/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^3), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 253 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 81 ms]
        (14) BEST
            (15) proven lower bound
                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
                (17) BOUNDS(n^3, INF)
            (18) typed CpxTrs
                (19) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
                (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
                (22) typed CpxTrs


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

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


The TRS R consists of the following rules:

   p(0) -> 0
   p(s(x)) -> x
   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(p(s(x)), y))
   plus(x, s(y)) -> s(plus(x, p(s(y))))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(zero(y), s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(zero(y), z))
   eq(0, 0) -> true
   eq(s(x), 0) -> false
   eq(0, s(y)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   divides(y, x) -> eq(x, times(div(x, y), y))
   prime(s(s(x))) -> pr(s(s(x)), s(x))
   pr(x, s(0)) -> true
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
   if(true, x, y) -> false
   if(false, x, y) -> pr(x, y)
   zero(div(x, x)) -> x
   zero(divides(x, x)) -> x
   zero(times(x, x)) -> x
   zero(quot(x, x, x)) -> x
   zero(s(x)) -> if(eq(x, s(0)), plus(zero(0), 0), s(plus(0, zero(0))))

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

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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   p(0') -> 0'
   p(s(x)) -> x
   plus(x, 0') -> x
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(p(s(x)), y))
   plus(x, s(y)) -> s(plus(x, p(s(y))))
   times(0', y) -> 0'
   times(s(0'), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0', y) -> 0'
   div(x, y) -> quot(x, y, y)
   quot(zero(y), s(y), z) -> 0'
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0', s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(zero(y), z))
   eq(0', 0') -> true
   eq(s(x), 0') -> false
   eq(0', s(y)) -> false
   eq(s(x), s(y)) -> eq(x, y)
   divides(y, x) -> eq(x, times(div(x, y), y))
   prime(s(s(x))) -> pr(s(s(x)), s(x))
   pr(x, s(0')) -> true
   pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
   if(true, x, y) -> false
   if(false, x, y) -> pr(x, y)
   zero(div(x, x)) -> x
   zero(divides(x, x)) -> x
   zero(times(x, x)) -> x
   zero(quot(x, x, x)) -> x
   zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

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

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

(4)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(zero(y), s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(zero(y), z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)
zero(div(x, x)) -> x
zero(divides(x, x)) -> x
zero(times(x, x)) -> x
zero(quot(x, x, x)) -> x
zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

Types:
p :: 0':s:true:false -> 0':s:true:false
0' :: 0':s:true:false
s :: 0':s:true:false -> 0':s:true:false
plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
zero :: 0':s:true:false -> 0':s:true:false
eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
true :: 0':s:true:false
false :: 0':s:true:false
divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
prime :: 0':s:true:false -> 0':s:true:false
pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
hole_0':s:true:false1_0 :: 0':s:true:false
gen_0':s:true:false2_0 :: Nat -> 0':s:true:false

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, times, div, quot, zero, eq, divides, pr, if

They will be analysed ascendingly in the following order:
plus < times
plus < zero
times < div
times < divides
div = quot
div = zero
div = divides
div = pr
div = if
quot = zero
quot = divides
quot = pr
quot = if
eq < zero
zero = divides
zero = pr
zero = if
eq < divides
divides = pr
divides = if
pr = if

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

(6)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(zero(y), s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(zero(y), z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)
zero(div(x, x)) -> x
zero(divides(x, x)) -> x
zero(times(x, x)) -> x
zero(quot(x, x, x)) -> x
zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

Types:
p :: 0':s:true:false -> 0':s:true:false
0' :: 0':s:true:false
s :: 0':s:true:false -> 0':s:true:false
plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
zero :: 0':s:true:false -> 0':s:true:false
eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
true :: 0':s:true:false
false :: 0':s:true:false
divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
prime :: 0':s:true:false -> 0':s:true:false
pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
hole_0':s:true:false1_0 :: 0':s:true:false
gen_0':s:true:false2_0 :: Nat -> 0':s:true:false


Generator Equations:
gen_0':s:true:false2_0(0) <=> 0'
gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x))


