WORST_CASE(Omega(n^1),?)
proof of input_HbmsLiF8ZI.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 4 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 396 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), 62 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 85 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 40 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
        (26) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(0, s(y)) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   mod(s(x), 0) -> 0
   mod(x, s(y)) -> help(x, s(y), 0)
   help(x, s(y), c) -> if(le(c, x), x, s(y), c)
   if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y)))
   if(false, x, s(y), c) -> minus(x, minus(c, s(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:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(0, s(z0)) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   mod(s(z0), 0) -> 0
   mod(z0, s(z1)) -> help(z0, s(z1), 0)
   help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
   if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))
Tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(z0, 0) -> c3
   MINUS(0, s(z0)) -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   PLUS(z0, 0) -> c6
   PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
   MOD(s(z0), 0) -> c8
   MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0))
   HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
   IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
S tuples:
   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(z0, 0) -> c3
   MINUS(0, s(z0)) -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   PLUS(z0, 0) -> c6
   PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
   MOD(s(z0), 0) -> c8
   MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0))
   HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
   IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
K tuples:none
Defined Rule Symbols:   le_2, minus_2, plus_2, mod_2, help_3, if_4

Defined Pair Symbols:   LE_2, MINUS_2, PLUS_2, MOD_2, HELP_3, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8, c9_1, c10_2, c11_2, c12_2


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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   LE(0, z0) -> c
   LE(s(z0), 0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(z0, 0) -> c3
   MINUS(0, s(z0)) -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   PLUS(z0, 0) -> c6
   PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
   MOD(s(z0), 0) -> c8
   MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0))
   HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
   IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))

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

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0) -> z0
   minus(0, s(z0)) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   mod(s(z0), 0) -> 0
   mod(z0, s(z1)) -> help(z0, s(z1), 0)
   help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
   if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   LE(0', z0) -> c
   LE(s(z0), 0') -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   MINUS(z0, 0') -> c3
   MINUS(0', s(z0)) -> c4
   MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
   PLUS(z0, 0') -> c6
   PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
   MOD(s(z0), 0') -> c8
   MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
   HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
   IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))

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

   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, 0') -> z0
   minus(0', s(z0)) -> 0'
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(z0, 0') -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   mod(s(z0), 0') -> 0'
   mod(z0, s(z1)) -> help(z0, s(z1), 0')
   help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
   if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LE, MINUS, PLUS, HELP, le, plus, minus, help

They will be analysed ascendingly in the following order:
LE < HELP
MINUS < HELP
PLUS < HELP
le < HELP
plus < HELP
minus < HELP
le < help
plus < help
minus < help

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

(10)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
LE, MINUS, PLUS, HELP, le, plus, minus, help

They will be analysed ascendingly in the following order:
LE < HELP
MINUS < HELP
PLUS < HELP
le < HELP
plus < HELP
minus < HELP
le < help
plus < help
minus < help

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s10_13(n14_13), gen_0':s10_13(n14_13)) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)

Induction Base:
LE(gen_0':s10_13(0), gen_0':s10_13(0)) ->_R^Omega(1)
c

Induction Step:
LE(gen_0':s10_13(+(n14_13, 1)), gen_0':s10_13(+(n14_13, 1))) ->_R^Omega(1)
c2(LE(gen_0':s10_13(n14_13), gen_0':s10_13(n14_13))) ->_IH
c2(gen_c:c1:c29_13(c15_13))

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

(12)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
LE, MINUS, PLUS, HELP, le, plus, minus, help

They will be analysed ascendingly in the following order:
LE < HELP
MINUS < HELP
PLUS < HELP
le < HELP
plus < HELP
minus < HELP
le < help
plus < help
minus < help

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
LE(gen_0':s10_13(n14_13), gen_0':s10_13(n14_13)) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


The following defined symbols remain to be analysed:
MINUS, PLUS, HELP, le, plus, minus, help

They will be analysed ascendingly in the following order:
MINUS < HELP
PLUS < HELP
le < HELP
plus < HELP
minus < HELP
le < help
plus < help
minus < help

