WORST_CASE(Omega(n^1),?)
proof of input_Y7BXK6YVeQ.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 356 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 28 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 103 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 66 ms]
        (22) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, 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)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   log(0) -> logError
   log(s(x)) -> loop(s(x), s(0), 0)
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
   if(true, x, y, z) -> z
   if(false, x, y, z) -> loop(x, double(y), s(z))

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

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   log(0) -> logError
   log(s(z0)) -> loop(s(z0), s(0), 0)
   loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
   if(true, z0, z1, z2) -> z2
   if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))
Tuples:
   LE(s(z0), 0) -> c
   LE(0, z0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(0) -> c3
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOG(0) -> c5
   LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(true, z0, z1, z2) -> c8
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
S tuples:
   LE(s(z0), 0) -> c
   LE(0, z0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(0) -> c3
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOG(0) -> c5
   LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(true, z0, z1, z2) -> c8
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
K tuples:none
Defined Rule Symbols:   le_2, double_1, log_1, loop_3, if_4

Defined Pair Symbols:   LE_2, DOUBLE_1, LOG_1, LOOP_3, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5, c6_1, c7_2, c8, c9_2


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

(3) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to 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:

   LE(s(z0), 0) -> c
   LE(0, z0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(0) -> c3
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOG(0) -> c5
   LOG(s(z0)) -> c6(LOOP(s(z0), s(0), 0))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(true, z0, z1, z2) -> c8
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))

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

   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   log(0) -> logError
   log(s(z0)) -> loop(s(z0), s(0), 0)
   loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
   if(true, z0, z1, z2) -> z2
   if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

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

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

(6)
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:

   LE(s(z0), 0') -> c
   LE(0', z0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   DOUBLE(0') -> c3
   DOUBLE(s(z0)) -> c4(DOUBLE(z0))
   LOG(0') -> c5
   LOG(s(z0)) -> c6(LOOP(s(z0), s(0'), 0'))
   LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
   IF(true, z0, z1, z2) -> c8
   IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))

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

   le(s(z0), 0') -> false
   le(0', z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   double(0') -> 0'
   double(s(z0)) -> s(s(double(z0)))
   log(0') -> logError
   log(s(z0)) -> loop(s(z0), s(0'), 0')
   loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
   if(true, z0, z1, z2) -> z2
   if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

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

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(8)
Obligation:
Innermost TRS:
Rules:
LE(s(z0), 0') -> c
LE(0', z0) -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
DOUBLE(0') -> c3
DOUBLE(s(z0)) -> c4(DOUBLE(z0))
LOG(0') -> c5
LOG(s(z0)) -> c6(LOOP(s(z0), s(0'), 0'))
LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
IF(true, z0, z1, z2) -> c8
IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
le(s(z0), 0') -> false
le(0', z0) -> true
le(s(z0), s(z1)) -> le(z0, z1)
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
log(0') -> logError
log(s(z0)) -> loop(s(z0), s(0'), 0')
loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
if(true, z0, z1, z2) -> z2
if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

Types:
LE :: s:0':logError -> s:0':logError -> c:c1:c2
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
DOUBLE :: s:0':logError -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LOG :: s:0':logError -> c5:c6
c5 :: c5:c6
c6 :: c7 -> c5:c6
LOOP :: s:0':logError -> s:0':logError -> s:0':logError -> c7
c7 :: c8:c9 -> c:c1:c2 -> c7
IF :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> c8:c9
le :: s:0':logError -> s:0':logError -> true:false
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c7 -> c3:c4 -> c8:c9
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_c:c1:c21_10 :: c:c1:c2
hole_s:0':logError2_10 :: s:0':logError
hole_c3:c43_10 :: c3:c4
hole_c5:c64_10 :: c5:c6
hole_c75_10 :: c7
hole_c8:c96_10 :: c8:c9
hole_true:false7_10 :: true:false
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_s:0':logError9_10 :: Nat -> s:0':logError
gen_c3:c410_10 :: Nat -> c3:c4

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LE, DOUBLE, LOOP, le, double, loop

They will be analysed ascendingly in the following order:
LE < LOOP
DOUBLE < LOOP
le < LOOP
double < LOOP
le < loop
double < loop

