WORST_CASE(Omega(n^1),?)
proof of input_H5FHMmLUMq.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 20 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 306 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), 122 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 10 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 18 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 402 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
        (26) 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:

   fst(0, Z) -> nil
   fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
   from(X) -> cons(X, from(s(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   len(nil) -> 0
   len(cons(X, Z)) -> s(len(Z))

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:
   fst(0, z0) -> nil
   fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
   from(z0) -> cons(z0, from(s(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   len(nil) -> 0
   len(cons(z0, z1)) -> s(len(z1))
Tuples:
   FST(0, z0) -> c
   FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ADD(0, z0) -> c3
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   LEN(nil) -> c5
   LEN(cons(z0, z1)) -> c6(LEN(z1))
S tuples:
   FST(0, z0) -> c
   FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ADD(0, z0) -> c3
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   LEN(nil) -> c5
   LEN(cons(z0, z1)) -> c6(LEN(z1))
K tuples:none
Defined Rule Symbols:   fst_2, from_1, add_2, len_1

Defined Pair Symbols:   FST_2, FROM_1, ADD_2, LEN_1

Compound Symbols:   c, c1_1, c2_1, c3, c4_1, c5, c6_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:

   FST(0, z0) -> c
   FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ADD(0, z0) -> c3
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   LEN(nil) -> c5
   LEN(cons(z0, z1)) -> c6(LEN(z1))

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

   fst(0, z0) -> nil
   fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
   from(z0) -> cons(z0, from(s(z0)))
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   len(nil) -> 0
   len(cons(z0, z1)) -> s(len(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:

   FST(0', z0) -> c
   FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ADD(0', z0) -> c3
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   LEN(nil) -> c5
   LEN(cons(z0, z1)) -> c6(LEN(z1))

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

   fst(0', z0) -> nil
   fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
   from(z0) -> cons(z0, from(s(z0)))
   add(0', z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   len(nil) -> 0'
   len(cons(z0, z1)) -> s(len(z1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
FST(0', z0) -> c
FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
ADD(0', z0) -> c3
ADD(s(z0), z1) -> c4(ADD(z0, z1))
LEN(nil) -> c5
LEN(cons(z0, z1)) -> c6(LEN(z1))
fst(0', z0) -> nil
fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
len(nil) -> 0'
len(cons(z0, z1)) -> s(len(z1))

Types:
FST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
ADD :: 0':s -> a -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LEN :: cons:nil -> c5:c6
nil :: cons:nil
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
fst :: 0':s -> cons:nil -> cons:nil
from :: 0':s -> cons:nil
add :: 0':s -> 0':s -> 0':s
len :: cons:nil -> 0':s
hole_c:c11_7 :: c:c1
hole_0':s2_7 :: 0':s
hole_cons:nil3_7 :: cons:nil
hole_c24_7 :: c2
hole_c3:c45_7 :: c3:c4
hole_a6_7 :: a
hole_c5:c67_7 :: c5:c6
gen_c:c18_7 :: Nat -> c:c1
gen_0':s9_7 :: Nat -> 0':s
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c211_7 :: Nat -> c2
gen_c3:c412_7 :: Nat -> c3:c4
gen_c5:c613_7 :: Nat -> c5:c6

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FST, FROM, ADD, LEN, fst, from, add, len
----------------------------------------

(10)
Obligation:
Innermost TRS:
Rules:
FST(0', z0) -> c
FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
ADD(0', z0) -> c3
ADD(s(z0), z1) -> c4(ADD(z0, z1))
LEN(nil) -> c5
LEN(cons(z0, z1)) -> c6(LEN(z1))
fst(0', z0) -> nil
fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
len(nil) -> 0'
len(cons(z0, z1)) -> s(len(z1))

Types:
FST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
ADD :: 0':s -> a -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LEN :: cons:nil -> c5:c6
nil :: cons:nil
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
fst :: 0':s -> cons:nil -> cons:nil
from :: 0':s -> cons:nil
add :: 0':s -> 0':s -> 0':s
len :: cons:nil -> 0':s
hole_c:c11_7 :: c:c1
hole_0':s2_7 :: 0':s
hole_cons:nil3_7 :: cons:nil
hole_c24_7 :: c2
hole_c3:c45_7 :: c3:c4
hole_a6_7 :: a
hole_c5:c67_7 :: c5:c6
gen_c:c18_7 :: Nat -> c:c1
gen_0':s9_7 :: Nat -> 0':s
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c211_7 :: Nat -> c2
gen_c3:c412_7 :: Nat -> c3:c4
gen_c5:c613_7 :: Nat -> c5:c6


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_0':s9_7(0) <=> 0'
gen_0':s9_7(+(x, 1)) <=> s(gen_0':s9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c211_7(0) <=> hole_c24_7
gen_c211_7(+(x, 1)) <=> c2(gen_c211_7(x))
gen_c3:c412_7(0) <=> c3
gen_c3:c412_7(+(x, 1)) <=> c4(gen_c3:c412_7(x))
gen_c5:c613_7(0) <=> c5
gen_c5:c613_7(+(x, 1)) <=> c6(gen_c5:c613_7(x))


The following defined symbols remain to be analysed:
FST, FROM, ADD, LEN, fst, from, add, len
----------------------------------------

