WORST_CASE(Omega(n^1),?)
proof of input_ch7pWjOWkQ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 1 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 95.0 s]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) 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:

   empty(nil) -> true
   empty(cons(x, y)) -> false
   tail(nil) -> nil
   tail(cons(x, y)) -> y
   head(cons(x, y)) -> x
   zero(0) -> true
   zero(s(x)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   intlist(x) -> if_intlist(empty(x), x)
   if_intlist(true, x) -> nil
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
   int(x, y) -> if_int(zero(x), zero(y), x, y)
   if_int(true, b, x, y) -> if1(b, x, y)
   if_int(false, b, x, y) -> if2(b, x, y)
   if1(true, x, y) -> cons(0, nil)
   if1(false, x, y) -> cons(0, int(s(0), y))
   if2(true, x, y) -> nil
   if2(false, x, y) -> intlist(int(p(x), p(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:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))
Tuples:
   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
S tuples:
   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
K tuples:none
Defined Rule Symbols:   empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3

Defined Pair Symbols:   EMPTY_1, TAIL_1, HEAD_1, ZERO_1, P_1, INTLIST_1, IF_INTLIST_2, INT_2, IF_INT_4, IF1_3, IF2_3

Compound Symbols:   c, c1, c2, c3, c4, c5, c6, c7, c8, c9_1, c10_2, c11, c12_1, c13_2, c14_2, c15_2, c16_1, c17_1, c18, c19_1, c20, c21_3, c22_3


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

(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:

   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))

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

   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(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:

   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0') -> c5
   ZERO(s(z0)) -> c6
   P(0') -> c7
   P(s(0')) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0'), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))

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

   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0') -> true
   zero(s(z0)) -> false
   p(0') -> 0'
   p(s(0')) -> 0'
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0', nil)
   if1(false, z0, z1) -> cons(0', int(s(0'), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
EMPTY(nil) -> c
EMPTY(cons(z0, z1)) -> c1
TAIL(nil) -> c2
TAIL(cons(z0, z1)) -> c3
HEAD(cons(z0, z1)) -> c4
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(0')) -> c8
