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


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 327 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   and(false, false) -> false
   and(true, false) -> false
   and(false, true) -> false
   and(true, true) -> true
   eq(nil, nil) -> true
   eq(cons(T, L), nil) -> false
   eq(nil, cons(T, L)) -> false
   eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
   eq(var(L), var(Lp)) -> eq(L, Lp)
   eq(var(L), apply(T, S)) -> false
   eq(var(L), lambda(X, T)) -> false
   eq(apply(T, S), var(L)) -> false
   eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
   eq(apply(T, S), lambda(X, Tp)) -> false
   eq(lambda(X, T), var(L)) -> false
   eq(lambda(X, T), apply(Tp, Sp)) -> false
   eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
   if(true, var(K), var(L)) -> var(K)
   if(false, var(K), var(L)) -> var(L)
   ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
   ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
   ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(false) -> false
   encArg(true) -> true
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(var(x_1)) -> var(encArg(x_1))
   encArg(apply(x_1, x_2)) -> apply(encArg(x_1), encArg(x_2))
   encArg(lambda(x_1, x_2)) -> lambda(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_ren(x_1, x_2, x_3)) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_var(x_1) -> var(encArg(x_1))
   encode_apply(x_1, x_2) -> apply(encArg(x_1), encArg(x_2))
   encode_lambda(x_1, x_2) -> lambda(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_ren(x_1, x_2, x_3) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   and(false, false) -> false
   and(true, false) -> false
   and(false, true) -> false
   and(true, true) -> true
   eq(nil, nil) -> true
   eq(cons(T, L), nil) -> false
   eq(nil, cons(T, L)) -> false
   eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
   eq(var(L), var(Lp)) -> eq(L, Lp)
   eq(var(L), apply(T, S)) -> false
   eq(var(L), lambda(X, T)) -> false
   eq(apply(T, S), var(L)) -> false
   eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
   eq(apply(T, S), lambda(X, Tp)) -> false
   eq(lambda(X, T), var(L)) -> false
   eq(lambda(X, T), apply(Tp, Sp)) -> false
   eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
   if(true, var(K), var(L)) -> var(K)
   if(false, var(K), var(L)) -> var(L)
   ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
   ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
   ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

The (relative) TRS S consists of the following rules:

   encArg(false) -> false
   encArg(true) -> true
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(var(x_1)) -> var(encArg(x_1))
   encArg(apply(x_1, x_2)) -> apply(encArg(x_1), encArg(x_2))
   encArg(lambda(x_1, x_2)) -> lambda(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_ren(x_1, x_2, x_3)) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_var(x_1) -> var(encArg(x_1))
   encode_apply(x_1, x_2) -> apply(encArg(x_1), encArg(x_2))
   encode_lambda(x_1, x_2) -> lambda(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_ren(x_1, x_2, x_3) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   and(false, false) -> false
   and(true, false) -> false
   and(false, true) -> false
   and(true, true) -> true
   eq(nil, nil) -> true
   eq(cons(T, L), nil) -> false
   eq(nil, cons(T, L)) -> false
   eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
   eq(var(L), var(Lp)) -> eq(L, Lp)
   eq(var(L), apply(T, S)) -> false
   eq(var(L), lambda(X, T)) -> false
   eq(apply(T, S), var(L)) -> false
   eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
   eq(apply(T, S), lambda(X, Tp)) -> false
   eq(lambda(X, T), var(L)) -> false
   eq(lambda(X, T), apply(Tp, Sp)) -> false
   eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
   if(true, var(K), var(L)) -> var(K)
   if(false, var(K), var(L)) -> var(L)
   ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
   ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
   ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

The (relative) TRS S consists of the following rules:

   encArg(false) -> false
   encArg(true) -> true
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(var(x_1)) -> var(encArg(x_1))
   encArg(apply(x_1, x_2)) -> apply(encArg(x_1), encArg(x_2))
   encArg(lambda(x_1, x_2)) -> lambda(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_ren(x_1, x_2, x_3)) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_var(x_1) -> var(encArg(x_1))
   encode_apply(x_1, x_2) -> apply(encArg(x_1), encArg(x_2))
   encode_lambda(x_1, x_2) -> lambda(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_ren(x_1, x_2, x_3) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   and(false, false) -> false
   and(true, false) -> false
   and(false, true) -> false
   and(true, true) -> true
   eq(nil, nil) -> true
   eq(cons(T, L), nil) -> false
   eq(nil, cons(T, L)) -> false
   eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
   eq(var(L), var(Lp)) -> eq(L, Lp)
   eq(var(L), apply(T, S)) -> false
   eq(var(L), lambda(X, T)) -> false
   eq(apply(T, S), var(L)) -> false
   eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
   eq(apply(T, S), lambda(X, Tp)) -> false
   eq(lambda(X, T), var(L)) -> false
   eq(lambda(X, T), apply(Tp, Sp)) -> false
   eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
   if(true, var(K), var(L)) -> var(K)
   if(false, var(K), var(L)) -> var(L)
   ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
   ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
   ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

