WORST_CASE(Omega(n^1),?)
proof of input_SLdA2ewG9M.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 16 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 375 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), 123 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 2214 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 419 ms]
        (22) BOUNDS(1, INF)


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

(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:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

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:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)
Tuples:
   FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)))
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(0, z0) -> c2
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(0, cons(z0, z1)) -> c4
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
S tuples:
   FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)))
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(0, z0) -> c2
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(0, cons(z0, z1)) -> c4
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
K tuples:none
Defined Rule Symbols:   fib_1, fib1_2, add_2, sel_2

Defined Pair Symbols:   FIB_1, FIB1_2, ADD_2, SEL_2

Compound Symbols:   c_2, c1_2, c2, c3_1, c4, c5_1


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

(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:

   FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)))
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(0, z0) -> c2
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(0, cons(z0, z1)) -> c4
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))

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

   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

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:

   FIB(z0) -> c(SEL(z0, fib1(s(0'), s(0'))), FIB1(s(0'), s(0')))
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(0', z0) -> c2
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(0', cons(z0, z1)) -> c4
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))

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

   fib(z0) -> sel(z0, fib1(s(0'), s(0')))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
   add(0', z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   sel(0', cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

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

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

(8)
Obligation:
Innermost TRS:
Rules:
FIB(z0) -> c(SEL(z0, fib1(s(0'), s(0'))), FIB1(s(0'), s(0')))
FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
ADD(0', z0) -> c2
ADD(s(z0), z1) -> c3(ADD(z0, z1))
SEL(0', cons(z0, z1)) -> c4
SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
fib(z0) -> sel(z0, fib1(s(0'), s(0')))
fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FIB :: 0':s -> c
c :: c4:c5 -> c1 -> c
SEL :: 0':s -> cons -> c4:c5
fib1 :: 0':s -> 0':s -> cons
s :: 0':s -> 0':s
0' :: 0':s
FIB1 :: 0':s -> 0':s -> c1
c1 :: c1 -> c2:c3 -> c1
add :: 0':s -> 0':s -> 0':s
ADD :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
cons :: 0':s -> cons -> cons
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
fib :: 0':s -> 0':s
sel :: 0':s -> cons -> 0':s
hole_c1_6 :: c
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_c14_6 :: c1
hole_cons5_6 :: cons
hole_c2:c36_6 :: c2:c3
gen_0':s7_6 :: Nat -> 0':s
gen_c4:c58_6 :: Nat -> c4:c5
gen_c19_6 :: Nat -> c1
gen_cons10_6 :: Nat -> cons
gen_c2:c311_6 :: Nat -> c2:c3

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SEL, fib1, FIB1, add, ADD, sel

They will be analysed ascendingly in the following order:
add < fib1
add < FIB1
ADD < FIB1

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

(10)
Obligation:
Innermost TRS:
Rules:
FIB(z0) -> c(SEL(z0, fib1(s(0'), s(0'))), FIB1(s(0'), s(0')))
FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
ADD(0', z0) -> c2
ADD(s(z0), z1) -> c3(ADD(z0, z1))
SEL(0', cons(z0, z1)) -> c4
SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
fib(z0) -> sel(z0, fib1(s(0'), s(0')))
fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FIB :: 0':s -> c
c :: c4:c5 -> c1 -> c
SEL :: 0':s -> cons -> c4:c5
fib1 :: 0':s -> 0':s -> cons
s :: 0':s -> 0':s
0' :: 0':s
FIB1 :: 0':s -> 0':s -> c1
c1 :: c1 -> c2:c3 -> c1
add :: 0':s -> 0':s -> 0':s
ADD :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
cons :: 0':s -> cons -> cons
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
fib :: 0':s -> 0':s
sel :: 0':s -> cons -> 0':s
hole_c1_6 :: c
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_c14_6 :: c1
hole_cons5_6 :: cons
hole_c2:c36_6 :: c2:c3
gen_0':s7_6 :: Nat -> 0':s
gen_c4:c58_6 :: Nat -> c4:c5
gen_c19_6 :: Nat -> c1
gen_cons10_6 :: Nat -> cons
gen_c2:c311_6 :: Nat -> c2:c3