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

(10)
Obligation:
Innermost TRS:
Rules:
LE(s(z0), 0') -> c
LE(0', z0) -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
DOUBLE(0') -> c3
DOUBLE(s(z0)) -> c4(DOUBLE(z0))
LOG(0') -> c5
LOG(s(z0)) -> c6(LOOP(s(z0), s(0'), 0'))
LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
IF(true, z0, z1, z2) -> c8
IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
le(s(z0), 0') -> false
le(0', z0) -> true
le(s(z0), s(z1)) -> le(z0, z1)
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
log(0') -> logError
log(s(z0)) -> loop(s(z0), s(0'), 0')
loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
if(true, z0, z1, z2) -> z2
if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

Types:
LE :: s:0':logError -> s:0':logError -> c:c1:c2
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
DOUBLE :: s:0':logError -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LOG :: s:0':logError -> c5:c6
c5 :: c5:c6
c6 :: c7 -> c5:c6
LOOP :: s:0':logError -> s:0':logError -> s:0':logError -> c7
c7 :: c8:c9 -> c:c1:c2 -> c7
IF :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> c8:c9
le :: s:0':logError -> s:0':logError -> true:false
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c7 -> c3:c4 -> c8:c9
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_c:c1:c21_10 :: c:c1:c2
hole_s:0':logError2_10 :: s:0':logError
hole_c3:c43_10 :: c3:c4
hole_c5:c64_10 :: c5:c6
hole_c75_10 :: c7
hole_c8:c96_10 :: c8:c9
hole_true:false7_10 :: true:false
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_s:0':logError9_10 :: Nat -> s:0':logError
gen_c3:c410_10 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c2(gen_c:c1:c28_10(x))
gen_s:0':logError9_10(0) <=> 0'
gen_s:0':logError9_10(+(x, 1)) <=> s(gen_s:0':logError9_10(x))
gen_c3:c410_10(0) <=> c3
gen_c3:c410_10(+(x, 1)) <=> c4(gen_c3:c410_10(x))


The following defined symbols remain to be analysed:
LE, DOUBLE, LOOP, le, double, loop

They will be analysed ascendingly in the following order:
LE < LOOP
DOUBLE < LOOP
le < LOOP
double < LOOP
le < loop
double < loop

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_s:0':logError9_10(+(1, n12_10)), gen_s:0':logError9_10(n12_10)) -> gen_c:c1:c28_10(n12_10), rt in Omega(1 + n12_10)

Induction Base:
LE(gen_s:0':logError9_10(+(1, 0)), gen_s:0':logError9_10(0)) ->_R^Omega(1)
c

Induction Step:
LE(gen_s:0':logError9_10(+(1, +(n12_10, 1))), gen_s:0':logError9_10(+(n12_10, 1))) ->_R^Omega(1)
c2(LE(gen_s:0':logError9_10(+(1, n12_10)), gen_s:0':logError9_10(n12_10))) ->_IH
c2(gen_c:c1:c28_10(c13_10))

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LE(s(z0), 0') -> c
LE(0', z0) -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
DOUBLE(0') -> c3
DOUBLE(s(z0)) -> c4(DOUBLE(z0))
LOG(0') -> c5
LOG(s(z0)) -> c6(LOOP(s(z0), s(0'), 0'))
LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
IF(true, z0, z1, z2) -> c8
IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
le(s(z0), 0') -> false
le(0', z0) -> true
le(s(z0), s(z1)) -> le(z0, z1)
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
log(0') -> logError
log(s(z0)) -> loop(s(z0), s(0'), 0')
loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
if(true, z0, z1, z2) -> z2
if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

Types:
LE :: s:0':logError -> s:0':logError -> c:c1:c2
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
DOUBLE :: s:0':logError -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LOG :: s:0':logError -> c5:c6
c5 :: c5:c6
c6 :: c7 -> c5:c6
LOOP :: s:0':logError -> s:0':logError -> s:0':logError -> c7
c7 :: c8:c9 -> c:c1:c2 -> c7
IF :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> c8:c9
le :: s:0':logError -> s:0':logError -> true:false
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c7 -> c3:c4 -> c8:c9
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_c:c1:c21_10 :: c:c1:c2
hole_s:0':logError2_10 :: s:0':logError
hole_c3:c43_10 :: c3:c4
hole_c5:c64_10 :: c5:c6
hole_c75_10 :: c7
hole_c8:c96_10 :: c8:c9
hole_true:false7_10 :: true:false
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_s:0':logError9_10 :: Nat -> s:0':logError
gen_c3:c410_10 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c2(gen_c:c1:c28_10(x))
gen_s:0':logError9_10(0) <=> 0'
gen_s:0':logError9_10(+(x, 1)) <=> s(gen_s:0':logError9_10(x))
gen_c3:c410_10(0) <=> c3
gen_c3:c410_10(+(x, 1)) <=> c4(gen_c3:c410_10(x))


