WORST_CASE(Omega(n^3),?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1276 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) SlicingProof [LOWER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 5 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 293 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), 763 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 219 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 65 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 655 ms]
        (24) BEST
            (25) proven lower bound
                (26) LowerBoundPropagationProof [FINISHED, 0 ms]
                (27) BOUNDS(n^2, INF)
            (28) typed CpxTrs
                (29) RewriteLemmaProof [LOWER BOUND(ID), 1450 ms]
                (30) BEST
                    (31) proven lower bound
                        (32) LowerBoundPropagationProof [FINISHED, 0 ms]
                        (33) BOUNDS(n^3, INF)
                    (34) typed CpxTrs
                        (35) RewriteLemmaProof [LOWER BOUND(ID), 378 ms]
                        (36) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(0, z0) -> c2
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   P(s(z0)) -> c4
   P(0) -> c5
   MINUS(z0, 0) -> c6
   MINUS(0, z0) -> c7
   MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
   ISZERO(0) -> c9
   ISZERO(s(z0)) -> c10
   FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), ISZERO(z0))
   FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0)))
   FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0))
   IF(true, z0, z1, z2) -> c14
   IF(false, z0, z1, z2) -> c15(FACITER(z0, z2))
   FACTORIAL(z0) -> c16(FACITER(z0, s(0)))

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

   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   p(s(z0)) -> z0
   p(0) -> 0
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(z0, s(z1)) -> p(minus(z0, z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))
   if(true, z0, z1, z2) -> z1
   if(false, z0, z1, z2) -> facIter(z0, z2)
   factorial(z0) -> facIter(z0, s(0))

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^3, INF).


The TRS R consists of the following rules:

   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(0, z0) -> c2
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   P(s(z0)) -> c4
   P(0) -> c5
   MINUS(z0, 0) -> c6
   MINUS(0, z0) -> c7
   MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
   ISZERO(0) -> c9
   ISZERO(s(z0)) -> c10
   FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), ISZERO(z0))
   FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), MINUS(z0, s(0)))
   FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0)), z1, times(z1, z0)), TIMES(z1, z0))
   IF(true, z0, z1, z2) -> c14
   IF(false, z0, z1, z2) -> c15(FACITER(z0, z2))
   FACTORIAL(z0) -> c16(FACITER(z0, s(0)))

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

   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   p(s(z0)) -> z0
   p(0) -> 0
   minus(z0, 0) -> z0
   minus(0, z0) -> 0
   minus(z0, s(z1)) -> p(minus(z0, z1))
   isZero(0) -> true
   isZero(s(z0)) -> false
   facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0)), z1, times(z1, z0))
   if(true, z0, z1, z2) -> z1
   if(false, z0, z1, z2) -> facIter(z0, z2)
   factorial(z0) -> facIter(z0, s(0))

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^3, INF).


The TRS R consists of the following rules:

   PLUS(0', z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(0', z0) -> c2
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   P(s(z0)) -> c4
   P(0') -> c5
   MINUS(z0, 0') -> c6
   MINUS(0', z0) -> c7
   MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
   ISZERO(0') -> c9
   ISZERO(s(z0)) -> c10
   FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), z1, times(z1, z0)), ISZERO(z0))
   FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), z1, times(z1, z0)), MINUS(z0, s(0')))
   FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), z1, times(z1, z0)), TIMES(z1, z0))
   IF(true, z0, z1, z2) -> c14
   IF(false, z0, z1, z2) -> c15(FACITER(z0, z2))
   FACTORIAL(z0) -> c16(FACITER(z0, s(0')))

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

   plus(0', z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0', z0) -> 0'
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   p(s(z0)) -> z0
   p(0') -> 0'
   minus(z0, 0') -> z0
   minus(0', z0) -> 0'
   minus(z0, s(z1)) -> p(minus(z0, z1))
   isZero(0') -> true
   isZero(s(z0)) -> false
   facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
   if(true, z0, z1, z2) -> z1
   if(false, z0, z1, z2) -> facIter(z0, z2)
   factorial(z0) -> facIter(z0, s(0'))

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

(5) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
IF/2

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   PLUS(0', z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(0', z0) -> c2
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   P(s(z0)) -> c4
   P(0') -> c5
   MINUS(z0, 0') -> c6
   MINUS(0', z0) -> c7
   MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
   ISZERO(0') -> c9
   ISZERO(s(z0)) -> c10
   FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
   FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
   FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
   IF(true, z0, z2) -> c14
   IF(false, z0, z2) -> c15(FACITER(z0, z2))
   FACTORIAL(z0) -> c16(FACITER(z0, s(0')))

