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


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


WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 260 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 67 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
        (16) BEST
            (17) proven lower bound
                (18) LowerBoundPropagationProof [FINISHED, 0 ms]
                (19) BOUNDS(n^2, INF)
            (20) typed CpxTrs


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

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


The TRS R consists of the following rules:

   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   log(x, 0) -> baseError
   log(x, s(0)) -> baseError
   log(0, s(s(b))) -> logZeroError
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0)
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
   if(true, x, b, y, z) -> z
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

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

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

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


The TRS R consists of the following rules:

   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(x, y))
   log(x, 0') -> baseError
   log(x, s(0')) -> baseError
   log(0', s(s(b))) -> logZeroError
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
   if(true, x, b, y, z) -> z
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

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

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

(4)
Obligation:
TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
log(x, 0') -> baseError
log(x, s(0')) -> baseError
log(0', s(s(b))) -> logZeroError
log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) -> z
if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true
s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, plus, times, loop

They will be analysed ascendingly in the following order:
le < loop
plus < times
times < loop

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

(6)
Obligation:
TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
log(x, 0') -> baseError
log(x, s(0')) -> baseError
log(0', s(s(b))) -> logZeroError
log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) -> z
if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true
s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError


Generator Equations:
gen_s:0':baseError:logZeroError3_0(0) <=> 0'
gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x))


The following defined symbols remain to be analysed:
le, plus, times, loop

They will be analysed ascendingly in the following order:
le < loop
plus < times
times < loop

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0)

Induction Base:
le(gen_s:0':baseError:logZeroError3_0(+(1, 0)), gen_s:0':baseError:logZeroError3_0(0)) ->_R^Omega(1)
false

Induction Step:
le(gen_s:0':baseError:logZeroError3_0(+(1, +(n5_0, 1))), gen_s:0':baseError:logZeroError3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) ->_IH
false

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

(8)
Complex Obligation (BEST)

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

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

TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
log(x, 0') -> baseError
log(x, s(0')) -> baseError
log(0', s(s(b))) -> logZeroError
log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) -> z
if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true
s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError


Generator Equations:
gen_s:0':baseError:logZeroError3_0(0) <=> 0'
gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x))


The following defined symbols remain to be analysed:
le, plus, times, loop

They will be analysed ascendingly in the following order:
le < loop
plus < times
times < loop

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
log(x, 0') -> baseError
log(x, s(0')) -> baseError
log(0', s(s(b))) -> logZeroError
log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) -> z
if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true
s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError


Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0':baseError:logZeroError3_0(0) <=> 0'
gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x))


The following defined symbols remain to be analysed:
plus, times, loop

They will be analysed ascendingly in the following order:
plus < times
times < loop

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n276_0, b)), rt in Omega(1 + n276_0)

Induction Base:
plus(gen_s:0':baseError:logZeroError3_0(0), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1)
gen_s:0':baseError:logZeroError3_0(b)

Induction Step:
plus(gen_s:0':baseError:logZeroError3_0(+(n276_0, 1)), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1)
s(plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b))) ->_IH
s(gen_s:0':baseError:logZeroError3_0(+(b, c277_0)))

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

(14)
Obligation:
TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
log(x, 0') -> baseError
log(x, s(0')) -> baseError
log(0', s(s(b))) -> logZeroError
log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) -> z
if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true
s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError


Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n276_0, b)), rt in Omega(1 + n276_0)


Generator Equations:
gen_s:0':baseError:logZeroError3_0(0) <=> 0'
gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x))


The following defined symbols remain to be analysed:
times, loop

They will be analysed ascendingly in the following order:
times < loop

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_s:0':baseError:logZeroError3_0(n873_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(*(n873_0, b)), rt in Omega(1 + b*n873_0 + n873_0)

Induction Base:
times(gen_s:0':baseError:logZeroError3_0(0), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_s:0':baseError:logZeroError3_0(+(n873_0, 1)), gen_s:0':baseError:logZeroError3_0(b)) ->_R^Omega(1)
plus(gen_s:0':baseError:logZeroError3_0(b), times(gen_s:0':baseError:logZeroError3_0(n873_0), gen_s:0':baseError:logZeroError3_0(b))) ->_IH
plus(gen_s:0':baseError:logZeroError3_0(b), gen_s:0':baseError:logZeroError3_0(*(c874_0, b))) ->_L^Omega(1 + b)
gen_s:0':baseError:logZeroError3_0(+(b, *(n873_0, b)))

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:

TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
log(x, 0') -> baseError
log(x, s(0')) -> baseError
log(0', s(s(b))) -> logZeroError
log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) -> z
if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true
s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError


Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n276_0, b)), rt in Omega(1 + n276_0)


Generator Equations:
gen_s:0':baseError:logZeroError3_0(0) <=> 0'
gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x))


The following defined symbols remain to be analysed:
times, loop

They will be analysed ascendingly in the following order:
times < loop

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

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

(19)
BOUNDS(n^2, INF)

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

(20)
Obligation:
TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
log(x, 0') -> baseError
log(x, s(0')) -> baseError
log(0', s(s(b))) -> logZeroError
log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
if(true, x, b, y, z) -> z
if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

Types:
le :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> false:true
s :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError
0' :: s:0':baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
times :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
log :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
baseError :: s:0':baseError:logZeroError
logZeroError :: s:0':baseError:logZeroError
loop :: s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
if :: false:true -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError -> s:0':baseError:logZeroError
hole_false:true1_0 :: false:true
hole_s:0':baseError:logZeroError2_0 :: s:0':baseError:logZeroError
gen_s:0':baseError:logZeroError3_0 :: Nat -> s:0':baseError:logZeroError


Lemmas:
le(gen_s:0':baseError:logZeroError3_0(+(1, n5_0)), gen_s:0':baseError:logZeroError3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
plus(gen_s:0':baseError:logZeroError3_0(n276_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(+(n276_0, b)), rt in Omega(1 + n276_0)
times(gen_s:0':baseError:logZeroError3_0(n873_0), gen_s:0':baseError:logZeroError3_0(b)) -> gen_s:0':baseError:logZeroError3_0(*(n873_0, b)), rt in Omega(1 + b*n873_0 + n873_0)


Generator Equations:
gen_s:0':baseError:logZeroError3_0(0) <=> 0'
gen_s:0':baseError:logZeroError3_0(+(x, 1)) <=> s(gen_s:0':baseError:logZeroError3_0(x))


The following defined symbols remain to be analysed:
loop