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


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


WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, 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), 228 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), 52 ms]
        (14) BEST
            (15) proven lower bound
                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
                (17) BOUNDS(n^2, INF)
            (18) typed CpxTrs
                (19) RewriteLemmaProof [LOWER BOUND(ID), 54 ms]
                (20) typed CpxTrs


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

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


The TRS R consists of the following rules:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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^2, INF).


The TRS R consists of the following rules:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0') -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0'
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0') -> 0'
   shorter(nil, y) -> true
   shorter(cons(x, l), 0') -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
   if(true, b, l) -> s(0')
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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

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

(4)
Obligation:
TRS:
Rules:
car(cons(x, l)) -> x
cddr(nil) -> nil
cddr(cons(x, nil)) -> nil
cddr(cons(x, cons(y, l))) -> l
cadr(cons(x, cons(y, l))) -> y
isZero(0') -> true
isZero(s(x)) -> false
plus(x, y) -> ifplus(isZero(x), x, y)
ifplus(true, x, y) -> y
ifplus(false, x, y) -> s(plus(p(x), y))
times(x, y) -> iftimes(isZero(x), x, y)
iftimes(true, x, y) -> 0'
iftimes(false, x, y) -> plus(y, times(p(x), y))
p(s(x)) -> x
p(0') -> 0'
shorter(nil, y) -> true
shorter(cons(x, l), 0') -> false
shorter(cons(x, l), s(y)) -> shorter(l, y)
prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) -> s(0')
if(false, b, l) -> if2(b, l)
if2(true, l) -> car(l)
if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0':s
isZero :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
plus :: 0':s -> 0':s -> 0':s
ifplus :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
iftimes :: true:false -> 0':s -> 0':s -> 0':s
shorter :: cons:nil -> 0':s -> true:false
prod :: cons:nil -> 0':s
if :: true:false -> true:false -> cons:nil -> 0':s
if2 :: true:false -> cons:nil -> 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, times, shorter, prod

They will be analysed ascendingly in the following order:
plus < times
times < prod
shorter < prod

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

(6)
Obligation:
TRS:
Rules:
car(cons(x, l)) -> x
cddr(nil) -> nil
cddr(cons(x, nil)) -> nil
cddr(cons(x, cons(y, l))) -> l
cadr(cons(x, cons(y, l))) -> y
isZero(0') -> true
isZero(s(x)) -> false
plus(x, y) -> ifplus(isZero(x), x, y)
ifplus(true, x, y) -> y
ifplus(false, x, y) -> s(plus(p(x), y))
times(x, y) -> iftimes(isZero(x), x, y)
iftimes(true, x, y) -> 0'
iftimes(false, x, y) -> plus(y, times(p(x), y))
p(s(x)) -> x
p(0') -> 0'
shorter(nil, y) -> true
shorter(cons(x, l), 0') -> false
shorter(cons(x, l), s(y)) -> shorter(l, y)
prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) -> s(0')
if(false, b, l) -> if2(b, l)
if2(true, l) -> car(l)
if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0':s
isZero :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
plus :: 0':s -> 0':s -> 0':s
ifplus :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
iftimes :: true:false -> 0':s -> 0':s -> 0':s
shorter :: cons:nil -> 0':s -> true:false
prod :: cons:nil -> 0':s
if :: true:false -> true:false -> cons:nil -> 0':s
if2 :: true:false -> cons:nil -> 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
plus, times, shorter, prod

They will be analysed ascendingly in the following order:
plus < times
times < prod
shorter < prod

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n7_0, b)), rt in Omega(1 + n7_0)

Induction Base:
plus(gen_0':s4_0(0), gen_0':s4_0(b)) ->_R^Omega(1)
ifplus(isZero(gen_0':s4_0(0)), gen_0':s4_0(0), gen_0':s4_0(b)) ->_R^Omega(1)
ifplus(true, gen_0':s4_0(0), gen_0':s4_0(b)) ->_R^Omega(1)
gen_0':s4_0(b)

