WORST_CASE(Omega(n^2),?)
proof of input_nmikTorEPz.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, 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), 9 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 361 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), 73 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 59 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 66 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 50 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 660 ms]
        (26) BEST
            (27) proven lower bound
                (28) LowerBoundPropagationProof [FINISHED, 0 ms]
                (29) BOUNDS(n^2, INF)
            (30) typed CpxTrs


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

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


The TRS R consists of the following rules:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   fac(x) -> loop(x, s(0), s(0))
   loop(x, c, y) -> if(lt(x, c), x, c, y)
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
   if(true, x, c, y) -> y

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:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   fac(z0) -> loop(z0, s(0), s(0))
   loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
   if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
   if(true, z0, z1, z2) -> z2
Tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(0, z0) -> c3
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(0, z0) -> c5
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   FAC(z0) -> c7(LOOP(z0, s(0), s(0)))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   IF(true, z0, z1, z2) -> c10
S tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(0, z0) -> c3
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(0, z0) -> c5
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   FAC(z0) -> c7(LOOP(z0, s(0), s(0)))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   IF(true, z0, z1, z2) -> c10
K tuples:none
Defined Rule Symbols:   lt_2, times_2, plus_2, fac_1, loop_3, if_4

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, FAC_1, LOOP_3, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4_2, c5, c6_1, c7_1, c8_2, c9_2, c10


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

(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^2, INF).


The TRS R consists of the following rules:

   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(0, z0) -> c3
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(0, z0) -> c5
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   FAC(z0) -> c7(LOOP(z0, s(0), s(0)))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   IF(true, z0, z1, z2) -> c10

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

   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   fac(z0) -> loop(z0, s(0), s(0))
   loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
   if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
   if(true, z0, z1, z2) -> 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^2, INF).


The TRS R consists of the following rules:

   LT(0', s(z0)) -> c
   LT(z0, 0') -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(0', z0) -> c3
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(0', z0) -> c5
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   IF(true, z0, z1, z2) -> c10

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

   lt(0', s(z0)) -> true
   lt(z0, 0') -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   times(0', z0) -> 0'
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0', z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   fac(z0) -> loop(z0, s(0'), s(0'))
   loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
   if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
   if(true, z0, z1, z2) -> z2

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

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

(8)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LT, TIMES, PLUS, times, LOOP, lt, plus, loop

They will be analysed ascendingly in the following order:
LT < LOOP
PLUS < TIMES
times < TIMES
TIMES < LOOP
times < LOOP
plus < times
times < loop
lt < LOOP
lt < loop

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

(10)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


The following defined symbols remain to be analysed:
LT, TIMES, PLUS, times, LOOP, lt, plus, loop

They will be analysed ascendingly in the following order:
LT < LOOP
PLUS < TIMES
times < TIMES
TIMES < LOOP
times < LOOP
plus < times
times < loop
lt < LOOP
lt < loop

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11))) -> gen_c:c1:c29_11(n14_11), rt in Omega(1 + n14_11)

Induction Base:
LT(gen_0':s10_11(0), gen_0':s10_11(+(1, 0))) ->_R^Omega(1)
c

Induction Step:
LT(gen_0':s10_11(+(n14_11, 1)), gen_0':s10_11(+(1, +(n14_11, 1)))) ->_R^Omega(1)
c2(LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11)))) ->_IH
c2(gen_c:c1:c29_11(c15_11))

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:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


The following defined symbols remain to be analysed:
LT, TIMES, PLUS, times, LOOP, lt, plus, loop

They will be analysed ascendingly in the following order:
LT < LOOP
PLUS < TIMES
times < TIMES
TIMES < LOOP
times < LOOP
plus < times
times < loop
lt < LOOP
lt < loop

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Lemmas:
LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11))) -> gen_c:c1:c29_11(n14_11), rt in Omega(1 + n14_11)


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


The following defined symbols remain to be analysed:
PLUS, TIMES, times, LOOP, lt, plus, loop

They will be analysed ascendingly in the following order:
PLUS < TIMES
times < TIMES
TIMES < LOOP
times < LOOP
plus < times
times < loop
lt < LOOP
lt < loop

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s10_11(n617_11), gen_0':s10_11(b)) -> gen_c5:c612_11(n617_11), rt in Omega(1 + n617_11)

Induction Base:
PLUS(gen_0':s10_11(0), gen_0':s10_11(b)) ->_R^Omega(1)
c5

Induction Step:
PLUS(gen_0':s10_11(+(n617_11, 1)), gen_0':s10_11(b)) ->_R^Omega(1)
c6(PLUS(gen_0':s10_11(n617_11), gen_0':s10_11(b))) ->_IH
c6(gen_c5:c612_11(c618_11))

