/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  !870 : [o * o] --> o 
  !dot : [o * o] --> o 
  admit : [o * o] --> o 
  carry : [o * o * o] --> o 
  cond : [o * o] --> o 
  nil : [] --> o 
  sum : [o * o * o] --> o 
  true : [] --> o 
  w : [] --> o 

  admit(X, nil) => nil 
  admit(X, !dot(Y, !dot(Z, !dot(w, U)))) => cond(!870(sum(X, Y, Z), w), !dot(Y, !dot(Z, !dot(w, admit(carry(X, Y, Z), U))))) 
  cond(true, X) => X 

As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95].  Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows:

  !870 : [fa * s] --> ya 
  !dot : [s * bb] --> bb 
  admit : [pa * bb] --> bb 
  carry : [pa * s * s] --> pa 
  cond : [ya * bb] --> bb 
  nil : [] --> bb 
  sum : [pa * s * s] --> fa 
  true : [] --> ya 
  w : [] --> s 

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

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

  admit(X, nil) >? nil 
  admit(X, !dot(Y, !dot(Z, !dot(w, U)))) >? cond(!870(sum(X, Y, Z), w), !dot(Y, !dot(Z, !dot(w, admit(carry(X, Y, Z), U))))) 
  cond(true, X) >? X 

about to try horpo

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

Argument functions:

  [[admit(x_1, x_2)]] = admit(x_2, x_1) 
  [[nil]] = _|_ 

We choose Lex = {admit} and Mul = {!870, !dot, carry, cond, sum, true, w}, and the following precedence: admit > !870 > !dot > cond > w > sum > carry = true

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

  admit(X, _|_) >= _|_ 
  admit(X, !dot(Y, !dot(Z, !dot(w, U)))) >= cond(!870(sum(X, Y, Z), w), !dot(Y, !dot(Z, !dot(w, admit(carry(X, Y, Z), U))))) 
  cond(true, X) > X 

With these choices, we have:

  1] admit(X, _|_) >= _|_  by (Bot) 

  2] admit(X, !dot(Y, !dot(Z, !dot(w, U)))) >= cond(!870(sum(X, Y, Z), w), !dot(Y, !dot(Z, !dot(w, admit(carry(X, Y, Z), U)))))  because [3], by (Star) 
  3] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= cond(!870(sum(X, Y, Z), w), !dot(Y, !dot(Z, !dot(w, admit(carry(X, Y, Z), U)))))  because admit > cond, [4] and [19], by (Copy) 
  4] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= !870(sum(X, Y, Z), w)  because admit > !870, [5] and [18], by (Copy) 
  5] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= sum(X, Y, Z)  because admit > sum, [6], [8] and [12], by (Copy) 
  6] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= X  because [7], by (Select) 
  7] X >= X  by (Meta) 
  8] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= Y  because [9], by (Select) 
  9] !dot(Y, !dot(Z, !dot(w, U))) >= Y  because [10], by (Star) 
  10] !dot*(Y, !dot(Z, !dot(w, U))) >= Y  because [11], by (Select) 
  11] Y >= Y  by (Meta) 
  12] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= Z  because [13], by (Select) 
  13] !dot(Y, !dot(Z, !dot(w, U))) >= Z  because [14], by (Star) 
  14] !dot*(Y, !dot(Z, !dot(w, U))) >= Z  because [15], by (Select) 
  15] !dot(Z, !dot(w, U)) >= Z  because [16], by (Star) 
  16] !dot*(Z, !dot(w, U)) >= Z  because [17], by (Select) 
  17] Z >= Z  by (Meta) 
  18] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= w  because admit > w, by (Copy) 
  19] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= !dot(Y, !dot(Z, !dot(w, admit(carry(X, Y, Z), U))))  because admit > !dot, [8] and [20], by (Copy) 
  20] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= !dot(Z, !dot(w, admit(carry(X, Y, Z), U)))  because admit > !dot, [12] and [21], by (Copy) 
  21] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= !dot(w, admit(carry(X, Y, Z), U))  because admit > !dot, [18] and [22], by (Copy) 
  22] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= admit(carry(X, Y, Z), U)  because [23], [30] and [31], by (Stat) 
  23] !dot(Y, !dot(Z, !dot(w, U))) > U  because [24], by definition 
  24] !dot*(Y, !dot(Z, !dot(w, U))) >= U  because [25], by (Select) 
  25] !dot(Z, !dot(w, U)) >= U  because [26], by (Star) 
  26] !dot*(Z, !dot(w, U)) >= U  because [27], by (Select) 
  27] !dot(w, U) >= U  because [28], by (Star) 
  28] !dot*(w, U) >= U  because [29], by (Select) 
  29] U >= U  by (Meta) 
  30] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= carry(X, Y, Z)  because admit > carry, [6], [8] and [12], by (Copy) 
  31] admit*(X, !dot(Y, !dot(Z, !dot(w, U)))) >= U  because [32], by (Select) 
  32] !dot(Y, !dot(Z, !dot(w, U))) >= U  because [24], by (Star) 

  33] cond(true, X) > X  because [34], by definition 
  34] cond*(true, X) >= X  because [35], by (Select) 
  35] X >= X  by (Meta) 