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

   plus(0', z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0', z0) -> 0'
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   p(s(z0)) -> z0
   p(0') -> 0'
   minus(z0, 0') -> z0
   minus(0', z0) -> 0'
   minus(z0, s(z1)) -> p(minus(z0, z1))
   isZero(0') -> true
   isZero(s(z0)) -> false
   facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
   if(true, z0, z1, z2) -> z1
   if(false, z0, z1, z2) -> facIter(z0, z2)
   factorial(z0) -> facIter(z0, s(0'))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
PLUS, TIMES, times, MINUS, minus, FACITER, plus, facIter

They will be analysed ascendingly in the following order:
PLUS < TIMES
times < TIMES
TIMES < FACITER
times < FACITER
plus < times
times < facIter
minus < MINUS
MINUS < FACITER
minus < FACITER
minus < facIter

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

(10)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
PLUS, TIMES, times, MINUS, minus, FACITER, plus, facIter

They will be analysed ascendingly in the following order:
PLUS < TIMES
times < TIMES
TIMES < FACITER
times < FACITER
plus < times
times < facIter
minus < MINUS
MINUS < FACITER
minus < FACITER
minus < facIter

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)

Induction Base:
PLUS(gen_0':s12_17(0), gen_0':s12_17(b)) ->_R^Omega(1)
c

Induction Step:
PLUS(gen_0':s12_17(+(n16_17, 1)), gen_0':s12_17(b)) ->_R^Omega(1)
c1(PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b))) ->_IH
c1(gen_c:c111_17(c17_17))

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:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
PLUS, TIMES, times, MINUS, minus, FACITER, plus, facIter

They will be analysed ascendingly in the following order:
PLUS < TIMES
times < TIMES
TIMES < FACITER
times < FACITER
plus < times
times < facIter
minus < MINUS
MINUS < FACITER
minus < FACITER
minus < facIter

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Lemmas:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
minus, TIMES, times, MINUS, FACITER, plus, facIter

They will be analysed ascendingly in the following order:
times < TIMES
TIMES < FACITER
times < FACITER
plus < times
times < facIter
minus < MINUS
MINUS < FACITER
minus < FACITER
minus < facIter

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17))) -> *15_17, rt in Omega(0)

Induction Base:
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, 0)))

Induction Step:
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, +(n747_17, 1)))) ->_R^Omega(0)
p(minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17)))) ->_IH
p(*15_17)

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

(18)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Lemmas:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17))) -> *15_17, rt in Omega(0)


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
MINUS, TIMES, times, FACITER, plus, facIter

They will be analysed ascendingly in the following order:
times < TIMES
TIMES < FACITER
times < FACITER
plus < times
times < facIter
MINUS < FACITER

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s12_17(n10714_17), gen_0':s12_17(b)) -> gen_0':s12_17(+(n10714_17, b)), rt in Omega(0)

Induction Base:
plus(gen_0':s12_17(0), gen_0':s12_17(b)) ->_R^Omega(0)
gen_0':s12_17(b)

Induction Step:
plus(gen_0':s12_17(+(n10714_17, 1)), gen_0':s12_17(b)) ->_R^Omega(0)
s(plus(gen_0':s12_17(n10714_17), gen_0':s12_17(b))) ->_IH
s(gen_0':s12_17(+(b, c10715_17)))

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:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Lemmas:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17))) -> *15_17, rt in Omega(0)
plus(gen_0':s12_17(n10714_17), gen_0':s12_17(b)) -> gen_0':s12_17(+(n10714_17, b)), rt in Omega(0)


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
times, TIMES, FACITER, facIter

They will be analysed ascendingly in the following order:
times < TIMES
TIMES < FACITER
times < FACITER
times < facIter

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s12_17(n12294_17), gen_0':s12_17(b)) -> gen_0':s12_17(*(n12294_17, b)), rt in Omega(0)

Induction Base:
times(gen_0':s12_17(0), gen_0':s12_17(b)) ->_R^Omega(0)
0'

Induction Step:
times(gen_0':s12_17(+(n12294_17, 1)), gen_0':s12_17(b)) ->_R^Omega(0)
plus(gen_0':s12_17(b), times(gen_0':s12_17(n12294_17), gen_0':s12_17(b))) ->_IH
plus(gen_0':s12_17(b), gen_0':s12_17(*(c12295_17, b))) ->_L^Omega(0)
gen_0':s12_17(+(b, *(n12294_17, b)))

