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


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


WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 145 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 293 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 23 ms]
        (16) BEST
            (17) proven lower bound
                (18) LowerBoundPropagationProof [FINISHED, 0 ms]
                (19) BOUNDS(n^2, INF)
            (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 24 ms]
                (22) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   power(x', S(x)) -> mult(x', power(x', x))
   mult(x', S(x)) -> add0(x', mult(x', x))
   add0(x', S(x)) -> +(S(0), add0(x', x))
   power(x, 0) -> S(0)
   mult(x, 0) -> 0
   add0(x, 0) -> x

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

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)

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

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

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


The TRS R consists of the following rules:

   power(x', S(x)) -> mult(x', power(x', x))
   mult(x', S(x)) -> add0(x', mult(x', x))
   add0(x', S(x)) -> +(S(0), add0(x', x))
   power(x, 0) -> S(0)
   mult(x, 0) -> 0
   add0(x, 0) -> x

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

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)

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

(3) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   power(x', S(x)) -> mult(x', power(x', x))
   mult(x', S(x)) -> add0(x', mult(x', x))
   add0(x', S(x)) -> +'(S(0'), add0(x', x))
   power(x, 0') -> S(0')
   mult(x, 0') -> 0'
   add0(x, 0') -> x

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

   +'(x, S(0')) -> S(x)
   +'(S(0'), y) -> S(y)

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

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
Obligation:
Innermost TRS:
Rules:
power(x', S(x)) -> mult(x', power(x', x))
mult(x', S(x)) -> add0(x', mult(x', x))
add0(x', S(x)) -> +'(S(0'), add0(x', x))
power(x, 0') -> S(0')
mult(x, 0') -> 0'
add0(x, 0') -> x
+'(x, S(0')) -> S(x)
+'(S(0'), y) -> S(y)

Types:
power :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
mult :: S:0' -> S:0' -> S:0'
add0 :: S:0' -> S:0' -> S:0'
+' :: S:0' -> S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
power, mult, add0

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

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

(8)
Obligation:
Innermost TRS:
Rules:
power(x', S(x)) -> mult(x', power(x', x))
mult(x', S(x)) -> add0(x', mult(x', x))
add0(x', S(x)) -> +'(S(0'), add0(x', x))
power(x, 0') -> S(0')
mult(x, 0') -> 0'
add0(x, 0') -> x
+'(x, S(0')) -> S(x)
+'(S(0'), y) -> S(y)

Types:
power :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
mult :: S:0' -> S:0' -> S:0'
add0 :: S:0' -> S:0' -> S:0'
+' :: S:0' -> S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'


Generator Equations:
gen_S:0'2_1(0) <=> 0'
gen_S:0'2_1(+(x, 1)) <=> S(gen_S:0'2_1(x))


The following defined symbols remain to be analysed:
add0, power, mult

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) -> gen_S:0'2_1(+(1, n4_1)), rt in Omega(1 + n4_1)

Induction Base:
add0(gen_S:0'2_1(1), gen_S:0'2_1(0)) ->_R^Omega(1)
gen_S:0'2_1(1)

Induction Step:
add0(gen_S:0'2_1(1), gen_S:0'2_1(+(n4_1, 1))) ->_R^Omega(1)
+'(S(0'), add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1))) ->_IH
+'(S(0'), gen_S:0'2_1(+(1, c5_1))) ->_R^Omega(0)
S(gen_S:0'2_1(+(1, n4_1)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
power(x', S(x)) -> mult(x', power(x', x))
mult(x', S(x)) -> add0(x', mult(x', x))
add0(x', S(x)) -> +'(S(0'), add0(x', x))
power(x, 0') -> S(0')
mult(x, 0') -> 0'
add0(x, 0') -> x
+'(x, S(0')) -> S(x)
+'(S(0'), y) -> S(y)

Types:
power :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
mult :: S:0' -> S:0' -> S:0'
add0 :: S:0' -> S:0' -> S:0'
+' :: S:0' -> S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'


Generator Equations:
gen_S:0'2_1(0) <=> 0'
gen_S:0'2_1(+(x, 1)) <=> S(gen_S:0'2_1(x))