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s10_13(n662_13), gen_0':s10_13(n662_13)) -> gen_c3:c4:c511_13(n662_13), rt in Omega(1 + n662_13)

Induction Base:
MINUS(gen_0':s10_13(0), gen_0':s10_13(0)) ->_R^Omega(1)
c3

Induction Step:
MINUS(gen_0':s10_13(+(n662_13, 1)), gen_0':s10_13(+(n662_13, 1))) ->_R^Omega(1)
c5(MINUS(gen_0':s10_13(n662_13), gen_0':s10_13(n662_13))) ->_IH
c5(gen_c3:c4:c511_13(c663_13))

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

(18)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
LE(gen_0':s10_13(n14_13), gen_0':s10_13(n14_13)) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
MINUS(gen_0':s10_13(n662_13), gen_0':s10_13(n662_13)) -> gen_c3:c4:c511_13(n662_13), rt in Omega(1 + n662_13)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s10_13(a), gen_0':s10_13(n1384_13)) -> gen_c6:c712_13(n1384_13), rt in Omega(1 + n1384_13)

Induction Base:
PLUS(gen_0':s10_13(a), gen_0':s10_13(0)) ->_R^Omega(1)
c6

Induction Step:
PLUS(gen_0':s10_13(a), gen_0':s10_13(+(n1384_13, 1))) ->_R^Omega(1)
c7(PLUS(gen_0':s10_13(a), gen_0':s10_13(n1384_13))) ->_IH
c7(gen_c6:c712_13(c1385_13))

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

(20)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
LE(gen_0':s10_13(n14_13), gen_0':s10_13(n14_13)) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
MINUS(gen_0':s10_13(n662_13), gen_0':s10_13(n662_13)) -> gen_c3:c4:c511_13(n662_13), rt in Omega(1 + n662_13)
PLUS(gen_0':s10_13(a), gen_0':s10_13(n1384_13)) -> gen_c6:c712_13(n1384_13), rt in Omega(1 + n1384_13)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s10_13(n2147_13), gen_0':s10_13(n2147_13)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s10_13(0), gen_0':s10_13(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s10_13(+(n2147_13, 1)), gen_0':s10_13(+(n2147_13, 1))) ->_R^Omega(0)
le(gen_0':s10_13(n2147_13), gen_0':s10_13(n2147_13)) ->_IH
true

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

(22)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
LE(gen_0':s10_13(n14_13), gen_0':s10_13(n14_13)) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
MINUS(gen_0':s10_13(n662_13), gen_0':s10_13(n662_13)) -> gen_c3:c4:c511_13(n662_13), rt in Omega(1 + n662_13)
PLUS(gen_0':s10_13(a), gen_0':s10_13(n1384_13)) -> gen_c6:c712_13(n1384_13), rt in Omega(1 + n1384_13)
le(gen_0':s10_13(n2147_13), gen_0':s10_13(n2147_13)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s10_13(a), gen_0':s10_13(n2580_13)) -> gen_0':s10_13(+(n2580_13, a)), rt in Omega(0)

Induction Base:
plus(gen_0':s10_13(a), gen_0':s10_13(0)) ->_R^Omega(0)
gen_0':s10_13(a)

Induction Step:
plus(gen_0':s10_13(a), gen_0':s10_13(+(n2580_13, 1))) ->_R^Omega(0)
s(plus(gen_0':s10_13(a), gen_0':s10_13(n2580_13))) ->_IH
s(gen_0':s10_13(+(a, c2581_13)))

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:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
LE(gen_0':s10_13(n14_13), gen_0':s10_13(n14_13)) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
MINUS(gen_0':s10_13(n662_13), gen_0':s10_13(n662_13)) -> gen_c3:c4:c511_13(n662_13), rt in Omega(1 + n662_13)
PLUS(gen_0':s10_13(a), gen_0':s10_13(n1384_13)) -> gen_c6:c712_13(n1384_13), rt in Omega(1 + n1384_13)
le(gen_0':s10_13(n2147_13), gen_0':s10_13(n2147_13)) -> true, rt in Omega(0)
plus(gen_0':s10_13(a), gen_0':s10_13(n2580_13)) -> gen_0':s10_13(+(n2580_13, a)), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