Induction Step:
plus(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
ifplus(isZero(gen_0':s4_0(+(n7_0, 1))), gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
ifplus(false, gen_0':s4_0(+(1, n7_0)), gen_0':s4_0(b)) ->_R^Omega(1)
s(plus(p(gen_0':s4_0(+(1, n7_0))), gen_0':s4_0(b))) ->_R^Omega(1)
s(plus(gen_0':s4_0(n7_0), gen_0':s4_0(b))) ->_IH
s(gen_0':s4_0(+(b, c8_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:
car(cons(x, l)) -> x
cddr(nil) -> nil
cddr(cons(x, nil)) -> nil
cddr(cons(x, cons(y, l))) -> l
cadr(cons(x, cons(y, l))) -> y
isZero(0') -> true
isZero(s(x)) -> false
plus(x, y) -> ifplus(isZero(x), x, y)
ifplus(true, x, y) -> y
ifplus(false, x, y) -> s(plus(p(x), y))
times(x, y) -> iftimes(isZero(x), x, y)
iftimes(true, x, y) -> 0'
iftimes(false, x, y) -> plus(y, times(p(x), y))
p(s(x)) -> x
p(0') -> 0'
shorter(nil, y) -> true
shorter(cons(x, l), 0') -> false
shorter(cons(x, l), s(y)) -> shorter(l, y)
prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) -> s(0')
if(false, b, l) -> if2(b, l)
if2(true, l) -> car(l)
if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0':s
isZero :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
plus :: 0':s -> 0':s -> 0':s
ifplus :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
iftimes :: true:false -> 0':s -> 0':s -> 0':s
shorter :: cons:nil -> 0':s -> true:false
prod :: cons:nil -> 0':s
if :: true:false -> true:false -> cons:nil -> 0':s
if2 :: true:false -> cons:nil -> 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
plus, times, shorter, prod

They will be analysed ascendingly in the following order:
plus < times
times < prod
shorter < prod

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
car(cons(x, l)) -> x
cddr(nil) -> nil
cddr(cons(x, nil)) -> nil
cddr(cons(x, cons(y, l))) -> l
cadr(cons(x, cons(y, l))) -> y
isZero(0') -> true
isZero(s(x)) -> false
plus(x, y) -> ifplus(isZero(x), x, y)
ifplus(true, x, y) -> y
ifplus(false, x, y) -> s(plus(p(x), y))
times(x, y) -> iftimes(isZero(x), x, y)
iftimes(true, x, y) -> 0'
iftimes(false, x, y) -> plus(y, times(p(x), y))
p(s(x)) -> x
p(0') -> 0'
shorter(nil, y) -> true
shorter(cons(x, l), 0') -> false
shorter(cons(x, l), s(y)) -> shorter(l, y)
prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) -> s(0')
if(false, b, l) -> if2(b, l)
if2(true, l) -> car(l)
if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0':s
isZero :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
plus :: 0':s -> 0':s -> 0':s
ifplus :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
iftimes :: true:false -> 0':s -> 0':s -> 0':s
shorter :: cons:nil -> 0':s -> true:false
prod :: cons:nil -> 0':s
if :: true:false -> true:false -> cons:nil -> 0':s
if2 :: true:false -> cons:nil -> 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n7_0, b)), rt in Omega(1 + n7_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
times, shorter, prod

They will be analysed ascendingly in the following order:
times < prod
shorter < prod

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s4_0(n1253_0), gen_0':s4_0(b)) -> gen_0':s4_0(*(n1253_0, b)), rt in Omega(1 + b*n1253_0 + n1253_0)

Induction Base:
times(gen_0':s4_0(0), gen_0':s4_0(b)) ->_R^Omega(1)
iftimes(isZero(gen_0':s4_0(0)), gen_0':s4_0(0), gen_0':s4_0(b)) ->_R^Omega(1)
iftimes(true, gen_0':s4_0(0), gen_0':s4_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_0':s4_0(+(n1253_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
iftimes(isZero(gen_0':s4_0(+(n1253_0, 1))), gen_0':s4_0(+(n1253_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
iftimes(false, gen_0':s4_0(+(1, n1253_0)), gen_0':s4_0(b)) ->_R^Omega(1)
plus(gen_0':s4_0(b), times(p(gen_0':s4_0(+(1, n1253_0))), gen_0':s4_0(b))) ->_R^Omega(1)
plus(gen_0':s4_0(b), times(gen_0':s4_0(n1253_0), gen_0':s4_0(b))) ->_IH
plus(gen_0':s4_0(b), gen_0':s4_0(*(c1254_0, b))) ->_L^Omega(1 + b)
gen_0':s4_0(+(b, *(n1253_0, b)))

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
car(cons(x, l)) -> x
cddr(nil) -> nil
cddr(cons(x, nil)) -> nil
cddr(cons(x, cons(y, l))) -> l
cadr(cons(x, cons(y, l))) -> y
isZero(0') -> true
isZero(s(x)) -> false
plus(x, y) -> ifplus(isZero(x), x, y)
ifplus(true, x, y) -> y
ifplus(false, x, y) -> s(plus(p(x), y))
times(x, y) -> iftimes(isZero(x), x, y)
iftimes(true, x, y) -> 0'
iftimes(false, x, y) -> plus(y, times(p(x), y))
p(s(x)) -> x
p(0') -> 0'
shorter(nil, y) -> true
shorter(cons(x, l), 0') -> false
shorter(cons(x, l), s(y)) -> shorter(l, y)
prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) -> s(0')
if(false, b, l) -> if2(b, l)
if2(true, l) -> car(l)
if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0':s
isZero :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
plus :: 0':s -> 0':s -> 0':s
ifplus :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
iftimes :: true:false -> 0':s -> 0':s -> 0':s
shorter :: cons:nil -> 0':s -> true:false
prod :: cons:nil -> 0':s
if :: true:false -> true:false -> cons:nil -> 0':s
if2 :: true:false -> cons:nil -> 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n7_0, b)), rt in Omega(1 + n7_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
times, shorter, prod

