y)  because [2], by (Star) 
  2] lt*(s(X), s(Y)) >= lt(X, Y)  because lt in Mul, [3] and [6], by (Stat) 
  3] s(X) >= X  because [4], by (Star) 
  4] s*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 
  6] s(Y) > Y  because [7], by definition 
  7] s*(Y) >= Y  because [8], by (Select) 
  8] Y >= Y  by (Meta) 

  9] lt(0, s(X)) >= _|_  by (Bot) 

  10] lt(X, 0) >= _|_  by (Bot) 

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

  13] eq(s(X), 0) >= _|_  by (Bot) 

  14] eq(0, s(X)) >= _|_  by (Bot) 

  15] member(X, null) >= _|_  by (Bot) 

  16] member(X, fork(Y, Z, U)) > if(lt(X, Z), member(X, Y), if(eq(X, Z), _|_, member(X, U)))  because [17], by definition 
  17] member*(X, fork(Y, Z, U)) >= if(lt(X, Z), member(X, Y), if(eq(X, Z), _|_, member(X, U)))  because member > if, [18], [23] and [27], by (Copy) 
  18] member*(X, fork(Y, Z, U)) >= lt(X, Z)  because member = lt, member in Mul, [19] and [20], by (Stat) 
  19] X >= X  by (Meta) 
  20] fork(Y, Z, U) > Z  because [21], by definition 
  21] fork*(Y, Z, U) >= Z  because [22], by (Select) 
  22] Z >= Z  by (Meta) 
  23] member*(X, fork(Y, Z, U)) >= member(X, Y)  because member in Mul, [19] and [24], by (Stat) 
  24] fork(Y, Z, U) > Y  because [25], by definition 
  25] fork*(Y, Z, U) >= Y  because [26], by (Select) 
  26] Y >= Y  by (Meta) 
  27] member*(X, fork(Y, Z, U)) >= if(eq(X, Z), _|_, member(X, U))  because member > if, [28], [29] and [30], by (Copy) 
  28] member*(X, fork(Y, Z, U)) >= eq(X, Z)  because member = eq, member in Mul, [19] and [20], by (Stat) 
  29] member*(X, fork(Y, Z, U)) >= _|_  by (Bot) 
  30] member*(X, fork(Y, Z, U)) >= member(X, U)  because member in Mul, [19] and [31], by (Stat) 
  31] fork(Y, Z, U) > U  because [32], by definition 
  32] fork*(Y, Z, U) >= U  because [33], by (Select) 
  33] U >= U  by (Meta) 

We can thus remove the following rules:

  eq(X, X) => true 
  member(X, fork(Y, Z, U)) => if(lt(X, Z), member(X, Y), if(eq(X, Z), true, member(X, U))) 

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

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

  lt(s(X), s(Y)) >? lt(X, Y) 
  lt(0, s(X)) >? true 
  lt(X, 0) >? false 
  eq(s(X), 0) >? false 
  eq(0, s(X)) >? false 
  member(X, null) >? false 

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

The following interpretation satisfies the requirements:

  0 = 3 
  eq = \y0y1.3 + 3y0 + 3y1 
  false = 0 
  lt = \y0y1.3 + y0 + y1 
  member = \y0y1.3 + y0 + 3y1 
  null = 3 
  s = \y0.3 + 3y0 
  true = 0 

Using this interpretation, the requirements translate to:

  [[lt(s(_x0), s(_x1))]] = 9 + 3x0 + 3x1 > 3 + x0 + x1 = [[lt(_x0, _x1)]] 
  [[lt(0, s(_x0))]] = 9 + 3x0 > 0 = [[true]] 
  [[lt(_x0, 0)]] = 6 + x0 > 0 = [[false]] 
  [[eq(s(_x0), 0)]] = 21 + 9x0 > 0 = [[false]] 
  [[eq(0, s(_x0))]] = 21 + 9x0 > 0 = [[false]] 
  [[member(_x0, null)]] = 12 + x0 > 0 = [[false]] 

We can thus remove the following rules:

  lt(s(X), s(Y)) => lt(X, Y) 
  lt(0, s(X)) => true 
  lt(X, 0) => false 
  eq(s(X), 0) => false 
  eq(0, s(X)) => false 
  member(X, null) => false 

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.