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

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s10_13(n3933_13), gen_0':s10_13(n3933_13)) -> gen_0':s10_13(0), rt in Omega(0)

Induction Base:
minus(gen_0':s10_13(0), gen_0':s10_13(0)) ->_R^Omega(0)
gen_0':s10_13(0)

Induction Step:
minus(gen_0':s10_13(+(n3933_13, 1)), gen_0':s10_13(+(n3933_13, 1))) ->_R^Omega(0)
minus(gen_0':s10_13(n3933_13), gen_0':s10_13(n3933_13)) ->_IH
gen_0':s10_13(0)

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

(26)
Obligation:
Innermost TRS:
Rules:
LE(0', z0) -> c
LE(s(z0), 0') -> c1
LE(s(z0), s(z1)) -> c2(LE(z0, z1))
MINUS(z0, 0') -> c3
MINUS(0', s(z0)) -> c4
MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1))
PLUS(z0, 0') -> c6
PLUS(z0, s(z1)) -> c7(PLUS(z0, z1))
MOD(s(z0), 0') -> c8
MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0'))
HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0))
IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1)))
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, 0') -> z0
minus(0', s(z0)) -> 0'
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(z0, 0') -> z0
plus(z0, s(z1)) -> s(plus(z0, z1))
mod(s(z0), 0') -> 0'
mod(z0, s(z1)) -> help(z0, s(z1), 0')
help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2)
if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1)))
if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1)))

Types:
LE :: 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
MINUS :: 0':s -> 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
PLUS :: 0':s -> 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
MOD :: 0':s -> 0':s -> c8:c9
c8 :: c8:c9
c9 :: c10 -> c8:c9
HELP :: 0':s -> 0':s -> 0':s -> c10
c10 :: c11:c12 -> c:c1:c2 -> c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
le :: 0':s -> 0':s -> true:false
true :: true:false
c11 :: c10 -> c6:c7 -> c11:c12
plus :: 0':s -> 0':s -> 0':s
false :: true:false
c12 :: c3:c4:c5 -> c3:c4:c5 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
mod :: 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_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c8:c95_13 :: c8:c9
hole_c106_13 :: c10
hole_c11:c127_13 :: c11:c12
hole_true:false8_13 :: true:false
gen_c:c1:c29_13 :: Nat -> c:c1:c2
gen_0':s10_13 :: Nat -> 0':s
gen_c3:c4:c511_13 :: Nat -> c3:c4:c5
gen_c6:c712_13 :: Nat -> c6:c7


Lemmas:
LE(gen_0':s10_13(n14_13), gen_0':s10_13(n14_13)) -> gen_c:c1:c29_13(n14_13), rt in Omega(1 + n14_13)
MINUS(gen_0':s10_13(n662_13), gen_0':s10_13(n662_13)) -> gen_c3:c4:c511_13(n662_13), rt in Omega(1 + n662_13)
PLUS(gen_0':s10_13(a), gen_0':s10_13(n1384_13)) -> gen_c6:c712_13(n1384_13), rt in Omega(1 + n1384_13)
le(gen_0':s10_13(n2147_13), gen_0':s10_13(n2147_13)) -> true, rt in Omega(0)
plus(gen_0':s10_13(a), gen_0':s10_13(n2580_13)) -> gen_0':s10_13(+(n2580_13, a)), rt in Omega(0)
minus(gen_0':s10_13(n3933_13), gen_0':s10_13(n3933_13)) -> gen_0':s10_13(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c29_13(0) <=> c
gen_c:c1:c29_13(+(x, 1)) <=> c2(gen_c:c1:c29_13(x))
gen_0':s10_13(0) <=> 0'
gen_0':s10_13(+(x, 1)) <=> s(gen_0':s10_13(x))
gen_c3:c4:c511_13(0) <=> c3
gen_c3:c4:c511_13(+(x, 1)) <=> c5(gen_c3:c4:c511_13(x))
gen_c6:c712_13(0) <=> c6
gen_c6:c712_13(+(x, 1)) <=> c7(gen_c6:c712_13(x))


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