The following defined symbols remain to be analysed:
plus, times, div, quot, zero, eq, divides, pr, if

They will be analysed ascendingly in the following order:
plus < times
plus < zero
times < div
times < divides
div = quot
div = zero
div = divides
div = pr
div = if
quot = zero
quot = divides
quot = pr
quot = if
eq < zero
zero = divides
zero = pr
zero = if
eq < divides
divides = pr
divides = if
pr = if

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(0)) ->_R^Omega(1)
gen_0':s:true:false2_0(a)

Induction Step:
plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(+(n4_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s:true:false2_0(a), p(s(gen_0':s:true:false2_0(n4_0))))) ->_R^Omega(1)
s(plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0))) ->_IH
s(gen_0':s:true:false2_0(+(a, c5_0)))

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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(zero(y), s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(zero(y), z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)
zero(div(x, x)) -> x
zero(divides(x, x)) -> x
zero(times(x, x)) -> x
zero(quot(x, x, x)) -> x
zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

Types:
p :: 0':s:true:false -> 0':s:true:false
0' :: 0':s:true:false
s :: 0':s:true:false -> 0':s:true:false
plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
zero :: 0':s:true:false -> 0':s:true:false
eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
true :: 0':s:true:false
false :: 0':s:true:false
divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
prime :: 0':s:true:false -> 0':s:true:false
pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
hole_0':s:true:false1_0 :: 0':s:true:false
gen_0':s:true:false2_0 :: Nat -> 0':s:true:false


Generator Equations:
gen_0':s:true:false2_0(0) <=> 0'
gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x))


The following defined symbols remain to be analysed:
plus, times, div, quot, zero, eq, divides, pr, if

They will be analysed ascendingly in the following order:
plus < times
plus < zero
times < div
times < divides
div = quot
div = zero
div = divides
div = pr
div = if
quot = zero
quot = divides
quot = pr
quot = if
eq < zero
zero = divides
zero = pr
zero = if
eq < divides
divides = pr
divides = if
pr = if

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(zero(y), s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(zero(y), z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)
zero(div(x, x)) -> x
zero(divides(x, x)) -> x
zero(times(x, x)) -> x
zero(quot(x, x, x)) -> x
zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

Types:
p :: 0':s:true:false -> 0':s:true:false
0' :: 0':s:true:false
s :: 0':s:true:false -> 0':s:true:false
plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
zero :: 0':s:true:false -> 0':s:true:false
eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
true :: 0':s:true:false
false :: 0':s:true:false
divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
prime :: 0':s:true:false -> 0':s:true:false
pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
hole_0':s:true:false1_0 :: 0':s:true:false
gen_0':s:true:false2_0 :: Nat -> 0':s:true:false


Lemmas:
plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:true:false2_0(0) <=> 0'
gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x))


The following defined symbols remain to be analysed:
times, div, quot, zero, eq, divides, pr, if

They will be analysed ascendingly in the following order:
times < div
times < divides
div = quot
div = zero
div = divides
div = pr
div = if
quot = zero
quot = divides
quot = pr
quot = if
eq < zero
zero = divides
zero = pr
zero = if
eq < divides
divides = pr
divides = if
pr = if

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b)) -> gen_0':s:true:false2_0(*(n983_0, b)), rt in Omega(1 + b*n983_0^2 + n983_0)

Induction Base:
times(gen_0':s:true:false2_0(0), gen_0':s:true:false2_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_0':s:true:false2_0(+(n983_0, 1)), gen_0':s:true:false2_0(b)) ->_R^Omega(1)
plus(gen_0':s:true:false2_0(b), times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b))) ->_IH
plus(gen_0':s:true:false2_0(b), gen_0':s:true:false2_0(*(c984_0, b))) ->_L^Omega(1 + b*n983_0)
gen_0':s:true:false2_0(+(*(n983_0, b), b))

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(zero(y), s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(zero(y), z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)
zero(div(x, x)) -> x
zero(divides(x, x)) -> x
zero(times(x, x)) -> x
zero(quot(x, x, x)) -> x
zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