The following defined symbols remain to be analysed:
add0, power, mult

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
Innermost TRS:
Rules:
power(x', S(x)) -> mult(x', power(x', x))
mult(x', S(x)) -> add0(x', mult(x', x))
add0(x', S(x)) -> +'(S(0'), add0(x', x))
power(x, 0') -> S(0')
mult(x, 0') -> 0'
add0(x, 0') -> x
+'(x, S(0')) -> S(x)
+'(S(0'), y) -> S(y)

Types:
power :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
mult :: S:0' -> S:0' -> S:0'
add0 :: S:0' -> S:0' -> S:0'
+' :: S:0' -> S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'


Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) -> gen_S:0'2_1(+(1, n4_1)), rt in Omega(1 + n4_1)


Generator Equations:
gen_S:0'2_1(0) <=> 0'
gen_S:0'2_1(+(x, 1)) <=> S(gen_S:0'2_1(x))


The following defined symbols remain to be analysed:
mult, power

They will be analysed ascendingly in the following order:
mult < power

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) -> gen_S:0'2_1(n487_1), rt in Omega(1 + n487_1 + n487_1^2)

Induction Base:
mult(gen_S:0'2_1(1), gen_S:0'2_1(0)) ->_R^Omega(1)
0'

Induction Step:
mult(gen_S:0'2_1(1), gen_S:0'2_1(+(n487_1, 1))) ->_R^Omega(1)
add0(gen_S:0'2_1(1), mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1))) ->_IH
add0(gen_S:0'2_1(1), gen_S:0'2_1(c488_1)) ->_L^Omega(1 + n487_1)
gen_S:0'2_1(+(1, n487_1))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
power(x', S(x)) -> mult(x', power(x', x))
mult(x', S(x)) -> add0(x', mult(x', x))
add0(x', S(x)) -> +'(S(0'), add0(x', x))
power(x, 0') -> S(0')
mult(x, 0') -> 0'
add0(x, 0') -> x
+'(x, S(0')) -> S(x)
+'(S(0'), y) -> S(y)

Types:
power :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
mult :: S:0' -> S:0' -> S:0'
add0 :: S:0' -> S:0' -> S:0'
+' :: S:0' -> S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'


Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) -> gen_S:0'2_1(+(1, n4_1)), rt in Omega(1 + n4_1)


Generator Equations:
gen_S:0'2_1(0) <=> 0'
gen_S:0'2_1(+(x, 1)) <=> S(gen_S:0'2_1(x))


The following defined symbols remain to be analysed:
mult, power

They will be analysed ascendingly in the following order:
mult < power

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

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

(19)
BOUNDS(n^2, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
power(x', S(x)) -> mult(x', power(x', x))
mult(x', S(x)) -> add0(x', mult(x', x))
add0(x', S(x)) -> +'(S(0'), add0(x', x))
power(x, 0') -> S(0')
mult(x, 0') -> 0'
add0(x, 0') -> x
+'(x, S(0')) -> S(x)
+'(S(0'), y) -> S(y)

Types:
power :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
mult :: S:0' -> S:0' -> S:0'
add0 :: S:0' -> S:0' -> S:0'
+' :: S:0' -> S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'


Lemmas:
add0(gen_S:0'2_1(1), gen_S:0'2_1(n4_1)) -> gen_S:0'2_1(+(1, n4_1)), rt in Omega(1 + n4_1)
mult(gen_S:0'2_1(1), gen_S:0'2_1(n487_1)) -> gen_S:0'2_1(n487_1), rt in Omega(1 + n487_1 + n487_1^2)


Generator Equations:
gen_S:0'2_1(0) <=> 0'
gen_S:0'2_1(+(x, 1)) <=> S(gen_S:0'2_1(x))


The following defined symbols remain to be analysed:
power
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
power(gen_S:0'2_1(1), gen_S:0'2_1(n780_1)) -> gen_S:0'2_1(1), rt in Omega(1 + n780_1)

Induction Base:
power(gen_S:0'2_1(1), gen_S:0'2_1(0)) ->_R^Omega(1)
S(0')

Induction Step:
power(gen_S:0'2_1(1), gen_S:0'2_1(+(n780_1, 1))) ->_R^Omega(1)
mult(gen_S:0'2_1(1), power(gen_S:0'2_1(1), gen_S:0'2_1(n780_1))) ->_IH
mult(gen_S:0'2_1(1), gen_S:0'2_1(1)) ->_L^Omega(3)
gen_S:0'2_1(1)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
BOUNDS(1, INF)