P(s(s(z0))) -> c9(P(s(z0)))
INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
IF_INTLIST(true, z0) -> c11
IF_INTLIST(false, z0) -> c12(HEAD(z0))
IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
IF1(true, z0, z1) -> c18
IF1(false, z0, z1) -> c19(INT(s(0'), z1))
IF2(true, z0, z1) -> c20
IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
empty(nil) -> true
empty(cons(z0, z1)) -> false
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
head(cons(z0, z1)) -> z0
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(z0))) -> s(p(s(z0)))
intlist(z0) -> if_intlist(empty(z0), z0)
if_intlist(true, z0) -> nil
if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
if1(true, z0, z1) -> cons(0', nil)
if1(false, z0, z1) -> cons(0', int(s(0'), z1))
if2(true, z0, z1) -> nil
if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Types:
EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c:c1
TAIL :: nil:cons -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:cons -> c4
c4 :: c4
ZERO :: 0':s -> c5:c6
0' :: 0':s
c5 :: c5:c6
s :: 0':s -> 0':s
c6 :: c5:c6
P :: 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
INTLIST :: nil:cons -> c10
c10 :: c11:c12:c13 -> c:c1 -> c10
IF_INTLIST :: true:false -> nil:cons -> c11:c12:c13
empty :: nil:cons -> true:false
true :: true:false
c11 :: c11:c12:c13
false :: true:false
c12 :: c4 -> c11:c12:c13
c13 :: c10 -> c2:c3 -> c11:c12:c13
tail :: nil:cons -> nil:cons
INT :: 0':s -> 0':s -> c14:c15
c14 :: c16:c17 -> c5:c6 -> c14:c15
IF_INT :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
zero :: 0':s -> true:false
c15 :: c16:c17 -> c5:c6 -> c14:c15
c16 :: c18:c19 -> c16:c17
IF1 :: true:false -> 0':s -> 0':s -> c18:c19
c17 :: c20:c21:c22 -> c16:c17
IF2 :: true:false -> 0':s -> 0':s -> c20:c21:c22
c18 :: c18:c19
c19 :: c14:c15 -> c18:c19
c20 :: c20:c21:c22
c21 :: c10 -> c14:c15 -> c7:c8:c9 -> c20:c21:c22
int :: 0':s -> 0':s -> nil:cons
p :: 0':s -> 0':s
c22 :: c10 -> c14:c15 -> c7:c8:c9 -> c20:c21:c22
head :: nil:cons -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_c:c11_23 :: c:c1
hole_nil:cons2_23 :: nil:cons
hole_0':s3_23 :: 0':s
hole_c2:c34_23 :: c2:c3
hole_c45_23 :: c4
hole_c5:c66_23 :: c5:c6
hole_c7:c8:c97_23 :: c7:c8:c9
hole_c108_23 :: c10
hole_c11:c12:c139_23 :: c11:c12:c13
hole_true:false10_23 :: true:false
hole_c14:c1511_23 :: c14:c15
hole_c16:c1712_23 :: c16:c17
hole_c18:c1913_23 :: c18:c19
hole_c20:c21:c2214_23 :: c20:c21:c22
gen_nil:cons15_23 :: Nat -> nil:cons
gen_0':s16_23 :: Nat -> 0':s
gen_c7:c8:c917_23 :: Nat -> c7:c8:c9

