YES
  TPA v.1.0

Result: TRS is terminating

Default interpretations for symbols are not printed. For polynomial interpretations
and semantic labelling over N\{0,1} defaults are 2 for constants, identity for unary
symbols and x+y-2 for binary symbols. For semantic labelling over {0,1} (booleans)
defaults are 0 for constants, identity for unary symbols and disjunction for binary symbols.

[1]  TRS loaded from input file:
(1) min(X,0) -> X
(2) min(s(X),s(Y)) -> min(X,Y)
(3) quot(0,s(Y)) -> 0
(4) quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
(5) log(s(0)) -> 0
(6) log(s(s(X))) -> s(log(s(quot(X,s(s(0))))))

[2]  Apply the Dependency Pair transformation resulting in the following dependency pairs:
<Min(s(X),s(Y)), Min(X,Y)>
<Quot(s(X),s(Y)), Quot(min(X,Y),s(Y))>
<Quot(s(X),s(Y)), Min(X,Y)>
<Log(s(s(X))), Log(s(quot(X,s(s(0)))))>
<Log(s(s(X))), Quot(X,s(s(0)))>
 and 3 SCCs in the dependency graph. Proofs for those SCCs follow.

[2a-1]  Consider the following SCC obtained from analysis of dependency graph:
(1) min(X,0) ->= X
(2) min(s(X),s(Y)) ->= min(X,Y)
(3) quot(0,s(Y)) ->= 0
(4) quot(s(X),s(Y)) ->= s(quot(min(X,Y),s(Y)))
(5) log(s(0)) ->= 0
(6) log(s(s(X))) ->= s(log(s(quot(X,s(s(0))))))
(DP4) Log(s(s(X))) -> Log(s(quot(X,s(s(0)))))

[2a-2]  Use following polynomial interpretation:
[min(x,y)] = x
[s(x)] = x + 1
[quot(x,y)] = x
rest default

  Remove rules with left hand side strictly bigger than right hand side:
(2), (5), (DP4)

[2a-3]  Since there are no remaining strict rules, relative termination is proved!

[2b-1]  Consider the following SCC obtained from analysis of dependency graph:
(1) min(X,0) ->= X
(2) min(s(X),s(Y)) ->= min(X,Y)
(3) quot(0,s(Y)) ->= 0
(4) quot(s(X),s(Y)) ->= s(quot(min(X,Y),s(Y)))
(5) log(s(0)) ->= 0
(6) log(s(s(X))) ->= s(log(s(quot(X,s(s(0))))))
(DP2) Quot(s(X),s(Y)) -> Quot(min(X,Y),s(Y))

[2b-2]  Use following polynomial interpretation:
[min(x,y)] = x
[s(x)] = x + 1
[quot(x,y)] = x
rest default

  Remove rules with left hand side strictly bigger than right hand side:
(2), (5), (DP2)

[2b-3]  Since there are no remaining strict rules, relative termination is proved!

[2c-1]  Consider the following SCC obtained from analysis of dependency graph:
(1) min(X,0) ->= X
(2) min(s(X),s(Y)) ->= min(X,Y)
(3) quot(0,s(Y)) ->= 0
(4) quot(s(X),s(Y)) ->= s(quot(min(X,Y),s(Y)))
(5) log(s(0)) ->= 0
(6) log(s(s(X))) ->= s(log(s(quot(X,s(s(0))))))
(DP1) Min(s(X),s(Y)) -> Min(X,Y)

[2c-2]  Use following polynomial interpretation:
[min(x,y)] = x
[s(x)] = x + 1
[quot(x,y)] = x
rest default

  Remove rules with left hand side strictly bigger than right hand side:
(2), (5), (DP1)

[2c-3]  Since there are no remaining strict rules, relative termination is proved!



../tpdb/TRS/Rubio/logarquot.trs, 0., Y
Couldn't open file <60>: 60: No such file or directory