Types:
p :: 0':s:true:false -> 0':s:true:false
0' :: 0':s:true:false
s :: 0':s:true:false -> 0':s:true:false
plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
zero :: 0':s:true:false -> 0':s:true:false
eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
true :: 0':s:true:false
false :: 0':s:true:false
divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
prime :: 0':s:true:false -> 0':s:true:false
pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
hole_0':s:true:false1_0 :: 0':s:true:false
gen_0':s:true:false2_0 :: Nat -> 0':s:true:false


Lemmas:
plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:true:false2_0(0) <=> 0'
gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x))


The following defined symbols remain to be analysed:
times, div, quot, zero, eq, divides, pr, if

They will be analysed ascendingly in the following order:
times < div
times < divides
div = quot
div = zero
div = divides
div = pr
div = if
quot = zero
quot = divides
quot = pr
quot = if
eq < zero
zero = divides
zero = pr
zero = if
eq < divides
divides = pr
divides = if
pr = if

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

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

(17)
BOUNDS(n^3, INF)

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

(18)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(zero(y), s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(zero(y), z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)
zero(div(x, x)) -> x
zero(divides(x, x)) -> x
zero(times(x, x)) -> x
zero(quot(x, x, x)) -> x
zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

Types:
p :: 0':s:true:false -> 0':s:true:false
0' :: 0':s:true:false
s :: 0':s:true:false -> 0':s:true:false
plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
zero :: 0':s:true:false -> 0':s:true:false
eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
true :: 0':s:true:false
false :: 0':s:true:false
divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
prime :: 0':s:true:false -> 0':s:true:false
pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
hole_0':s:true:false1_0 :: 0':s:true:false
gen_0':s:true:false2_0 :: Nat -> 0':s:true:false


Lemmas:
plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b)) -> gen_0':s:true:false2_0(*(n983_0, b)), rt in Omega(1 + b*n983_0^2 + n983_0)


Generator Equations:
gen_0':s:true:false2_0(0) <=> 0'
gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x))


The following defined symbols remain to be analysed:
eq, div, quot, zero, divides, pr, if

They will be analysed ascendingly in the following order:
div = quot
div = zero
div = divides
div = pr
div = if
quot = zero
quot = divides
quot = pr
quot = if
eq < zero
zero = divides
zero = pr
zero = if
eq < divides
divides = pr
divides = if
pr = if

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s:true:false2_0(n2234_0), gen_0':s:true:false2_0(n2234_0)) -> true, rt in Omega(1 + n2234_0)

Induction Base:
eq(gen_0':s:true:false2_0(0), gen_0':s:true:false2_0(0)) ->_R^Omega(1)
true

Induction Step:
eq(gen_0':s:true:false2_0(+(n2234_0, 1)), gen_0':s:true:false2_0(+(n2234_0, 1))) ->_R^Omega(1)
eq(gen_0':s:true:false2_0(n2234_0), gen_0':s:true:false2_0(n2234_0)) ->_IH
true

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

(20)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(zero(y), s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(zero(y), z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)
zero(div(x, x)) -> x
zero(divides(x, x)) -> x
zero(times(x, x)) -> x
zero(quot(x, x, x)) -> x
zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

Types:
p :: 0':s:true:false -> 0':s:true:false
0' :: 0':s:true:false
s :: 0':s:true:false -> 0':s:true:false
plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
zero :: 0':s:true:false -> 0':s:true:false
eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
true :: 0':s:true:false
false :: 0':s:true:false
divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
prime :: 0':s:true:false -> 0':s:true:false
pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
hole_0':s:true:false1_0 :: 0':s:true:false
gen_0':s:true:false2_0 :: Nat -> 0':s:true:false


Lemmas:
plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b)) -> gen_0':s:true:false2_0(*(n983_0, b)), rt in Omega(1 + b*n983_0^2 + n983_0)
eq(gen_0':s:true:false2_0(n2234_0), gen_0':s:true:false2_0(n2234_0)) -> true, rt in Omega(1 + n2234_0)