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:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Lemmas:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17))) -> *15_17, rt in Omega(0)
plus(gen_0':s12_17(n10714_17), gen_0':s12_17(b)) -> gen_0':s12_17(+(n10714_17, b)), rt in Omega(0)
times(gen_0':s12_17(n12294_17), gen_0':s12_17(b)) -> gen_0':s12_17(*(n12294_17, b)), rt in Omega(0)


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
TIMES, FACITER, facIter

They will be analysed ascendingly in the following order:
TIMES < FACITER

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
TIMES(gen_0':s12_17(n15292_17), gen_0':s12_17(b)) -> *15_17, rt in Omega(b*n15292_17 + n15292_17)

Induction Base:
TIMES(gen_0':s12_17(0), gen_0':s12_17(b))

Induction Step:
TIMES(gen_0':s12_17(+(n15292_17, 1)), gen_0':s12_17(b)) ->_R^Omega(1)
c3(PLUS(gen_0':s12_17(b), times(gen_0':s12_17(n15292_17), gen_0':s12_17(b))), TIMES(gen_0':s12_17(n15292_17), gen_0':s12_17(b))) ->_L^Omega(0)
c3(PLUS(gen_0':s12_17(b), gen_0':s12_17(*(n15292_17, b))), TIMES(gen_0':s12_17(n15292_17), gen_0':s12_17(b))) ->_L^Omega(1 + b)
c3(gen_c:c111_17(b), TIMES(gen_0':s12_17(n15292_17), gen_0':s12_17(*(n15292_17, b)))) ->_IH
c3(gen_c:c111_17(*(n15292_17, b)), *15_17)

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Lemmas:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17))) -> *15_17, rt in Omega(0)
plus(gen_0':s12_17(n10714_17), gen_0':s12_17(b)) -> gen_0':s12_17(+(n10714_17, b)), rt in Omega(0)
times(gen_0':s12_17(n12294_17), gen_0':s12_17(b)) -> gen_0':s12_17(*(n12294_17, b)), rt in Omega(0)


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
TIMES, FACITER, facIter

They will be analysed ascendingly in the following order:
TIMES < FACITER

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

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

(27)
BOUNDS(n^2, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Lemmas:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17))) -> *15_17, rt in Omega(0)
plus(gen_0':s12_17(n10714_17), gen_0':s12_17(b)) -> gen_0':s12_17(+(n10714_17, b)), rt in Omega(0)
times(gen_0':s12_17(n12294_17), gen_0':s12_17(b)) -> gen_0':s12_17(*(n12294_17, b)), rt in Omega(0)
TIMES(gen_0':s12_17(n15292_17), gen_0':s12_17(b)) -> *15_17, rt in Omega(b*n15292_17 + n15292_17)


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
FACITER, facIter
----------------------------------------

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
FACITER(gen_0':s12_17(n44219_17), gen_0':s12_17(b)) -> *15_17, rt in Omega(n44219_17 + n44219_17^2 + n44219_17^3)

Induction Base:
FACITER(gen_0':s12_17(0), gen_0':s12_17(b))