Generator Equations:
gen_0':s7_6(0) <=> 0'
gen_0':s7_6(+(x, 1)) <=> s(gen_0':s7_6(x))
gen_c4:c58_6(0) <=> c4
gen_c4:c58_6(+(x, 1)) <=> c5(gen_c4:c58_6(x))
gen_c19_6(0) <=> hole_c14_6
gen_c19_6(+(x, 1)) <=> c1(gen_c19_6(x), c2)
gen_cons10_6(0) <=> hole_cons5_6
gen_cons10_6(+(x, 1)) <=> cons(0', gen_cons10_6(x))
gen_c2:c311_6(0) <=> c2
gen_c2:c311_6(+(x, 1)) <=> c3(gen_c2:c311_6(x))


The following defined symbols remain to be analysed:
SEL, fib1, FIB1, add, ADD, sel

They will be analysed ascendingly in the following order:
add < fib1
add < FIB1
ADD < FIB1

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SEL(gen_0':s7_6(n13_6), gen_cons10_6(+(1, n13_6))) -> gen_c4:c58_6(n13_6), rt in Omega(1 + n13_6)

Induction Base:
SEL(gen_0':s7_6(0), gen_cons10_6(+(1, 0))) ->_R^Omega(1)
c4

Induction Step:
SEL(gen_0':s7_6(+(n13_6, 1)), gen_cons10_6(+(1, +(n13_6, 1)))) ->_R^Omega(1)
c5(SEL(gen_0':s7_6(n13_6), gen_cons10_6(+(1, n13_6)))) ->_IH
c5(gen_c4:c58_6(c14_6))

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

(12)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
FIB(z0) -> c(SEL(z0, fib1(s(0'), s(0'))), FIB1(s(0'), s(0')))
FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
ADD(0', z0) -> c2
ADD(s(z0), z1) -> c3(ADD(z0, z1))
SEL(0', cons(z0, z1)) -> c4
SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
fib(z0) -> sel(z0, fib1(s(0'), s(0')))
fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FIB :: 0':s -> c
c :: c4:c5 -> c1 -> c
SEL :: 0':s -> cons -> c4:c5
fib1 :: 0':s -> 0':s -> cons
s :: 0':s -> 0':s
0' :: 0':s
FIB1 :: 0':s -> 0':s -> c1
c1 :: c1 -> c2:c3 -> c1
add :: 0':s -> 0':s -> 0':s
ADD :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
cons :: 0':s -> cons -> cons
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
fib :: 0':s -> 0':s
sel :: 0':s -> cons -> 0':s
hole_c1_6 :: c
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_c14_6 :: c1
hole_cons5_6 :: cons
hole_c2:c36_6 :: c2:c3
gen_0':s7_6 :: Nat -> 0':s
gen_c4:c58_6 :: Nat -> c4:c5
gen_c19_6 :: Nat -> c1
gen_cons10_6 :: Nat -> cons
gen_c2:c311_6 :: Nat -> c2:c3


Generator Equations:
gen_0':s7_6(0) <=> 0'
gen_0':s7_6(+(x, 1)) <=> s(gen_0':s7_6(x))
gen_c4:c58_6(0) <=> c4
gen_c4:c58_6(+(x, 1)) <=> c5(gen_c4:c58_6(x))
gen_c19_6(0) <=> hole_c14_6
gen_c19_6(+(x, 1)) <=> c1(gen_c19_6(x), c2)
gen_cons10_6(0) <=> hole_cons5_6
gen_cons10_6(+(x, 1)) <=> cons(0', gen_cons10_6(x))
gen_c2:c311_6(0) <=> c2
gen_c2:c311_6(+(x, 1)) <=> c3(gen_c2:c311_6(x))


The following defined symbols remain to be analysed:
SEL, fib1, FIB1, add, ADD, sel

They will be analysed ascendingly in the following order:
add < fib1
add < FIB1
ADD < FIB1

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
FIB(z0) -> c(SEL(z0, fib1(s(0'), s(0'))), FIB1(s(0'), s(0')))
FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
ADD(0', z0) -> c2
ADD(s(z0), z1) -> c3(ADD(z0, z1))
SEL(0', cons(z0, z1)) -> c4
SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
fib(z0) -> sel(z0, fib1(s(0'), s(0')))
fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FIB :: 0':s -> c
c :: c4:c5 -> c1 -> c
SEL :: 0':s -> cons -> c4:c5
fib1 :: 0':s -> 0':s -> cons
s :: 0':s -> 0':s
0' :: 0':s
FIB1 :: 0':s -> 0':s -> c1
c1 :: c1 -> c2:c3 -> c1
add :: 0':s -> 0':s -> 0':s
ADD :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
cons :: 0':s -> cons -> cons
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
fib :: 0':s -> 0':s
sel :: 0':s -> cons -> 0':s
hole_c1_6 :: c
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_c14_6 :: c1
hole_cons5_6 :: cons
hole_c2:c36_6 :: c2:c3
gen_0':s7_6 :: Nat -> 0':s
gen_c4:c58_6 :: Nat -> c4:c5
gen_c19_6 :: Nat -> c1
gen_cons10_6 :: Nat -> cons
gen_c2:c311_6 :: Nat -> c2:c3


Lemmas:
SEL(gen_0':s7_6(n13_6), gen_cons10_6(+(1, n13_6))) -> gen_c4:c58_6(n13_6), rt in Omega(1 + n13_6)


Generator Equations:
gen_0':s7_6(0) <=> 0'
gen_0':s7_6(+(x, 1)) <=> s(gen_0':s7_6(x))
gen_c4:c58_6(0) <=> c4
gen_c4:c58_6(+(x, 1)) <=> c5(gen_c4:c58_6(x))
gen_c19_6(0) <=> hole_c14_6
gen_c19_6(+(x, 1)) <=> c1(gen_c19_6(x), c2)
gen_cons10_6(0) <=> hole_cons5_6
gen_cons10_6(+(x, 1)) <=> cons(0', gen_cons10_6(x))
gen_c2:c311_6(0) <=> c2
gen_c2:c311_6(+(x, 1)) <=> c3(gen_c2:c311_6(x))


The following defined symbols remain to be analysed:
add, fib1, FIB1, ADD, sel

They will be analysed ascendingly in the following order:
add < fib1
add < FIB1
ADD < FIB1

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s7_6(n416_6), gen_0':s7_6(b)) -> gen_0':s7_6(+(n416_6, b)), rt in Omega(0)

Induction Base:
add(gen_0':s7_6(0), gen_0':s7_6(b)) ->_R^Omega(0)
gen_0':s7_6(b)

Induction Step:
add(gen_0':s7_6(+(n416_6, 1)), gen_0':s7_6(b)) ->_R^Omega(0)
s(add(gen_0':s7_6(n416_6), gen_0':s7_6(b))) ->_IH
s(gen_0':s7_6(+(b, c417_6)))

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

(18)
Obligation:
Innermost TRS:
Rules:
FIB(z0) -> c(SEL(z0, fib1(s(0'), s(0'))), FIB1(s(0'), s(0')))
FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
ADD(0', z0) -> c2
ADD(s(z0), z1) -> c3(ADD(z0, z1))
SEL(0', cons(z0, z1)) -> c4
SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
fib(z0) -> sel(z0, fib1(s(0'), s(0')))
fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FIB :: 0':s -> c
c :: c4:c5 -> c1 -> c
SEL :: 0':s -> cons -> c4:c5
fib1 :: 0':s -> 0':s -> cons
s :: 0':s -> 0':s
0' :: 0':s
FIB1 :: 0':s -> 0':s -> c1
c1 :: c1 -> c2:c3 -> c1
add :: 0':s -> 0':s -> 0':s
ADD :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
cons :: 0':s -> cons -> cons
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
fib :: 0':s -> 0':s
sel :: 0':s -> cons -> 0':s
hole_c1_6 :: c
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_c14_6 :: c1
hole_cons5_6 :: cons
hole_c2:c36_6 :: c2:c3
gen_0':s7_6 :: Nat -> 0':s
gen_c4:c58_6 :: Nat -> c4:c5
gen_c19_6 :: Nat -> c1
gen_cons10_6 :: Nat -> cons
gen_c2:c311_6 :: Nat -> c2:c3


Lemmas:
SEL(gen_0':s7_6(n13_6), gen_cons10_6(+(1, n13_6))) -> gen_c4:c58_6(n13_6), rt in Omega(1 + n13_6)
add(gen_0':s7_6(n416_6), gen_0':s7_6(b)) -> gen_0':s7_6(+(n416_6, b)), rt in Omega(0)


Generator Equations:
gen_0':s7_6(0) <=> 0'
gen_0':s7_6(+(x, 1)) <=> s(gen_0':s7_6(x))
gen_c4:c58_6(0) <=> c4
gen_c4:c58_6(+(x, 1)) <=> c5(gen_c4:c58_6(x))
gen_c19_6(0) <=> hole_c14_6
gen_c19_6(+(x, 1)) <=> c1(gen_c19_6(x), c2)
gen_cons10_6(0) <=> hole_cons5_6
gen_cons10_6(+(x, 1)) <=> cons(0', gen_cons10_6(x))
gen_c2:c311_6(0) <=> c2
gen_c2:c311_6(+(x, 1)) <=> c3(gen_c2:c311_6(x))


