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


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NO

** BEGIN proof argument **
The following rule was generated while unfolding the analyzed TRS:
[iteration = 9] U41(tt,n__cons(n__0,n__zeros)) -> U41(tt,n__cons(n__0,n__zeros))
Let l be the left-hand side and r be the right-hand side of this rule.
Let p = epsilon, theta1 = {} and theta2 = {}.
We have r|p = U41(tt,n__cons(n__0,n__zeros)) and theta2(theta1(l)) = theta1(r|p).
Hence, the term theta1(l) = U41(tt,n__cons(n__0,n__zeros)) loops w.r.t. the analyzed TRS.
** END proof argument **

** BEGIN proof description **
## Searching for a generalized rewrite rule
(a rule whose right-hand side contains a variable
that does not occur in the left-hand side)...
No generalized rewrite rule found!

## Applying the DP framework...
## 2 initial DP problems to solve.
## First, we try to decompose these problems into smaller problems.
## Round 1 [2 DP problems]:
  ## DP problem:
  Dependency pairs = [U41^#(tt,_0) -> isNatIList^#(activate(_0)), isNatIList^#(n__cons(_0,_1)) -> U41^#(isNat(activate(_0)),activate(_1))]
  TRS = {zeros -> cons(0,n__zeros), U11(tt) -> tt, U21(tt) -> tt, U31(tt) -> tt, U41(tt,_0) -> U42(isNatIList(activate(_0))), U42(tt) -> tt, U51(tt,_0) -> U52(isNatList(activate(_0))), U52(tt) -> tt, U61(tt,_0,_1) -> U62(isNat(activate(_1)),activate(_0)), U62(tt,_0) -> s(length(activate(_0))), isNat(n__0) -> tt, isNat(n__length(_0)) -> U11(isNatList(activate(_0))), isNat(n__s(_0)) -> U21(isNat(activate(_0))), isNatIList(_0) -> U31(isNatList(activate(_0))), isNatIList(n__zeros) -> tt, isNatIList(n__cons(_0,_1)) -> U41(isNat(activate(_0)),activate(_1)), isNatList(n__nil) -> tt, isNatList(n__cons(_0,_1)) -> U51(isNat(activate(_0)),activate(_1)), length(nil) -> 0, length(cons(_0,_1)) -> U61(isNatList(activate(_1)),activate(_1),_0), zeros -> n__zeros, 0 -> n__0, length(_0) -> n__length(_0), s(_0) -> n__s(_0), cons(_0,_1) -> n__cons(_0,_1), nil -> n__nil, activate(n__zeros) -> zeros, activate(n__0) -> 0, activate(n__length(_0)) -> length(activate(_0)), activate(n__s(_0)) -> s(activate(_0)), activate(n__cons(_0,_1)) -> cons(activate(_0),_1), activate(n__nil) -> nil, activate(_0) -> _0}
    ## Trying with homeomorphic embeddings... Failed!
    ## Trying with polynomial interpretations... This DP problem is too complex! Aborting!
    ## Trying with lexicographic path orders... Too many argument filtering possibilities (7962624)! Aborting!
    ## Trying with Knuth-Bendix orders... This DP problem is too complex! Aborting!
  Don't know whether this DP problem is finite.
  ## DP problem:
  Dependency pairs = [isNatList^#(n__cons(_0,_1)) -> U51^#(isNat(activate(_0)),activate(_1)), isNat^#(n__length(_0)) -> isNatList^#(activate(_0)), U61^#(tt,_0,_1) -> isNat^#(activate(_1)), length^#(cons(_0,_1)) -> U61^#(isNatList(activate(_1)),activate(_1),_0), activate^#(n__length(_0)) -> length^#(activate(_0)), activate^#(n__length(_0)) -> activate^#(_0), activate^#(n__s(_0)) -> activate^#(_0), activate^#(n__cons(_0,_1)) -> activate^#(_0), length^#(cons(_0,_1)) -> activate^#(_1), length^#(cons(_0,_1)) -> activate^#(_1), isNatList^#(n__cons(_0,_1)) -> activate^#(_1), isNatList^#(n__cons(_0,_1)) -> activate^#(_0), isNat^#(n__s(_0)) -> activate^#(_0), isNat^#(n__length(_0)) -> activate^#(_0), isNat^#(n__s(_0)) -> isNat^#(activate(_0)), isNatList^#(n__cons(_0,_1)) -> isNat^#(activate(_0)), length^#(cons(_0,_1)) -> isNatList^#(activate(_1)), U62^#(tt,_0) -> activate^#(_0), U61^#(tt,_0,_1) -> activate^#(_0), U61^#(tt,_0,_1) -> activate^#(_1), U51^#(tt,_0) -> activate^#(_0), U62^#(tt,_0) -> length^#(activate(_0)), U61^#(tt,_0,_1) -> U62^#(isNat(activate(_1)),activate(_0)), U51^#(tt,_0) -> isNatList^#(activate(_0))]