They will be analysed ascendingly in the following order:
times < prod
shorter < prod

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
Obligation:
TRS:
Rules:
car(cons(x, l)) -> x
cddr(nil) -> nil
cddr(cons(x, nil)) -> nil
cddr(cons(x, cons(y, l))) -> l
cadr(cons(x, cons(y, l))) -> y
isZero(0') -> true
isZero(s(x)) -> false
plus(x, y) -> ifplus(isZero(x), x, y)
ifplus(true, x, y) -> y
ifplus(false, x, y) -> s(plus(p(x), y))
times(x, y) -> iftimes(isZero(x), x, y)
iftimes(true, x, y) -> 0'
iftimes(false, x, y) -> plus(y, times(p(x), y))
p(s(x)) -> x
p(0') -> 0'
shorter(nil, y) -> true
shorter(cons(x, l), 0') -> false
shorter(cons(x, l), s(y)) -> shorter(l, y)
prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) -> s(0')
if(false, b, l) -> if2(b, l)
if2(true, l) -> car(l)
if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0':s
isZero :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
plus :: 0':s -> 0':s -> 0':s
ifplus :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
iftimes :: true:false -> 0':s -> 0':s -> 0':s
shorter :: cons:nil -> 0':s -> true:false
prod :: cons:nil -> 0':s
if :: true:false -> true:false -> cons:nil -> 0':s
if2 :: true:false -> cons:nil -> 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n7_0, b)), rt in Omega(1 + n7_0)
times(gen_0':s4_0(n1253_0), gen_0':s4_0(b)) -> gen_0':s4_0(*(n1253_0, b)), rt in Omega(1 + b*n1253_0 + n1253_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
shorter, prod

They will be analysed ascendingly in the following order:
shorter < prod

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shorter(gen_cons:nil5_0(n3479_0), gen_0':s4_0(n3479_0)) -> true, rt in Omega(1 + n3479_0)

Induction Base:
shorter(gen_cons:nil5_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
true

Induction Step:
shorter(gen_cons:nil5_0(+(n3479_0, 1)), gen_0':s4_0(+(n3479_0, 1))) ->_R^Omega(1)
shorter(gen_cons:nil5_0(n3479_0), gen_0':s4_0(n3479_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(20)
Obligation:
TRS:
Rules:
car(cons(x, l)) -> x
cddr(nil) -> nil
cddr(cons(x, nil)) -> nil
cddr(cons(x, cons(y, l))) -> l
cadr(cons(x, cons(y, l))) -> y
isZero(0') -> true
isZero(s(x)) -> false
plus(x, y) -> ifplus(isZero(x), x, y)
ifplus(true, x, y) -> y
ifplus(false, x, y) -> s(plus(p(x), y))
times(x, y) -> iftimes(isZero(x), x, y)
iftimes(true, x, y) -> 0'
iftimes(false, x, y) -> plus(y, times(p(x), y))
p(s(x)) -> x
p(0') -> 0'
shorter(nil, y) -> true
shorter(cons(x, l), 0') -> false
shorter(cons(x, l), s(y)) -> shorter(l, y)
prod(l) -> if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) -> s(0')
if(false, b, l) -> if2(b, l)
if2(true, l) -> car(l)
if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
cddr :: cons:nil -> cons:nil
nil :: cons:nil
cadr :: cons:nil -> 0':s
isZero :: 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
plus :: 0':s -> 0':s -> 0':s
ifplus :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
iftimes :: true:false -> 0':s -> 0':s -> 0':s
shorter :: cons:nil -> 0':s -> true:false
prod :: cons:nil -> 0':s
if :: true:false -> true:false -> cons:nil -> 0':s
if2 :: true:false -> cons:nil -> 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_cons:nil5_0 :: Nat -> cons:nil


Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n7_0, b)), rt in Omega(1 + n7_0)
times(gen_0':s4_0(n1253_0), gen_0':s4_0(b)) -> gen_0':s4_0(*(n1253_0, b)), rt in Omega(1 + b*n1253_0 + n1253_0)
shorter(gen_cons:nil5_0(n3479_0), gen_0':s4_0(n3479_0)) -> true, rt in Omega(1 + n3479_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_cons:nil5_0(0) <=> nil
gen_cons:nil5_0(+(x, 1)) <=> cons(0', gen_cons:nil5_0(x))


The following defined symbols remain to be analysed:
prod