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:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Lemmas:
LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11))) -> gen_c:c1:c29_11(n14_11), rt in Omega(1 + n14_11)
PLUS(gen_0':s10_11(n617_11), gen_0':s10_11(b)) -> gen_c5:c612_11(n617_11), rt in Omega(1 + n617_11)


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s10_11(n1303_11), gen_0':s10_11(+(1, n1303_11))) -> true, rt in Omega(0)

Induction Base:
lt(gen_0':s10_11(0), gen_0':s10_11(+(1, 0))) ->_R^Omega(0)
true

Induction Step:
lt(gen_0':s10_11(+(n1303_11, 1)), gen_0':s10_11(+(1, +(n1303_11, 1)))) ->_R^Omega(0)
lt(gen_0':s10_11(n1303_11), gen_0':s10_11(+(1, n1303_11))) ->_IH
true

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:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Lemmas:
LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11))) -> gen_c:c1:c29_11(n14_11), rt in Omega(1 + n14_11)
PLUS(gen_0':s10_11(n617_11), gen_0':s10_11(b)) -> gen_c5:c612_11(n617_11), rt in Omega(1 + n617_11)
lt(gen_0':s10_11(n1303_11), gen_0':s10_11(+(1, n1303_11))) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s10_11(n1689_11), gen_0':s10_11(b)) -> gen_0':s10_11(+(n1689_11, b)), rt in Omega(0)

Induction Base:
plus(gen_0':s10_11(0), gen_0':s10_11(b)) ->_R^Omega(0)
gen_0':s10_11(b)

Induction Step:
plus(gen_0':s10_11(+(n1689_11, 1)), gen_0':s10_11(b)) ->_R^Omega(0)
s(plus(gen_0':s10_11(n1689_11), gen_0':s10_11(b))) ->_IH
s(gen_0':s10_11(+(b, c1690_11)))

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:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Lemmas:
LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11))) -> gen_c:c1:c29_11(n14_11), rt in Omega(1 + n14_11)
PLUS(gen_0':s10_11(n617_11), gen_0':s10_11(b)) -> gen_c5:c612_11(n617_11), rt in Omega(1 + n617_11)
lt(gen_0':s10_11(n1303_11), gen_0':s10_11(+(1, n1303_11))) -> true, rt in Omega(0)
plus(gen_0':s10_11(n1689_11), gen_0':s10_11(b)) -> gen_0':s10_11(+(n1689_11, b)), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s10_11(n2936_11), gen_0':s10_11(b)) -> gen_0':s10_11(*(n2936_11, b)), rt in Omega(0)

Induction Base:
times(gen_0':s10_11(0), gen_0':s10_11(b)) ->_R^Omega(0)
0'

Induction Step:
times(gen_0':s10_11(+(n2936_11, 1)), gen_0':s10_11(b)) ->_R^Omega(0)
plus(gen_0':s10_11(b), times(gen_0':s10_11(n2936_11), gen_0':s10_11(b))) ->_IH
plus(gen_0':s10_11(b), gen_0':s10_11(*(c2937_11, b))) ->_L^Omega(0)
gen_0':s10_11(+(b, *(n2936_11, b)))

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

(24)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Lemmas:
LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11))) -> gen_c:c1:c29_11(n14_11), rt in Omega(1 + n14_11)
PLUS(gen_0':s10_11(n617_11), gen_0':s10_11(b)) -> gen_c5:c612_11(n617_11), rt in Omega(1 + n617_11)
lt(gen_0':s10_11(n1303_11), gen_0':s10_11(+(1, n1303_11))) -> true, rt in Omega(0)
plus(gen_0':s10_11(n1689_11), gen_0':s10_11(b)) -> gen_0':s10_11(+(n1689_11, b)), rt in Omega(0)
times(gen_0':s10_11(n2936_11), gen_0':s10_11(b)) -> gen_0':s10_11(*(n2936_11, b)), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


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

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
TIMES(gen_0':s10_11(n4472_11), gen_0':s10_11(b)) -> *13_11, rt in Omega(b*n4472_11 + n4472_11)

Induction Base:
TIMES(gen_0':s10_11(0), gen_0':s10_11(b))

Induction Step:
TIMES(gen_0':s10_11(+(n4472_11, 1)), gen_0':s10_11(b)) ->_R^Omega(1)
c4(PLUS(gen_0':s10_11(b), times(gen_0':s10_11(n4472_11), gen_0':s10_11(b))), TIMES(gen_0':s10_11(n4472_11), gen_0':s10_11(b))) ->_L^Omega(0)
c4(PLUS(gen_0':s10_11(b), gen_0':s10_11(*(n4472_11, b))), TIMES(gen_0':s10_11(n4472_11), gen_0':s10_11(b))) ->_L^Omega(1 + b)
c4(gen_c5:c612_11(b), TIMES(gen_0':s10_11(n4472_11), gen_0':s10_11(*(n4472_11, b)))) ->_IH
c4(gen_c5:c612_11(*(n4472_11, b)), *13_11)

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

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Lemmas:
LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11))) -> gen_c:c1:c29_11(n14_11), rt in Omega(1 + n14_11)
PLUS(gen_0':s10_11(n617_11), gen_0':s10_11(b)) -> gen_c5:c612_11(n617_11), rt in Omega(1 + n617_11)
lt(gen_0':s10_11(n1303_11), gen_0':s10_11(+(1, n1303_11))) -> true, rt in Omega(0)
plus(gen_0':s10_11(n1689_11), gen_0':s10_11(b)) -> gen_0':s10_11(+(n1689_11, b)), rt in Omega(0)
times(gen_0':s10_11(n2936_11), gen_0':s10_11(b)) -> gen_0':s10_11(*(n2936_11, b)), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


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

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

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

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

(29)
BOUNDS(n^2, INF)

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

(30)
Obligation:
Innermost TRS:
Rules:
LT(0', s(z0)) -> c
LT(z0, 0') -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
TIMES(0', z0) -> c3
TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
PLUS(0', z0) -> c5
PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
FAC(z0) -> c7(LOOP(z0, s(0'), s(0')))
LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
IF(true, z0, z1, z2) -> c10
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
times(0', z0) -> 0'
times(s(z0), z1) -> plus(z1, times(z0, z1))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
fac(z0) -> loop(z0, s(0'), s(0'))
loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
if(true, z0, z1, z2) -> z2

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
s :: 0':s -> 0':s
c :: c:c1:c2
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
TIMES :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c5:c6 -> c3:c4 -> c3:c4
PLUS :: 0':s -> 0':s -> c5:c6
times :: 0':s -> 0':s -> 0':s
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
FAC :: 0':s -> c7
c7 :: c8 -> c7
LOOP :: 0':s -> 0':s -> 0':s -> c8
c8 :: c9:c10 -> c:c1:c2 -> c8
IF :: false:true -> 0':s -> 0':s -> 0':s -> c9:c10
lt :: 0':s -> 0':s -> false:true
false :: false:true
c9 :: c8 -> c3:c4 -> c9:c10
true :: false:true
c10 :: c9:c10
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_11 :: c:c1:c2
hole_0':s2_11 :: 0':s
hole_c3:c43_11 :: c3:c4
hole_c5:c64_11 :: c5:c6
hole_c75_11 :: c7
hole_c86_11 :: c8
hole_c9:c107_11 :: c9:c10
hole_false:true8_11 :: false:true
gen_c:c1:c29_11 :: Nat -> c:c1:c2
gen_0':s10_11 :: Nat -> 0':s
gen_c3:c411_11 :: Nat -> c3:c4
gen_c5:c612_11 :: Nat -> c5:c6


Lemmas:
LT(gen_0':s10_11(n14_11), gen_0':s10_11(+(1, n14_11))) -> gen_c:c1:c29_11(n14_11), rt in Omega(1 + n14_11)
PLUS(gen_0':s10_11(n617_11), gen_0':s10_11(b)) -> gen_c5:c612_11(n617_11), rt in Omega(1 + n617_11)
lt(gen_0':s10_11(n1303_11), gen_0':s10_11(+(1, n1303_11))) -> true, rt in Omega(0)
plus(gen_0':s10_11(n1689_11), gen_0':s10_11(b)) -> gen_0':s10_11(+(n1689_11, b)), rt in Omega(0)
times(gen_0':s10_11(n2936_11), gen_0':s10_11(b)) -> gen_0':s10_11(*(n2936_11, b)), rt in Omega(0)
TIMES(gen_0':s10_11(n4472_11), gen_0':s10_11(b)) -> *13_11, rt in Omega(b*n4472_11 + n4472_11)


Generator Equations:
gen_c:c1:c29_11(0) <=> c
gen_c:c1:c29_11(+(x, 1)) <=> c2(gen_c:c1:c29_11(x))
gen_0':s10_11(0) <=> 0'
gen_0':s10_11(+(x, 1)) <=> s(gen_0':s10_11(x))
gen_c3:c411_11(0) <=> c3
gen_c3:c411_11(+(x, 1)) <=> c4(c5, gen_c3:c411_11(x))
gen_c5:c612_11(0) <=> c5
gen_c5:c612_11(+(x, 1)) <=> c6(gen_c5:c612_11(x))


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