We can thus remove the following rules:

  cond(true, 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]):

  admit(X, nil) >? nil 
  admit(X, !dot(Y, !dot(Z, !dot(w, U)))) >? cond(!870(sum(X, Y, Z), w), !dot(Y, !dot(Z, !dot(w, admit(carry(X, Y, Z), U))))) 

about to try horpo

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

Argument functions:

  [[admit(x_1, x_2)]] = admit(x_2, x_1) 
  [[w]] = _|_ 

We choose Lex = {admit} and Mul = {!870, !dot, carry, cond, nil, sum}, and the following precedence: admit > carry > cond > nil > !870 > !dot > sum

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

  admit(X, nil) >= nil 
  admit(X, !dot(Y, !dot(Z, !dot(_|_, U)))) > cond(!870(sum(X, Y, Z), _|_), !dot(Y, !dot(Z, !dot(_|_, admit(carry(X, Y, Z), U))))) 

With these choices, we have:

  1] admit(X, nil) >= nil  because [2], by (Star) 
  2] admit*(X, nil) >= nil  because admit > nil, by (Copy) 

  3] admit(X, !dot(Y, !dot(Z, !dot(_|_, U)))) > cond(!870(sum(X, Y, Z), _|_), !dot(Y, !dot(Z, !dot(_|_, admit(carry(X, Y, Z), U)))))  because [4], by definition 
  4] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= cond(!870(sum(X, Y, Z), _|_), !dot(Y, !dot(Z, !dot(_|_, admit(carry(X, Y, Z), U)))))  because admit > cond, [5] and [20], by (Copy) 
  5] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= !870(sum(X, Y, Z), _|_)  because admit > !870, [6] and [19], by (Copy) 
  6] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= sum(X, Y, Z)  because admit > sum, [7], [9] and [13], by (Copy) 
  7] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= X  because [8], by (Select) 
  8] X >= X  by (Meta) 
  9] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= Y  because [10], by (Select) 
  10] !dot(Y, !dot(Z, !dot(_|_, U))) >= Y  because [11], by (Star) 
  11] !dot*(Y, !dot(Z, !dot(_|_, U))) >= Y  because [12], by (Select) 
  12] Y >= Y  by (Meta) 
  13] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= Z  because [14], by (Select) 
  14] !dot(Y, !dot(Z, !dot(_|_, U))) >= Z  because [15], by (Star) 
  15] !dot*(Y, !dot(Z, !dot(_|_, U))) >= Z  because [16], by (Select) 
  16] !dot(Z, !dot(_|_, U)) >= Z  because [17], by (Star) 
  17] !dot*(Z, !dot(_|_, U)) >= Z  because [18], by (Select) 
  18] Z >= Z  by (Meta) 
  19] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= _|_  by (Bot) 
  20] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= !dot(Y, !dot(Z, !dot(_|_, admit(carry(X, Y, Z), U))))  because admit > !dot, [9] and [21], by (Copy) 
  21] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= !dot(Z, !dot(_|_, admit(carry(X, Y, Z), U)))  because admit > !dot, [13] and [22], by (Copy) 
  22] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= !dot(_|_, admit(carry(X, Y, Z), U))  because admit > !dot, [19] and [23], by (Copy) 
  23] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= admit(carry(X, Y, Z), U)  because [24], [31] and [32], by (Stat) 
  24] !dot(Y, !dot(Z, !dot(_|_, U))) > U  because [25], by definition 
  25] !dot*(Y, !dot(Z, !dot(_|_, U))) >= U  because [26], by (Select) 
  26] !dot(Z, !dot(_|_, U)) >= U  because [27], by (Star) 
  27] !dot*(Z, !dot(_|_, U)) >= U  because [28], by (Select) 
  28] !dot(_|_, U) >= U  because [29], by (Star) 
  29] !dot*(_|_, U) >= U  because [30], by (Select) 
  30] U >= U  by (Meta) 
  31] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= carry(X, Y, Z)  because admit > carry, [7], [9] and [13], by (Copy) 
  32] admit*(X, !dot(Y, !dot(Z, !dot(_|_, U)))) >= U  because [33], by (Select) 
  33] !dot(Y, !dot(Z, !dot(_|_, U))) >= U  because [25], by (Star) 

We can thus remove the following rules:

  admit(X, !dot(Y, !dot(Z, !dot(w, U)))) => cond(!870(sum(X, Y, Z), w), !dot(Y, !dot(Z, !dot(w, admit(carry(X, Y, Z), 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]):

  admit(X, nil) >? nil 

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

The following interpretation satisfies the requirements:

  admit = \y0y1.3 + y0 + 3y1 
  nil = 0 

Using this interpretation, the requirements translate to:

  [[admit(_x0, nil)]] = 3 + x0 > 0 = [[nil]] 

We can thus remove the following rules:

  admit(X, nil) => nil 

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 +++

[FuhGieParSchSwi11]  C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski.  Proving Termination by Dependency Pairs and Inductive Theorem Proving.  In volume 47(2) of Journal of Automated Reasoning.  133--160, 2011.
[Gra95]  B. Gramlich.  Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems.  In volume 24(1-2) of Fundamentae Informaticae.  3--23, 1995.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.