The following defined symbols remain to be analysed:
LE, DOUBLE, LOOP, le, double, loop

They will be analysed ascendingly in the following order:
LE < LOOP
DOUBLE < LOOP
le < LOOP
double < LOOP
le < loop
double < loop

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
LE(s(z0), 0') -> c
LE(0', z0) -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
DOUBLE(0') -> c3
DOUBLE(s(z0)) -> c4(DOUBLE(z0))
LOG(0') -> c5
LOG(s(z0)) -> c6(LOOP(s(z0), s(0'), 0'))
LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
IF(true, z0, z1, z2) -> c8
IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
le(s(z0), 0') -> false
le(0', z0) -> true
le(s(z0), s(z1)) -> le(z0, z1)
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
log(0') -> logError
log(s(z0)) -> loop(s(z0), s(0'), 0')
loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
if(true, z0, z1, z2) -> z2
if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

Types:
LE :: s:0':logError -> s:0':logError -> c:c1:c2
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
DOUBLE :: s:0':logError -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LOG :: s:0':logError -> c5:c6
c5 :: c5:c6
c6 :: c7 -> c5:c6
LOOP :: s:0':logError -> s:0':logError -> s:0':logError -> c7
c7 :: c8:c9 -> c:c1:c2 -> c7
IF :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> c8:c9
le :: s:0':logError -> s:0':logError -> true:false
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c7 -> c3:c4 -> c8:c9
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_c:c1:c21_10 :: c:c1:c2
hole_s:0':logError2_10 :: s:0':logError
hole_c3:c43_10 :: c3:c4
hole_c5:c64_10 :: c5:c6
hole_c75_10 :: c7
hole_c8:c96_10 :: c8:c9
hole_true:false7_10 :: true:false
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_s:0':logError9_10 :: Nat -> s:0':logError
gen_c3:c410_10 :: Nat -> c3:c4


Lemmas:
LE(gen_s:0':logError9_10(+(1, n12_10)), gen_s:0':logError9_10(n12_10)) -> gen_c:c1:c28_10(n12_10), rt in Omega(1 + n12_10)


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c2(gen_c:c1:c28_10(x))
gen_s:0':logError9_10(0) <=> 0'
gen_s:0':logError9_10(+(x, 1)) <=> s(gen_s:0':logError9_10(x))
gen_c3:c410_10(0) <=> c3
gen_c3:c410_10(+(x, 1)) <=> c4(gen_c3:c410_10(x))


The following defined symbols remain to be analysed:
DOUBLE, LOOP, le, double, loop

They will be analysed ascendingly in the following order:
DOUBLE < LOOP
le < LOOP
double < LOOP
le < loop
double < loop

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
DOUBLE(gen_s:0':logError9_10(n588_10)) -> gen_c3:c410_10(n588_10), rt in Omega(1 + n588_10)

Induction Base:
DOUBLE(gen_s:0':logError9_10(0)) ->_R^Omega(1)
c3

Induction Step:
DOUBLE(gen_s:0':logError9_10(+(n588_10, 1))) ->_R^Omega(1)
c4(DOUBLE(gen_s:0':logError9_10(n588_10))) ->_IH
c4(gen_c3:c410_10(c589_10))

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

(18)
Obligation:
Innermost TRS:
Rules:
LE(s(z0), 0') -> c
LE(0', z0) -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
DOUBLE(0') -> c3
DOUBLE(s(z0)) -> c4(DOUBLE(z0))
LOG(0') -> c5
LOG(s(z0)) -> c6(LOOP(s(z0), s(0'), 0'))
LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
IF(true, z0, z1, z2) -> c8
IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
le(s(z0), 0') -> false
le(0', z0) -> true
le(s(z0), s(z1)) -> le(z0, z1)
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
log(0') -> logError
log(s(z0)) -> loop(s(z0), s(0'), 0')
loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
if(true, z0, z1, z2) -> z2
if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