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
P, INTLIST, INT, int, p, intlist

They will be analysed ascendingly in the following order:
P < INT
INTLIST < INT
int < INT
p < INT
p < int
intlist < int

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

(10)
Obligation:
Innermost TRS:
Rules:
EMPTY(nil) -> c
EMPTY(cons(z0, z1)) -> c1
TAIL(nil) -> c2
TAIL(cons(z0, z1)) -> c3
HEAD(cons(z0, z1)) -> c4
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(0')) -> c8
P(s(s(z0))) -> c9(P(s(z0)))
INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
IF_INTLIST(true, z0) -> c11
IF_INTLIST(false, z0) -> c12(HEAD(z0))
IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
IF1(true, z0, z1) -> c18
IF1(false, z0, z1) -> c19(INT(s(0'), z1))
IF2(true, z0, z1) -> c20
IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
empty(nil) -> true
empty(cons(z0, z1)) -> false
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
head(cons(z0, z1)) -> z0
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(z0))) -> s(p(s(z0)))
intlist(z0) -> if_intlist(empty(z0), z0)
if_intlist(true, z0) -> nil
if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
if1(true, z0, z1) -> cons(0', nil)
if1(false, z0, z1) -> cons(0', int(s(0'), z1))
if2(true, z0, z1) -> nil
if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Types:
EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c:c1
TAIL :: nil:cons -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:cons -> c4
c4 :: c4
ZERO :: 0':s -> c5:c6
0' :: 0':s
c5 :: c5:c6
s :: 0':s -> 0':s
c6 :: c5:c6
P :: 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
INTLIST :: nil:cons -> c10
c10 :: c11:c12:c13 -> c:c1 -> c10
IF_INTLIST :: true:false -> nil:cons -> c11:c12:c13
empty :: nil:cons -> true:false
true :: true:false
c11 :: c11:c12:c13
false :: true:false
c12 :: c4 -> c11:c12:c13
c13 :: c10 -> c2:c3 -> c11:c12:c13
tail :: nil:cons -> nil:cons
INT :: 0':s -> 0':s -> c14:c15
c14 :: c16:c17 -> c5:c6 -> c14:c15
IF_INT :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
zero :: 0':s -> true:false
c15 :: c16:c17 -> c5:c6 -> c14:c15
c16 :: c18:c19 -> c16:c17
IF1 :: true:false -> 0':s -> 0':s -> c18:c19
c17 :: c20:c21:c22 -> c16:c17
IF2 :: true:false -> 0':s -> 0':s -> c20:c21:c22
c18 :: c18:c19
c19 :: c14:c15 -> c18:c19
c20 :: c20:c21:c22
c21 :: c10 -> c14:c15 -> c7:c8:c9 -> c20:c21:c22
int :: 0':s -> 0':s -> nil:cons
p :: 0':s -> 0':s
c22 :: c10 -> c14:c15 -> c7:c8:c9 -> c20:c21:c22
head :: nil:cons -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_c:c11_23 :: c:c1
hole_nil:cons2_23 :: nil:cons
hole_0':s3_23 :: 0':s
hole_c2:c34_23 :: c2:c3
hole_c45_23 :: c4
hole_c5:c66_23 :: c5:c6
hole_c7:c8:c97_23 :: c7:c8:c9
hole_c108_23 :: c10
hole_c11:c12:c139_23 :: c11:c12:c13
hole_true:false10_23 :: true:false
hole_c14:c1511_23 :: c14:c15
hole_c16:c1712_23 :: c16:c17
hole_c18:c1913_23 :: c18:c19
hole_c20:c21:c2214_23 :: c20:c21:c22
gen_nil:cons15_23 :: Nat -> nil:cons
gen_0':s16_23 :: Nat -> 0':s
gen_c7:c8:c917_23 :: Nat -> c7:c8:c9