The (relative) TRS S consists of the following rules:

   encArg(false) -> false
   encArg(true) -> true
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(var(x_1)) -> var(encArg(x_1))
   encArg(apply(x_1, x_2)) -> apply(encArg(x_1), encArg(x_2))
   encArg(lambda(x_1, x_2)) -> lambda(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_ren(x_1, x_2, x_3)) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_var(x_1) -> var(encArg(x_1))
   encode_apply(x_1, x_2) -> apply(encArg(x_1), encArg(x_2))
   encode_lambda(x_1, x_2) -> lambda(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_ren(x_1, x_2, x_3) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

eq(lambda(X, T), lambda(Xp, Tp)) ->^+ and(eq(T, Tp), eq(X, Xp))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [T / lambda(X, T), Tp / lambda(Xp, Tp)].

The result substitution is [ ].




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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   and(false, false) -> false
   and(true, false) -> false
   and(false, true) -> false
   and(true, true) -> true
   eq(nil, nil) -> true
   eq(cons(T, L), nil) -> false
   eq(nil, cons(T, L)) -> false
   eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
   eq(var(L), var(Lp)) -> eq(L, Lp)
   eq(var(L), apply(T, S)) -> false
   eq(var(L), lambda(X, T)) -> false
   eq(apply(T, S), var(L)) -> false
   eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
   eq(apply(T, S), lambda(X, Tp)) -> false
   eq(lambda(X, T), var(L)) -> false
   eq(lambda(X, T), apply(Tp, Sp)) -> false
   eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
   if(true, var(K), var(L)) -> var(K)
   if(false, var(K), var(L)) -> var(L)
   ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
   ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
   ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

The (relative) TRS S consists of the following rules:

   encArg(false) -> false
   encArg(true) -> true
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(var(x_1)) -> var(encArg(x_1))
   encArg(apply(x_1, x_2)) -> apply(encArg(x_1), encArg(x_2))
   encArg(lambda(x_1, x_2)) -> lambda(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_ren(x_1, x_2, x_3)) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_var(x_1) -> var(encArg(x_1))
   encode_apply(x_1, x_2) -> apply(encArg(x_1), encArg(x_2))
   encode_lambda(x_1, x_2) -> lambda(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_ren(x_1, x_2, x_3) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   and(false, false) -> false
   and(true, false) -> false
   and(false, true) -> false
   and(true, true) -> true
   eq(nil, nil) -> true
   eq(cons(T, L), nil) -> false
   eq(nil, cons(T, L)) -> false
   eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
   eq(var(L), var(Lp)) -> eq(L, Lp)
   eq(var(L), apply(T, S)) -> false
   eq(var(L), lambda(X, T)) -> false
   eq(apply(T, S), var(L)) -> false
   eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
   eq(apply(T, S), lambda(X, Tp)) -> false
   eq(lambda(X, T), var(L)) -> false
   eq(lambda(X, T), apply(Tp, Sp)) -> false
   eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
   if(true, var(K), var(L)) -> var(K)
   if(false, var(K), var(L)) -> var(L)
   ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
   ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
   ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

The (relative) TRS S consists of the following rules:

   encArg(false) -> false
   encArg(true) -> true
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(var(x_1)) -> var(encArg(x_1))
   encArg(apply(x_1, x_2)) -> apply(encArg(x_1), encArg(x_2))
   encArg(lambda(x_1, x_2)) -> lambda(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_ren(x_1, x_2, x_3)) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_true -> true
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_var(x_1) -> var(encArg(x_1))
   encode_apply(x_1, x_2) -> apply(encArg(x_1), encArg(x_2))
   encode_lambda(x_1, x_2) -> lambda(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_ren(x_1, x_2, x_3) -> ren(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST