WORST_CASE(Omega(n^1),?)
proof of input_YSEKERbVYQ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 388 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), 101 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 79 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:

   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   quot(x, s(y)) -> help(x, s(y), 0)
   help(x, s(y), c) -> if(lt(c, x), x, s(y), c)
   if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y))))
   if(false, x, s(y), c) -> 0

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(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   quot(z0, s(z1)) -> help(z0, s(z1), 0)
   help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
   if(false, z0, s(z1), z2) -> 0
Tuples:
   LT(z0, 0) -> c
   LT(0, s(z0)) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, 0) -> c3
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c8
S tuples:
   LT(z0, 0) -> c
   LT(0, s(z0)) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, 0) -> c3
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c8
K tuples:none
Defined Rule Symbols:   lt_2, plus_2, quot_2, help_3, if_4

Defined Pair Symbols:   LT_2, PLUS_2, QUOT_2, HELP_3, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5_1, c6_2, c7_2, c8


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

(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:

   LT(z0, 0) -> c
   LT(0, s(z0)) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, 0) -> c3
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c8

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

   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   quot(z0, s(z1)) -> help(z0, s(z1), 0)
   help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
   if(false, z0, s(z1), z2) -> 0

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:

   LT(z0, 0') -> c
   LT(0', s(z0)) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, 0') -> c3
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0'))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c8

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

   lt(z0, 0') -> false
   lt(0', s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, 0') -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   quot(z0, s(z1)) -> help(z0, s(z1), 0')
   help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
   if(false, z0, s(z1), z2) -> 0'

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

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

(8)
Obligation:
Innermost TRS:
Rules:
LT(z0, 0') -> c
LT(0', s(z0)) -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
PLUS(z0, 0') -> c3
PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c8
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
quot(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
if(false, z0, s(z1), z2) -> 0'

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
QUOT :: 0':s -> 0':s -> c5
c5 :: c6 -> c5
HELP :: 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c:c1:c2 -> c6
IF :: true:false -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> true:false
true :: true:false
c7 :: c6 -> c3:c4 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c8 :: c7:c8
quot :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c43_9 :: c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_true:false7_9 :: true:false
gen_c:c1:c28_9 :: Nat -> c:c1:c2
gen_0':s9_9 :: Nat -> 0':s
gen_c3:c410_9 :: Nat -> c3:c4

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LT, PLUS, HELP, lt, plus, help

They will be analysed ascendingly in the following order:
LT < HELP
PLUS < HELP
lt < HELP
plus < HELP
lt < help
plus < help

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

(10)
Obligation:
Innermost TRS:
Rules:
LT(z0, 0') -> c
LT(0', s(z0)) -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
PLUS(z0, 0') -> c3
PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c8
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
quot(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
if(false, z0, s(z1), z2) -> 0'

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
QUOT :: 0':s -> 0':s -> c5
c5 :: c6 -> c5
HELP :: 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c:c1:c2 -> c6
IF :: true:false -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> true:false
true :: true:false
c7 :: c6 -> c3:c4 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c8 :: c7:c8
quot :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c43_9 :: c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_true:false7_9 :: true:false
gen_c:c1:c28_9 :: Nat -> c:c1:c2
gen_0':s9_9 :: Nat -> 0':s
gen_c3:c410_9 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c28_9(0) <=> c
gen_c:c1:c28_9(+(x, 1)) <=> c2(gen_c:c1:c28_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c3:c410_9(0) <=> c3
gen_c3:c410_9(+(x, 1)) <=> c4(gen_c3:c410_9(x))


The following defined symbols remain to be analysed:
LT, PLUS, HELP, lt, plus, help

They will be analysed ascendingly in the following order:
LT < HELP
PLUS < HELP
lt < HELP
plus < HELP
lt < help
plus < help

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LT(gen_0':s9_9(n12_9), gen_0':s9_9(n12_9)) -> gen_c:c1:c28_9(n12_9), rt in Omega(1 + n12_9)

Induction Base:
LT(gen_0':s9_9(0), gen_0':s9_9(0)) ->_R^Omega(1)
c

Induction Step:
LT(gen_0':s9_9(+(n12_9, 1)), gen_0':s9_9(+(n12_9, 1))) ->_R^Omega(1)
c2(LT(gen_0':s9_9(n12_9), gen_0':s9_9(n12_9))) ->_IH
c2(gen_c:c1:c28_9(c13_9))

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(z0, 0') -> c
LT(0', s(z0)) -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
PLUS(z0, 0') -> c3
PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c8
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
quot(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
if(false, z0, s(z1), z2) -> 0'

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
QUOT :: 0':s -> 0':s -> c5
c5 :: c6 -> c5
HELP :: 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c:c1:c2 -> c6
IF :: true:false -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> true:false
true :: true:false
c7 :: c6 -> c3:c4 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c8 :: c7:c8
quot :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c43_9 :: c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_true:false7_9 :: true:false
gen_c:c1:c28_9 :: Nat -> c:c1:c2
gen_0':s9_9 :: Nat -> 0':s
gen_c3:c410_9 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c28_9(0) <=> c
gen_c:c1:c28_9(+(x, 1)) <=> c2(gen_c:c1:c28_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c3:c410_9(0) <=> c3
gen_c3:c410_9(+(x, 1)) <=> c4(gen_c3:c410_9(x))


The following defined symbols remain to be analysed:
LT, PLUS, HELP, lt, plus, help

They will be analysed ascendingly in the following order:
LT < HELP
PLUS < HELP
lt < HELP
plus < HELP
lt < help
plus < help

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
LT(z0, 0') -> c
LT(0', s(z0)) -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
PLUS(z0, 0') -> c3
PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c8
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
quot(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
if(false, z0, s(z1), z2) -> 0'

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
QUOT :: 0':s -> 0':s -> c5
c5 :: c6 -> c5
HELP :: 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c:c1:c2 -> c6
IF :: true:false -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> true:false
true :: true:false
c7 :: c6 -> c3:c4 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c8 :: c7:c8
quot :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c43_9 :: c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_true:false7_9 :: true:false
gen_c:c1:c28_9 :: Nat -> c:c1:c2
gen_0':s9_9 :: Nat -> 0':s
gen_c3:c410_9 :: Nat -> c3:c4


Lemmas:
LT(gen_0':s9_9(n12_9), gen_0':s9_9(n12_9)) -> gen_c:c1:c28_9(n12_9), rt in Omega(1 + n12_9)


Generator Equations:
gen_c:c1:c28_9(0) <=> c
gen_c:c1:c28_9(+(x, 1)) <=> c2(gen_c:c1:c28_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c3:c410_9(0) <=> c3
gen_c3:c410_9(+(x, 1)) <=> c4(gen_c3:c410_9(x))


The following defined symbols remain to be analysed:
PLUS, HELP, lt, plus, help

They will be analysed ascendingly in the following order:
PLUS < HELP
lt < HELP
plus < HELP
lt < help
plus < help

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s9_9(a), gen_0':s9_9(n569_9)) -> gen_c3:c410_9(n569_9), rt in Omega(1 + n569_9)

Induction Base:
PLUS(gen_0':s9_9(a), gen_0':s9_9(0)) ->_R^Omega(1)
c3

Induction Step:
PLUS(gen_0':s9_9(a), gen_0':s9_9(+(n569_9, 1))) ->_R^Omega(1)
c4(PLUS(gen_0':s9_9(a), gen_0':s9_9(n569_9))) ->_IH
c4(gen_c3:c410_9(c570_9))

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(z0, 0') -> c
LT(0', s(z0)) -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
PLUS(z0, 0') -> c3
PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c8
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
quot(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
if(false, z0, s(z1), z2) -> 0'

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
QUOT :: 0':s -> 0':s -> c5
c5 :: c6 -> c5
HELP :: 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c:c1:c2 -> c6
IF :: true:false -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> true:false
true :: true:false
c7 :: c6 -> c3:c4 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c8 :: c7:c8
quot :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c43_9 :: c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_true:false7_9 :: true:false
gen_c:c1:c28_9 :: Nat -> c:c1:c2
gen_0':s9_9 :: Nat -> 0':s
gen_c3:c410_9 :: Nat -> c3:c4


Lemmas:
LT(gen_0':s9_9(n12_9), gen_0':s9_9(n12_9)) -> gen_c:c1:c28_9(n12_9), rt in Omega(1 + n12_9)
PLUS(gen_0':s9_9(a), gen_0':s9_9(n569_9)) -> gen_c3:c410_9(n569_9), rt in Omega(1 + n569_9)


Generator Equations:
gen_c:c1:c28_9(0) <=> c
gen_c:c1:c28_9(+(x, 1)) <=> c2(gen_c:c1:c28_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c3:c410_9(0) <=> c3
gen_c3:c410_9(+(x, 1)) <=> c4(gen_c3:c410_9(x))


The following defined symbols remain to be analysed:
lt, HELP, plus, help

They will be analysed ascendingly in the following order:
lt < HELP
plus < HELP
lt < help
plus < help