Generator Equations:
gen_nil:cons15_23(0) <=> nil
gen_nil:cons15_23(+(x, 1)) <=> cons(0', gen_nil:cons15_23(x))
gen_0':s16_23(0) <=> 0'
gen_0':s16_23(+(x, 1)) <=> s(gen_0':s16_23(x))
gen_c7:c8:c917_23(0) <=> c7
gen_c7:c8:c917_23(+(x, 1)) <=> c9(gen_c7:c8:c917_23(x))


The following defined symbols remain to be analysed:
P, INTLIST, INT, int, p, intlist

They will be analysed ascendingly in the following order:
P < INT
INTLIST < INT
int < INT
p < INT
p < int
intlist < int

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
P(gen_0':s16_23(+(2, n19_23))) -> *18_23, rt in Omega(n19_23)

Induction Base:
P(gen_0':s16_23(+(2, 0)))

Induction Step:
P(gen_0':s16_23(+(2, +(n19_23, 1)))) ->_R^Omega(1)
c9(P(s(gen_0':s16_23(+(1, n19_23))))) ->_IH
c9(*18_23)

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:
EMPTY(nil) -> c
EMPTY(cons(z0, z1)) -> c1
TAIL(nil) -> c2
TAIL(cons(z0, z1)) -> c3
HEAD(cons(z0, z1)) -> c4
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(0')) -> c8
P(s(s(z0))) -> c9(P(s(z0)))
INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
IF_INTLIST(true, z0) -> c11
IF_INTLIST(false, z0) -> c12(HEAD(z0))
IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
IF1(true, z0, z1) -> c18
IF1(false, z0, z1) -> c19(INT(s(0'), z1))
IF2(true, z0, z1) -> c20
IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
empty(nil) -> true
empty(cons(z0, z1)) -> false
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
head(cons(z0, z1)) -> z0
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(z0))) -> s(p(s(z0)))
intlist(z0) -> if_intlist(empty(z0), z0)
if_intlist(true, z0) -> nil
if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
if1(true, z0, z1) -> cons(0', nil)
if1(false, z0, z1) -> cons(0', int(s(0'), z1))
if2(true, z0, z1) -> nil
if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Types:
EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c:c1
TAIL :: nil:cons -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:cons -> c4
c4 :: c4
ZERO :: 0':s -> c5:c6
0' :: 0':s
c5 :: c5:c6
s :: 0':s -> 0':s
c6 :: c5:c6
P :: 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
INTLIST :: nil:cons -> c10
c10 :: c11:c12:c13 -> c:c1 -> c10
IF_INTLIST :: true:false -> nil:cons -> c11:c12:c13
empty :: nil:cons -> true:false
true :: true:false
c11 :: c11:c12:c13
false :: true:false
c12 :: c4 -> c11:c12:c13
c13 :: c10 -> c2:c3 -> c11:c12:c13
tail :: nil:cons -> nil:cons
INT :: 0':s -> 0':s -> c14:c15
c14 :: c16:c17 -> c5:c6 -> c14:c15
IF_INT :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
zero :: 0':s -> true:false
c15 :: c16:c17 -> c5:c6 -> c14:c15
c16 :: c18:c19 -> c16:c17
IF1 :: true:false -> 0':s -> 0':s -> c18:c19
c17 :: c20:c21:c22 -> c16:c17
IF2 :: true:false -> 0':s -> 0':s -> c20:c21:c22
c18 :: c18:c19
c19 :: c14:c15 -> c18:c19
c20 :: c20:c21:c22
c21 :: c10 -> c14:c15 -> c7:c8:c9 -> c20:c21:c22
int :: 0':s -> 0':s -> nil:cons
p :: 0':s -> 0':s
c22 :: c10 -> c14:c15 -> c7:c8:c9 -> c20:c21:c22
head :: nil:cons -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_c:c11_23 :: c:c1
hole_nil:cons2_23 :: nil:cons
hole_0':s3_23 :: 0':s
hole_c2:c34_23 :: c2:c3
hole_c45_23 :: c4
hole_c5:c66_23 :: c5:c6
hole_c7:c8:c97_23 :: c7:c8:c9
hole_c108_23 :: c10
hole_c11:c12:c139_23 :: c11:c12:c13
hole_true:false10_23 :: true:false
hole_c14:c1511_23 :: c14:c15
hole_c16:c1712_23 :: c16:c17
hole_c18:c1913_23 :: c18:c19
hole_c20:c21:c2214_23 :: c20:c21:c22
gen_nil:cons15_23 :: Nat -> nil:cons
gen_0':s16_23 :: Nat -> 0':s
gen_c7:c8:c917_23 :: Nat -> c7:c8:c9


Generator Equations:
gen_nil:cons15_23(0) <=> nil
gen_nil:cons15_23(+(x, 1)) <=> cons(0', gen_nil:cons15_23(x))
gen_0':s16_23(0) <=> 0'
gen_0':s16_23(+(x, 1)) <=> s(gen_0':s16_23(x))
gen_c7:c8:c917_23(0) <=> c7
gen_c7:c8:c917_23(+(x, 1)) <=> c9(gen_c7:c8:c917_23(x))


The following defined symbols remain to be analysed:
P, INTLIST, INT, int, p, intlist

They will be analysed ascendingly in the following order:
P < INT
INTLIST < INT
int < INT
p < INT
p < int
intlist < int