Types:
LE :: s:0':logError -> s:0':logError -> c:c1:c2
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
DOUBLE :: s:0':logError -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LOG :: s:0':logError -> c5:c6
c5 :: c5:c6
c6 :: c7 -> c5:c6
LOOP :: s:0':logError -> s:0':logError -> s:0':logError -> c7
c7 :: c8:c9 -> c:c1:c2 -> c7
IF :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> c8:c9
le :: s:0':logError -> s:0':logError -> true:false
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c7 -> c3:c4 -> c8:c9
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_c:c1:c21_10 :: c:c1:c2
hole_s:0':logError2_10 :: s:0':logError
hole_c3:c43_10 :: c3:c4
hole_c5:c64_10 :: c5:c6
hole_c75_10 :: c7
hole_c8:c96_10 :: c8:c9
hole_true:false7_10 :: true:false
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_s:0':logError9_10 :: Nat -> s:0':logError
gen_c3:c410_10 :: Nat -> c3:c4


Lemmas:
LE(gen_s:0':logError9_10(+(1, n12_10)), gen_s:0':logError9_10(n12_10)) -> gen_c:c1:c28_10(n12_10), rt in Omega(1 + n12_10)
DOUBLE(gen_s:0':logError9_10(n588_10)) -> gen_c3:c410_10(n588_10), rt in Omega(1 + n588_10)


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c2(gen_c:c1:c28_10(x))
gen_s:0':logError9_10(0) <=> 0'
gen_s:0':logError9_10(+(x, 1)) <=> s(gen_s:0':logError9_10(x))
gen_c3:c410_10(0) <=> c3
gen_c3:c410_10(+(x, 1)) <=> c4(gen_c3:c410_10(x))


The following defined symbols remain to be analysed:
le, LOOP, double, loop

They will be analysed ascendingly in the following order:
le < LOOP
double < LOOP
le < loop
double < loop

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_s:0':logError9_10(+(1, n925_10)), gen_s:0':logError9_10(n925_10)) -> false, rt in Omega(0)

Induction Base:
le(gen_s:0':logError9_10(+(1, 0)), gen_s:0':logError9_10(0)) ->_R^Omega(0)
false

Induction Step:
le(gen_s:0':logError9_10(+(1, +(n925_10, 1))), gen_s:0':logError9_10(+(n925_10, 1))) ->_R^Omega(0)
le(gen_s:0':logError9_10(+(1, n925_10)), gen_s:0':logError9_10(n925_10)) ->_IH
false

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
LE(s(z0), 0') -> c
LE(0', z0) -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
DOUBLE(0') -> c3
DOUBLE(s(z0)) -> c4(DOUBLE(z0))
LOG(0') -> c5
LOG(s(z0)) -> c6(LOOP(s(z0), s(0'), 0'))
LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
IF(true, z0, z1, z2) -> c8
IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
le(s(z0), 0') -> false
le(0', z0) -> true
le(s(z0), s(z1)) -> le(z0, z1)
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
log(0') -> logError
log(s(z0)) -> loop(s(z0), s(0'), 0')
loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
if(true, z0, z1, z2) -> z2
if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

Types:
LE :: s:0':logError -> s:0':logError -> c:c1:c2
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
DOUBLE :: s:0':logError -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LOG :: s:0':logError -> c5:c6
c5 :: c5:c6
c6 :: c7 -> c5:c6
LOOP :: s:0':logError -> s:0':logError -> s:0':logError -> c7
c7 :: c8:c9 -> c:c1:c2 -> c7
IF :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> c8:c9
le :: s:0':logError -> s:0':logError -> true:false
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c7 -> c3:c4 -> c8:c9
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_c:c1:c21_10 :: c:c1:c2
hole_s:0':logError2_10 :: s:0':logError
hole_c3:c43_10 :: c3:c4
hole_c5:c64_10 :: c5:c6
hole_c75_10 :: c7
hole_c8:c96_10 :: c8:c9
hole_true:false7_10 :: true:false
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_s:0':logError9_10 :: Nat -> s:0':logError
gen_c3:c410_10 :: Nat -> c3:c4