Induction Step:
FACITER(gen_0':s12_17(+(n44219_17, 1)), gen_0':s12_17(b)) ->_R^Omega(1)
c13(IF(isZero(gen_0':s12_17(+(n44219_17, 1))), minus(gen_0':s12_17(+(n44219_17, 1)), s(0')), times(gen_0':s12_17(b), gen_0':s12_17(+(n44219_17, 1)))), TIMES(gen_0':s12_17(b), gen_0':s12_17(+(n44219_17, 1)))) ->_R^Omega(0)
c13(IF(false, minus(gen_0':s12_17(+(1, n44219_17)), s(0')), times(gen_0':s12_17(b), gen_0':s12_17(+(1, n44219_17)))), TIMES(gen_0':s12_17(b), gen_0':s12_17(+(1, n44219_17)))) ->_R^Omega(0)
c13(IF(false, p(minus(gen_0':s12_17(+(1, n44219_17)), 0')), times(gen_0':s12_17(b), gen_0':s12_17(+(1, n44219_17)))), TIMES(gen_0':s12_17(b), gen_0':s12_17(+(1, n44219_17)))) ->_R^Omega(0)
c13(IF(false, p(gen_0':s12_17(+(1, n44219_17))), times(gen_0':s12_17(b), gen_0':s12_17(+(1, n44219_17)))), TIMES(gen_0':s12_17(b), gen_0':s12_17(+(1, n44219_17)))) ->_R^Omega(0)
c13(IF(false, gen_0':s12_17(n44219_17), times(gen_0':s12_17(b), gen_0':s12_17(+(1, n44219_17)))), TIMES(gen_0':s12_17(b), gen_0':s12_17(+(1, n44219_17)))) ->_L^Omega(0)
c13(IF(false, gen_0':s12_17(n44219_17), gen_0':s12_17(*(b, +(1, n44219_17)))), TIMES(gen_0':s12_17(+(1, n44219_17)), gen_0':s12_17(+(1, n44219_17)))) ->_R^Omega(1)
c13(c15(FACITER(gen_0':s12_17(n44219_17), gen_0':s12_17(+(b, *(b, n44219_17))))), TIMES(gen_0':s12_17(+(1, n44219_17)), gen_0':s12_17(+(1, n44219_17)))) ->_IH
c13(c15(*15_17), TIMES(gen_0':s12_17(+(1, n44219_17)), gen_0':s12_17(+(1, n44219_17)))) ->_R^Omega(1)
c13(c15(*15_17), c3(PLUS(gen_0':s12_17(+(1, n44219_17)), times(gen_0':s12_17(n44219_17), gen_0':s12_17(+(1, n44219_17)))), TIMES(gen_0':s12_17(n44219_17), gen_0':s12_17(+(1, n44219_17))))) ->_L^Omega(0)
c13(c15(*15_17), c3(PLUS(gen_0':s12_17(+(1, n44219_17)), gen_0':s12_17(*(n44219_17, +(1, n44219_17)))), TIMES(gen_0':s12_17(n44219_17), gen_0':s12_17(+(1, n44219_17))))) ->_L^Omega(2 + n44219_17)
c13(c15(*15_17), c3(gen_c:c111_17(+(1, n44219_17)), TIMES(gen_0':s12_17(n44219_17), gen_0':s12_17(+(1, n44219_17))))) ->_L^Omega(2*n44219_17 + n44219_17^2)
c13(c15(*15_17), c3(gen_c:c111_17(+(1, n44219_17)), *15_17))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(30)
Complex Obligation (BEST)

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

(31)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Lemmas:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17))) -> *15_17, rt in Omega(0)
plus(gen_0':s12_17(n10714_17), gen_0':s12_17(b)) -> gen_0':s12_17(+(n10714_17, b)), rt in Omega(0)
times(gen_0':s12_17(n12294_17), gen_0':s12_17(b)) -> gen_0':s12_17(*(n12294_17, b)), rt in Omega(0)
TIMES(gen_0':s12_17(n15292_17), gen_0':s12_17(b)) -> *15_17, rt in Omega(b*n15292_17 + n15292_17)


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


The following defined symbols remain to be analysed:
FACITER, facIter
----------------------------------------

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

(33)
BOUNDS(n^3, INF)

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

(34)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
TIMES(0', z0) -> c2
TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
P(s(z0)) -> c4
P(0') -> c5
MINUS(z0, 0') -> c6
MINUS(0', z0) -> c7
MINUS(z0, s(z1)) -> c8(P(minus(z0, z1)), MINUS(z0, z1))
ISZERO(0') -> c9
ISZERO(s(z0)) -> c10
FACITER(z0, z1) -> c11(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), ISZERO(z0))
FACITER(z0, z1) -> c12(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), MINUS(z0, s(0')))
FACITER(z0, z1) -> c13(IF(isZero(z0), minus(z0, s(0')), times(z1, z0)), TIMES(z1, z0))
IF(true, z0, z2) -> c14
IF(false, z0, z2) -> c15(FACITER(z0, z2))
FACTORIAL(z0) -> c16(FACITER(z0, s(0')))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
p(s(z0)) -> z0
p(0') -> 0'
minus(z0, 0') -> z0
minus(0', z0) -> 0'
minus(z0, s(z1)) -> p(minus(z0, z1))
isZero(0') -> true
isZero(s(z0)) -> false
facIter(z0, z1) -> if(isZero(z0), minus(z0, s(0')), z1, times(z1, z0))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> facIter(z0, z2)
factorial(z0) -> facIter(z0, s(0'))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
TIMES :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c:c1 -> c2:c3 -> c2:c3
times :: 0':s -> 0':s -> 0':s
P :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
MINUS :: 0':s -> 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8
c8 :: c4:c5 -> c6:c7:c8 -> c6:c7:c8
minus :: 0':s -> 0':s -> 0':s
ISZERO :: 0':s -> c9:c10
c9 :: c9:c10
c10 :: c9:c10
FACITER :: 0':s -> 0':s -> c11:c12:c13
c11 :: c14:c15 -> c9:c10 -> c11:c12:c13
IF :: true:false -> 0':s -> 0':s -> c14:c15
isZero :: 0':s -> true:false
c12 :: c14:c15 -> c6:c7:c8 -> c11:c12:c13
c13 :: c14:c15 -> c2:c3 -> c11:c12:c13
true :: true:false
c14 :: c14:c15
false :: true:false
c15 :: c11:c12:c13 -> c14:c15
FACTORIAL :: 0':s -> c16
c16 :: c11:c12:c13 -> c16
plus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
facIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
factorial :: 0':s -> 0':s
hole_c:c11_17 :: c:c1
hole_0':s2_17 :: 0':s
hole_c2:c33_17 :: c2:c3
hole_c4:c54_17 :: c4:c5
hole_c6:c7:c85_17 :: c6:c7:c8
hole_c9:c106_17 :: c9:c10
hole_c11:c12:c137_17 :: c11:c12:c13
hole_c14:c158_17 :: c14:c15
hole_true:false9_17 :: true:false
hole_c1610_17 :: c16
gen_c:c111_17 :: Nat -> c:c1
gen_0':s12_17 :: Nat -> 0':s
gen_c2:c313_17 :: Nat -> c2:c3
gen_c6:c7:c814_17 :: Nat -> c6:c7:c8