  TRS = {zeros -> cons(0,n__zeros), U11(tt) -> tt, U21(tt) -> tt, U31(tt) -> tt, U41(tt,_0) -> U42(isNatIList(activate(_0))), U42(tt) -> tt, U51(tt,_0) -> U52(isNatList(activate(_0))), U52(tt) -> tt, U61(tt,_0,_1) -> U62(isNat(activate(_1)),activate(_0)), U62(tt,_0) -> s(length(activate(_0))), isNat(n__0) -> tt, isNat(n__length(_0)) -> U11(isNatList(activate(_0))), isNat(n__s(_0)) -> U21(isNat(activate(_0))), isNatIList(_0) -> U31(isNatList(activate(_0))), isNatIList(n__zeros) -> tt, isNatIList(n__cons(_0,_1)) -> U41(isNat(activate(_0)),activate(_1)), isNatList(n__nil) -> tt, isNatList(n__cons(_0,_1)) -> U51(isNat(activate(_0)),activate(_1)), length(nil) -> 0, length(cons(_0,_1)) -> U61(isNatList(activate(_1)),activate(_1),_0), zeros -> n__zeros, 0 -> n__0, length(_0) -> n__length(_0), s(_0) -> n__s(_0), cons(_0,_1) -> n__cons(_0,_1), nil -> n__nil, activate(n__zeros) -> zeros, activate(n__0) -> 0, activate(n__length(_0)) -> length(activate(_0)), activate(n__s(_0)) -> s(activate(_0)), activate(n__cons(_0,_1)) -> cons(activate(_0),_1), activate(n__nil) -> nil, activate(_0) -> _0}
    ## Trying with homeomorphic embeddings... Failed!
    ## Trying with polynomial interpretations... This DP problem is too complex! Aborting!
    ## Trying with lexicographic path orders... Too many argument filtering possibilities (7962624)! Aborting!
    ## Trying with Knuth-Bendix orders... This DP problem is too complex! Aborting!
  Don't know whether this DP problem is finite.
## Some DP problems could not be proved finite.
## Now, we try to prove that one of these problems is infinite.
    ## Trying to find a loop (forward=true, backward=false, max=-1)
      # max_depth=3, unfold_variables=false:
        # Iteration 0: no loop found, 2 unfolded rules generated.
        # Iteration 1: no loop found, 2 unfolded rules generated.
        # Iteration 2: no loop found, 13 unfolded rules generated.
        # Iteration 3: no loop found, 7 unfolded rules generated.
        # Iteration 4: no loop found, 4 unfolded rules generated.
        # Iteration 5: no loop found, 6 unfolded rules generated.
        # Iteration 6: no loop found, 30 unfolded rules generated.
        # Iteration 7: no loop found, 44 unfolded rules generated.
        # Iteration 8: no loop found, 79 unfolded rules generated.
        # Iteration 9: success, found a loop, 64 unfolded rules generated.
        Here is the successful unfolding. Let IR be the TRS under analysis.
        L0 = U41^#(tt,_0) -> isNatIList^#(activate(_0)) [trans] is in U_IR^0.
        D = isNatIList^#(n__cons(_0,_1)) -> U41^#(isNat(activate(_0)),activate(_1)) is a dependency pair of IR.
        We build a composed triple from L0 and D.
        ==> L1 = [U41^#(tt,_0) -> isNatIList^#(activate(_0)), isNatIList^#(n__cons(_1,_2)) -> U41^#(isNat(activate(_1)),activate(_2))] [comp] is in U_IR^1.
        Let p1 = [0].
        We unfold the first rule of L1 forwards at position p1
        with the rule activate(_0) -> _0.
        ==> L2 = U41^#(tt,n__cons(_0,_1)) -> U41^#(isNat(activate(_0)),activate(_1)) [trans] is in U_IR^2.
        We build a unit triple from L2.
        ==> L3 = U41^#(tt,n__cons(_0,_1)) -> U41^#(isNat(activate(_0)),activate(_1)) [unit] is in U_IR^3.
        Let p3 = [0, 0].
        We unfold the rule of L3 forwards at position p3
        with the rule activate(_0) -> _0.
        ==> L4 = U41^#(tt,n__cons(_0,_1)) -> U41^#(isNat(_0),activate(_1)) [unit] is in U_IR^4.
        Let p4 = [0].
        We unfold the rule of L4 forwards at position p4
        with the rule isNat(n__0) -> tt.
        ==> L5 = U41^#(tt,n__cons(n__0,_0)) -> U41^#(tt,activate(_0)) [unit] is in U_IR^5.
        Let p5 = [1].
        We unfold the rule of L5 forwards at position p5
        with the rule activate(n__zeros) -> zeros.
        ==> L6 = U41^#(tt,n__cons(n__0,n__zeros)) -> U41^#(tt,zeros) [unit] is in U_IR^6.
        Let p6 = [1].
        We unfold the rule of L6 forwards at position p6
        with the rule zeros -> cons(0,n__zeros).
        ==> L7 = U41^#(tt,n__cons(n__0,n__zeros)) -> U41^#(tt,cons(0,n__zeros)) [unit] is in U_IR^7.
        Let p7 = [1].
        We unfold the rule of L7 forwards at position p7
        with the rule cons(_0,_1) -> n__cons(_0,_1).
        ==> L8 = U41^#(tt,n__cons(n__0,n__zeros)) -> U41^#(tt,n__cons(0,n__zeros)) [unit] is in U_IR^8.
        Let p8 = [1, 0].
        We unfold the rule of L8 forwards at position p8
        with the rule 0 -> n__0.
        ==> L9 = U41^#(tt,n__cons(n__0,n__zeros)) -> U41^#(tt,n__cons(n__0,n__zeros)) [unit] is in U_IR^9.
  This DP problem is infinite.
Proof run on Linux version 3.10.0-1160.25.1.el7.x86_64 for amd64
using Java version 1.8.0_292
** END proof description **
Total number of generated unfolded rules = 2232