The following defined symbols remain to be analysed:
fib1, FIB1, ADD, sel

They will be analysed ascendingly in the following order:
ADD < FIB1

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ADD(gen_0':s7_6(n2018_6), gen_0':s7_6(b)) -> gen_c2:c311_6(n2018_6), rt in Omega(1 + n2018_6)

Induction Base:
ADD(gen_0':s7_6(0), gen_0':s7_6(b)) ->_R^Omega(1)
c2

Induction Step:
ADD(gen_0':s7_6(+(n2018_6, 1)), gen_0':s7_6(b)) ->_R^Omega(1)
c3(ADD(gen_0':s7_6(n2018_6), gen_0':s7_6(b))) ->_IH
c3(gen_c2:c311_6(c2019_6))

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:
FIB(z0) -> c(SEL(z0, fib1(s(0'), s(0'))), FIB1(s(0'), s(0')))
FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
ADD(0', z0) -> c2
ADD(s(z0), z1) -> c3(ADD(z0, z1))
SEL(0', cons(z0, z1)) -> c4
SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
fib(z0) -> sel(z0, fib1(s(0'), s(0')))
fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
sel(0', cons(z0, z1)) -> z0
sel(s(z0), cons(z1, z2)) -> sel(z0, z2)

Types:
FIB :: 0':s -> c
c :: c4:c5 -> c1 -> c
SEL :: 0':s -> cons -> c4:c5
fib1 :: 0':s -> 0':s -> cons
s :: 0':s -> 0':s
0' :: 0':s
FIB1 :: 0':s -> 0':s -> c1
c1 :: c1 -> c2:c3 -> c1
add :: 0':s -> 0':s -> 0':s
ADD :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
cons :: 0':s -> cons -> cons
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
fib :: 0':s -> 0':s
sel :: 0':s -> cons -> 0':s
hole_c1_6 :: c
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_c14_6 :: c1
hole_cons5_6 :: cons
hole_c2:c36_6 :: c2:c3
gen_0':s7_6 :: Nat -> 0':s
gen_c4:c58_6 :: Nat -> c4:c5
gen_c19_6 :: Nat -> c1
gen_cons10_6 :: Nat -> cons
gen_c2:c311_6 :: Nat -> c2:c3


Lemmas:
SEL(gen_0':s7_6(n13_6), gen_cons10_6(+(1, n13_6))) -> gen_c4:c58_6(n13_6), rt in Omega(1 + n13_6)
add(gen_0':s7_6(n416_6), gen_0':s7_6(b)) -> gen_0':s7_6(+(n416_6, b)), rt in Omega(0)
ADD(gen_0':s7_6(n2018_6), gen_0':s7_6(b)) -> gen_c2:c311_6(n2018_6), rt in Omega(1 + n2018_6)


Generator Equations:
gen_0':s7_6(0) <=> 0'
gen_0':s7_6(+(x, 1)) <=> s(gen_0':s7_6(x))
gen_c4:c58_6(0) <=> c4
gen_c4:c58_6(+(x, 1)) <=> c5(gen_c4:c58_6(x))
gen_c19_6(0) <=> hole_c14_6
gen_c19_6(+(x, 1)) <=> c1(gen_c19_6(x), c2)
gen_cons10_6(0) <=> hole_cons5_6
gen_cons10_6(+(x, 1)) <=> cons(0', gen_cons10_6(x))
gen_c2:c311_6(0) <=> c2
gen_c2:c311_6(+(x, 1)) <=> c3(gen_c2:c311_6(x))


The following defined symbols remain to be analysed:
FIB1, sel
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sel(gen_0':s7_6(n3399_6), gen_cons10_6(+(1, n3399_6))) -> gen_0':s7_6(0), rt in Omega(0)

Induction Base:
sel(gen_0':s7_6(0), gen_cons10_6(+(1, 0))) ->_R^Omega(0)
0'

Induction Step:
sel(gen_0':s7_6(+(n3399_6, 1)), gen_cons10_6(+(1, +(n3399_6, 1)))) ->_R^Omega(0)
sel(gen_0':s7_6(n3399_6), gen_cons10_6(+(1, n3399_6))) ->_IH
gen_0':s7_6(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(22)
BOUNDS(1, INF)