Lemmas:
PLUS(gen_0':s12_17(n16_17), gen_0':s12_17(b)) -> gen_c:c111_17(n16_17), rt in Omega(1 + n16_17)
minus(gen_0':s12_17(a), gen_0':s12_17(+(1, n747_17))) -> *15_17, rt in Omega(0)
plus(gen_0':s12_17(n10714_17), gen_0':s12_17(b)) -> gen_0':s12_17(+(n10714_17, b)), rt in Omega(0)
times(gen_0':s12_17(n12294_17), gen_0':s12_17(b)) -> gen_0':s12_17(*(n12294_17, b)), rt in Omega(0)
TIMES(gen_0':s12_17(n15292_17), gen_0':s12_17(b)) -> *15_17, rt in Omega(b*n15292_17 + n15292_17)
FACITER(gen_0':s12_17(n44219_17), gen_0':s12_17(b)) -> *15_17, rt in Omega(n44219_17 + n44219_17^2 + n44219_17^3)


Generator Equations:
gen_c:c111_17(0) <=> c
gen_c:c111_17(+(x, 1)) <=> c1(gen_c:c111_17(x))
gen_0':s12_17(0) <=> 0'
gen_0':s12_17(+(x, 1)) <=> s(gen_0':s12_17(x))
gen_c2:c313_17(0) <=> c2
gen_c2:c313_17(+(x, 1)) <=> c3(c, gen_c2:c313_17(x))
gen_c6:c7:c814_17(0) <=> c6
gen_c6:c7:c814_17(+(x, 1)) <=> c8(c4, gen_c6:c7:c814_17(x))


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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
facIter(gen_0':s12_17(n81456_17), gen_0':s12_17(b)) -> *15_17, rt in Omega(0)

Induction Base:
facIter(gen_0':s12_17(0), gen_0':s12_17(b))

Induction Step:
facIter(gen_0':s12_17(+(n81456_17, 1)), gen_0':s12_17(b)) ->_R^Omega(0)
if(isZero(gen_0':s12_17(+(n81456_17, 1))), minus(gen_0':s12_17(+(n81456_17, 1)), s(0')), gen_0':s12_17(b), times(gen_0':s12_17(b), gen_0':s12_17(+(n81456_17, 1)))) ->_R^Omega(0)
if(false, minus(gen_0':s12_17(+(1, n81456_17)), s(0')), gen_0':s12_17(b), times(gen_0':s12_17(b), gen_0':s12_17(+(1, n81456_17)))) ->_R^Omega(0)
if(false, p(minus(gen_0':s12_17(+(1, n81456_17)), 0')), gen_0':s12_17(b), times(gen_0':s12_17(b), gen_0':s12_17(+(1, n81456_17)))) ->_R^Omega(0)
if(false, p(gen_0':s12_17(+(1, n81456_17))), gen_0':s12_17(b), times(gen_0':s12_17(b), gen_0':s12_17(+(1, n81456_17)))) ->_R^Omega(0)
if(false, gen_0':s12_17(n81456_17), gen_0':s12_17(b), times(gen_0':s12_17(b), gen_0':s12_17(+(1, n81456_17)))) ->_L^Omega(0)
if(false, gen_0':s12_17(n81456_17), gen_0':s12_17(+(1, n81456_17)), gen_0':s12_17(*(b, +(1, n81456_17)))) ->_R^Omega(0)
facIter(gen_0':s12_17(n81456_17), gen_0':s12_17(+(b, *(b, n81456_17)))) ->_IH
*15_17

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

(36)
BOUNDS(1, INF)