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s9_9(n1141_9), gen_0':s9_9(n1141_9)) -> false, rt in Omega(0)

Induction Base:
lt(gen_0':s9_9(0), gen_0':s9_9(0)) ->_R^Omega(0)
false

Induction Step:
lt(gen_0':s9_9(+(n1141_9, 1)), gen_0':s9_9(+(n1141_9, 1))) ->_R^Omega(0)
lt(gen_0':s9_9(n1141_9), gen_0':s9_9(n1141_9)) ->_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:
LT(z0, 0') -> c
LT(0', s(z0)) -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
PLUS(z0, 0') -> c3
PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c8
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
quot(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
if(false, z0, s(z1), z2) -> 0'

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
QUOT :: 0':s -> 0':s -> c5
c5 :: c6 -> c5
HELP :: 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c:c1:c2 -> c6
IF :: true:false -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> true:false
true :: true:false
c7 :: c6 -> c3:c4 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c8 :: c7:c8
quot :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c43_9 :: c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_true:false7_9 :: true:false
gen_c:c1:c28_9 :: Nat -> c:c1:c2
gen_0':s9_9 :: Nat -> 0':s
gen_c3:c410_9 :: Nat -> c3:c4


Lemmas:
LT(gen_0':s9_9(n12_9), gen_0':s9_9(n12_9)) -> gen_c:c1:c28_9(n12_9), rt in Omega(1 + n12_9)
PLUS(gen_0':s9_9(a), gen_0':s9_9(n569_9)) -> gen_c3:c410_9(n569_9), rt in Omega(1 + n569_9)
lt(gen_0':s9_9(n1141_9), gen_0':s9_9(n1141_9)) -> false, rt in Omega(0)


Generator Equations:
gen_c:c1:c28_9(0) <=> c
gen_c:c1:c28_9(+(x, 1)) <=> c2(gen_c:c1:c28_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c3:c410_9(0) <=> c3
gen_c3:c410_9(+(x, 1)) <=> c4(gen_c3:c410_9(x))


The following defined symbols remain to be analysed:
plus, HELP, help

They will be analysed ascendingly in the following order:
plus < HELP
plus < help

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s9_9(a), gen_0':s9_9(n1491_9)) -> gen_0':s9_9(+(n1491_9, a)), rt in Omega(0)

Induction Base:
plus(gen_0':s9_9(a), gen_0':s9_9(0)) ->_R^Omega(0)
gen_0':s9_9(a)

Induction Step:
plus(gen_0':s9_9(a), gen_0':s9_9(+(n1491_9, 1))) ->_R^Omega(0)
s(plus(gen_0':s9_9(a), gen_0':s9_9(n1491_9))) ->_IH
s(gen_0':s9_9(+(a, c1492_9)))

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(z0, 0') -> c
LT(0', s(z0)) -> c1
LT(s(z0), s(z1)) -> c2(LT(z0, z1))
PLUS(z0, 0') -> c3
PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c8
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
quot(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
if(false, z0, s(z1), z2) -> 0'

Types:
LT :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
QUOT :: 0':s -> 0':s -> c5
c5 :: c6 -> c5
HELP :: 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c:c1:c2 -> c6
IF :: true:false -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> true:false
true :: true:false
c7 :: c6 -> c3:c4 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c8 :: c7:c8
quot :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c43_9 :: c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_true:false7_9 :: true:false
gen_c:c1:c28_9 :: Nat -> c:c1:c2
gen_0':s9_9 :: Nat -> 0':s
gen_c3:c410_9 :: Nat -> c3:c4


Lemmas:
LT(gen_0':s9_9(n12_9), gen_0':s9_9(n12_9)) -> gen_c:c1:c28_9(n12_9), rt in Omega(1 + n12_9)
PLUS(gen_0':s9_9(a), gen_0':s9_9(n569_9)) -> gen_c3:c410_9(n569_9), rt in Omega(1 + n569_9)
lt(gen_0':s9_9(n1141_9), gen_0':s9_9(n1141_9)) -> false, rt in Omega(0)
plus(gen_0':s9_9(a), gen_0':s9_9(n1491_9)) -> gen_0':s9_9(+(n1491_9, a)), rt in Omega(0)


Generator Equations:
gen_c:c1:c28_9(0) <=> c
gen_c:c1:c28_9(+(x, 1)) <=> c2(gen_c:c1:c28_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c3:c410_9(0) <=> c3
gen_c3:c410_9(+(x, 1)) <=> c4(gen_c3:c410_9(x))


The following defined symbols remain to be analysed:
HELP, help