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
EMPTY(nil) -> c
EMPTY(cons(z0, z1)) -> c1
TAIL(nil) -> c2
TAIL(cons(z0, z1)) -> c3
HEAD(cons(z0, z1)) -> c4
ZERO(0') -> c5
ZERO(s(z0)) -> c6
P(0') -> c7
P(s(0')) -> c8
P(s(s(z0))) -> c9(P(s(z0)))
INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
IF_INTLIST(true, z0) -> c11
IF_INTLIST(false, z0) -> c12(HEAD(z0))
IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
IF1(true, z0, z1) -> c18
IF1(false, z0, z1) -> c19(INT(s(0'), z1))
IF2(true, z0, z1) -> c20
IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))
empty(nil) -> true
empty(cons(z0, z1)) -> false
tail(nil) -> nil
tail(cons(z0, z1)) -> z1
head(cons(z0, z1)) -> z0
zero(0') -> true
zero(s(z0)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(z0))) -> s(p(s(z0)))
intlist(z0) -> if_intlist(empty(z0), z0)
if_intlist(true, z0) -> nil
if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
if1(true, z0, z1) -> cons(0', nil)
if1(false, z0, z1) -> cons(0', int(s(0'), z1))
if2(true, z0, z1) -> nil
if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Types:
EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: 0':s -> nil:cons -> nil:cons
c1 :: c:c1
TAIL :: nil:cons -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:cons -> c4
c4 :: c4
ZERO :: 0':s -> c5:c6
0' :: 0':s
c5 :: c5:c6
s :: 0':s -> 0':s
c6 :: c5:c6
P :: 0':s -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
INTLIST :: nil:cons -> c10
c10 :: c11:c12:c13 -> c:c1 -> c10
IF_INTLIST :: true:false -> nil:cons -> c11:c12:c13
empty :: nil:cons -> true:false
true :: true:false
c11 :: c11:c12:c13
false :: true:false
c12 :: c4 -> c11:c12:c13
c13 :: c10 -> c2:c3 -> c11:c12:c13
tail :: nil:cons -> nil:cons
INT :: 0':s -> 0':s -> c14:c15
c14 :: c16:c17 -> c5:c6 -> c14:c15
IF_INT :: true:false -> true:false -> 0':s -> 0':s -> c16:c17
zero :: 0':s -> true:false
c15 :: c16:c17 -> c5:c6 -> c14:c15
c16 :: c18:c19 -> c16:c17
IF1 :: true:false -> 0':s -> 0':s -> c18:c19
c17 :: c20:c21:c22 -> c16:c17
IF2 :: true:false -> 0':s -> 0':s -> c20:c21:c22
c18 :: c18:c19
c19 :: c14:c15 -> c18:c19
c20 :: c20:c21:c22
c21 :: c10 -> c14:c15 -> c7:c8:c9 -> c20:c21:c22
int :: 0':s -> 0':s -> nil:cons
p :: 0':s -> 0':s
c22 :: c10 -> c14:c15 -> c7:c8:c9 -> c20:c21:c22
head :: nil:cons -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_c:c11_23 :: c:c1
hole_nil:cons2_23 :: nil:cons
hole_0':s3_23 :: 0':s
hole_c2:c34_23 :: c2:c3
hole_c45_23 :: c4
hole_c5:c66_23 :: c5:c6
hole_c7:c8:c97_23 :: c7:c8:c9
hole_c108_23 :: c10
hole_c11:c12:c139_23 :: c11:c12:c13
hole_true:false10_23 :: true:false
hole_c14:c1511_23 :: c14:c15
hole_c16:c1712_23 :: c16:c17
hole_c18:c1913_23 :: c18:c19
hole_c20:c21:c2214_23 :: c20:c21:c22
gen_nil:cons15_23 :: Nat -> nil:cons
gen_0':s16_23 :: Nat -> 0':s
gen_c7:c8:c917_23 :: Nat -> c7:c8:c9


Lemmas:
P(gen_0':s16_23(+(2, n19_23))) -> *18_23, rt in Omega(n19_23)


Generator Equations:
gen_nil:cons15_23(0) <=> nil
gen_nil:cons15_23(+(x, 1)) <=> cons(0', gen_nil:cons15_23(x))
gen_0':s16_23(0) <=> 0'
gen_0':s16_23(+(x, 1)) <=> s(gen_0':s16_23(x))
gen_c7:c8:c917_23(0) <=> c7
gen_c7:c8:c917_23(+(x, 1)) <=> c9(gen_c7:c8:c917_23(x))


The following defined symbols remain to be analysed:
INTLIST, INT, int, p, intlist

They will be analysed ascendingly in the following order:
INTLIST < INT
int < INT
p < INT
p < int
intlist < int