Generator Equations:
gen_0':s:true:false2_0(0) <=> 0'
gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x))


The following defined symbols remain to be analysed:
quot, div, zero, divides, pr, if

They will be analysed ascendingly in the following order:
div = quot
div = zero
div = divides
div = pr
div = if
quot = zero
quot = divides
quot = pr
quot = if
zero = divides
zero = pr
zero = if
divides = pr
divides = if
pr = if

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(1)) -> gen_0':s:true:false2_0(1), rt in Omega(1 + n2831_0)

Induction Base:
quot(gen_0':s:true:false2_0(0), gen_0':s:true:false2_0(0), gen_0':s:true:false2_0(1)) ->_R^Omega(1)
s(div(gen_0':s:true:false2_0(0), s(gen_0':s:true:false2_0(0)))) ->_R^Omega(1)
s(0')

Induction Step:
quot(gen_0':s:true:false2_0(+(n2831_0, 1)), gen_0':s:true:false2_0(+(n2831_0, 1)), gen_0':s:true:false2_0(1)) ->_R^Omega(1)
quot(gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(1)) ->_IH
gen_0':s:true:false2_0(1)

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

(22)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> s(plus(p(s(x)), y))
plus(x, s(y)) -> s(plus(x, p(s(y))))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(zero(y), s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(zero(y), z))
eq(0', 0') -> true
eq(s(x), 0') -> false
eq(0', s(y)) -> false
eq(s(x), s(y)) -> eq(x, y)
divides(y, x) -> eq(x, times(div(x, y), y))
prime(s(s(x))) -> pr(s(s(x)), s(x))
pr(x, s(0')) -> true
pr(x, s(s(y))) -> if(divides(s(s(y)), x), x, s(y))
if(true, x, y) -> false
if(false, x, y) -> pr(x, y)
zero(div(x, x)) -> x
zero(divides(x, x)) -> x
zero(times(x, x)) -> x
zero(quot(x, x, x)) -> x
zero(s(x)) -> if(eq(x, s(0')), plus(zero(0'), 0'), s(plus(0', zero(0'))))

Types:
p :: 0':s:true:false -> 0':s:true:false
0' :: 0':s:true:false
s :: 0':s:true:false -> 0':s:true:false
plus :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
times :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
div :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
quot :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
zero :: 0':s:true:false -> 0':s:true:false
eq :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
true :: 0':s:true:false
false :: 0':s:true:false
divides :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
prime :: 0':s:true:false -> 0':s:true:false
pr :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false
if :: 0':s:true:false -> 0':s:true:false -> 0':s:true:false -> 0':s:true:false
hole_0':s:true:false1_0 :: 0':s:true:false
gen_0':s:true:false2_0 :: Nat -> 0':s:true:false


Lemmas:
plus(gen_0':s:true:false2_0(a), gen_0':s:true:false2_0(n4_0)) -> gen_0':s:true:false2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
times(gen_0':s:true:false2_0(n983_0), gen_0':s:true:false2_0(b)) -> gen_0':s:true:false2_0(*(n983_0, b)), rt in Omega(1 + b*n983_0^2 + n983_0)
eq(gen_0':s:true:false2_0(n2234_0), gen_0':s:true:false2_0(n2234_0)) -> true, rt in Omega(1 + n2234_0)
quot(gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(n2831_0), gen_0':s:true:false2_0(1)) -> gen_0':s:true:false2_0(1), rt in Omega(1 + n2831_0)


Generator Equations:
gen_0':s:true:false2_0(0) <=> 0'
gen_0':s:true:false2_0(+(x, 1)) <=> s(gen_0':s:true:false2_0(x))


The following defined symbols remain to be analysed:
div, zero, divides, pr, if

They will be analysed ascendingly in the following order:
div = quot
div = zero
div = divides
div = pr
div = if
quot = zero
quot = divides
quot = pr
quot = if
zero = divides
zero = pr
zero = if
divides = pr
divides = if
pr = if