Lemmas:
LE(gen_s:0':logError9_10(+(1, n12_10)), gen_s:0':logError9_10(n12_10)) -> gen_c:c1:c28_10(n12_10), rt in Omega(1 + n12_10)
DOUBLE(gen_s:0':logError9_10(n588_10)) -> gen_c3:c410_10(n588_10), rt in Omega(1 + n588_10)
le(gen_s:0':logError9_10(+(1, n925_10)), gen_s:0':logError9_10(n925_10)) -> false, rt in Omega(0)


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c2(gen_c:c1:c28_10(x))
gen_s:0':logError9_10(0) <=> 0'
gen_s:0':logError9_10(+(x, 1)) <=> s(gen_s:0':logError9_10(x))
gen_c3:c410_10(0) <=> c3
gen_c3:c410_10(+(x, 1)) <=> c4(gen_c3:c410_10(x))


The following defined symbols remain to be analysed:
double, LOOP, loop

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_s:0':logError9_10(n1292_10)) -> gen_s:0':logError9_10(*(2, n1292_10)), rt in Omega(0)

Induction Base:
double(gen_s:0':logError9_10(0)) ->_R^Omega(0)
0'

Induction Step:
double(gen_s:0':logError9_10(+(n1292_10, 1))) ->_R^Omega(0)
s(s(double(gen_s:0':logError9_10(n1292_10)))) ->_IH
s(s(gen_s:0':logError9_10(*(2, c1293_10))))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
LE(s(z0), 0') -> c
LE(0', z0) -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
DOUBLE(0') -> c3
DOUBLE(s(z0)) -> c4(DOUBLE(z0))
LOG(0') -> c5
LOG(s(z0)) -> c6(LOOP(s(z0), s(0'), 0'))
LOOP(z0, s(z1), z2) -> c7(IF(le(z0, s(z1)), z0, s(z1), z2), LE(z0, s(z1)))
IF(true, z0, z1, z2) -> c8
IF(false, z0, z1, z2) -> c9(LOOP(z0, double(z1), s(z2)), DOUBLE(z1))
le(s(z0), 0') -> false
le(0', z0) -> true
le(s(z0), s(z1)) -> le(z0, z1)
double(0') -> 0'
double(s(z0)) -> s(s(double(z0)))
log(0') -> logError
log(s(z0)) -> loop(s(z0), s(0'), 0')
loop(z0, s(z1), z2) -> if(le(z0, s(z1)), z0, s(z1), z2)
if(true, z0, z1, z2) -> z2
if(false, z0, z1, z2) -> loop(z0, double(z1), s(z2))

Types:
LE :: s:0':logError -> s:0':logError -> c:c1:c2
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
DOUBLE :: s:0':logError -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LOG :: s:0':logError -> c5:c6
c5 :: c5:c6
c6 :: c7 -> c5:c6
LOOP :: s:0':logError -> s:0':logError -> s:0':logError -> c7
c7 :: c8:c9 -> c:c1:c2 -> c7
IF :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> c8:c9
le :: s:0':logError -> s:0':logError -> true:false
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c7 -> c3:c4 -> c8:c9
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: true:false -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_c:c1:c21_10 :: c:c1:c2
hole_s:0':logError2_10 :: s:0':logError
hole_c3:c43_10 :: c3:c4
hole_c5:c64_10 :: c5:c6
hole_c75_10 :: c7
hole_c8:c96_10 :: c8:c9
hole_true:false7_10 :: true:false
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_s:0':logError9_10 :: Nat -> s:0':logError
gen_c3:c410_10 :: Nat -> c3:c4


Lemmas:
LE(gen_s:0':logError9_10(+(1, n12_10)), gen_s:0':logError9_10(n12_10)) -> gen_c:c1:c28_10(n12_10), rt in Omega(1 + n12_10)
DOUBLE(gen_s:0':logError9_10(n588_10)) -> gen_c3:c410_10(n588_10), rt in Omega(1 + n588_10)
le(gen_s:0':logError9_10(+(1, n925_10)), gen_s:0':logError9_10(n925_10)) -> false, rt in Omega(0)
double(gen_s:0':logError9_10(n1292_10)) -> gen_s:0':logError9_10(*(2, n1292_10)), rt in Omega(0)


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c2(gen_c:c1:c28_10(x))
gen_s:0':logError9_10(0) <=> 0'
gen_s:0':logError9_10(+(x, 1)) <=> s(gen_s:0':logError9_10(x))
gen_c3:c410_10(0) <=> c3
gen_c3:c410_10(+(x, 1)) <=> c4(gen_c3:c410_10(x))


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