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
FST(gen_0':s9_7(n15_7), gen_cons:nil10_7(n15_7)) -> gen_c:c18_7(n15_7), rt in Omega(1 + n15_7)

Induction Base:
FST(gen_0':s9_7(0), gen_cons:nil10_7(0)) ->_R^Omega(1)
c

Induction Step:
FST(gen_0':s9_7(+(n15_7, 1)), gen_cons:nil10_7(+(n15_7, 1))) ->_R^Omega(1)
c1(FST(gen_0':s9_7(n15_7), gen_cons:nil10_7(n15_7))) ->_IH
c1(gen_c:c18_7(c16_7))

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:
FST(0', z0) -> c
FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
ADD(0', z0) -> c3
ADD(s(z0), z1) -> c4(ADD(z0, z1))
LEN(nil) -> c5
LEN(cons(z0, z1)) -> c6(LEN(z1))
fst(0', z0) -> nil
fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
len(nil) -> 0'
len(cons(z0, z1)) -> s(len(z1))

Types:
FST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
ADD :: 0':s -> a -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LEN :: cons:nil -> c5:c6
nil :: cons:nil
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
fst :: 0':s -> cons:nil -> cons:nil
from :: 0':s -> cons:nil
add :: 0':s -> 0':s -> 0':s
len :: cons:nil -> 0':s
hole_c:c11_7 :: c:c1
hole_0':s2_7 :: 0':s
hole_cons:nil3_7 :: cons:nil
hole_c24_7 :: c2
hole_c3:c45_7 :: c3:c4
hole_a6_7 :: a
hole_c5:c67_7 :: c5:c6
gen_c:c18_7 :: Nat -> c:c1
gen_0':s9_7 :: Nat -> 0':s
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c211_7 :: Nat -> c2
gen_c3:c412_7 :: Nat -> c3:c4
gen_c5:c613_7 :: Nat -> c5:c6


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_0':s9_7(0) <=> 0'
gen_0':s9_7(+(x, 1)) <=> s(gen_0':s9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c211_7(0) <=> hole_c24_7
gen_c211_7(+(x, 1)) <=> c2(gen_c211_7(x))
gen_c3:c412_7(0) <=> c3
gen_c3:c412_7(+(x, 1)) <=> c4(gen_c3:c412_7(x))
gen_c5:c613_7(0) <=> c5
gen_c5:c613_7(+(x, 1)) <=> c6(gen_c5:c613_7(x))


The following defined symbols remain to be analysed:
FST, FROM, ADD, LEN, fst, from, add, len
----------------------------------------

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
FST(0', z0) -> c
FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
ADD(0', z0) -> c3
ADD(s(z0), z1) -> c4(ADD(z0, z1))
LEN(nil) -> c5
LEN(cons(z0, z1)) -> c6(LEN(z1))
fst(0', z0) -> nil
fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
len(nil) -> 0'
len(cons(z0, z1)) -> s(len(z1))

Types:
FST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
ADD :: 0':s -> a -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LEN :: cons:nil -> c5:c6
nil :: cons:nil
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
fst :: 0':s -> cons:nil -> cons:nil
from :: 0':s -> cons:nil
add :: 0':s -> 0':s -> 0':s
len :: cons:nil -> 0':s
hole_c:c11_7 :: c:c1
hole_0':s2_7 :: 0':s
hole_cons:nil3_7 :: cons:nil
hole_c24_7 :: c2
hole_c3:c45_7 :: c3:c4
hole_a6_7 :: a
hole_c5:c67_7 :: c5:c6
gen_c:c18_7 :: Nat -> c:c1
gen_0':s9_7 :: Nat -> 0':s
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c211_7 :: Nat -> c2
gen_c3:c412_7 :: Nat -> c3:c4
gen_c5:c613_7 :: Nat -> c5:c6


Lemmas:
FST(gen_0':s9_7(n15_7), gen_cons:nil10_7(n15_7)) -> gen_c:c18_7(n15_7), rt in Omega(1 + n15_7)


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_0':s9_7(0) <=> 0'
gen_0':s9_7(+(x, 1)) <=> s(gen_0':s9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c211_7(0) <=> hole_c24_7
gen_c211_7(+(x, 1)) <=> c2(gen_c211_7(x))
gen_c3:c412_7(0) <=> c3
gen_c3:c412_7(+(x, 1)) <=> c4(gen_c3:c412_7(x))
gen_c5:c613_7(0) <=> c5
gen_c5:c613_7(+(x, 1)) <=> c6(gen_c5:c613_7(x))


The following defined symbols remain to be analysed:
FROM, ADD, LEN, fst, from, add, len
----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ADD(gen_0':s9_7(n530_7), hole_a6_7) -> gen_c3:c412_7(n530_7), rt in Omega(1 + n530_7)

Induction Base:
ADD(gen_0':s9_7(0), hole_a6_7) ->_R^Omega(1)
c3

Induction Step:
ADD(gen_0':s9_7(+(n530_7, 1)), hole_a6_7) ->_R^Omega(1)
c4(ADD(gen_0':s9_7(n530_7), hole_a6_7)) ->_IH
c4(gen_c3:c412_7(c531_7))

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:
FST(0', z0) -> c
FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
ADD(0', z0) -> c3
ADD(s(z0), z1) -> c4(ADD(z0, z1))
LEN(nil) -> c5
LEN(cons(z0, z1)) -> c6(LEN(z1))
fst(0', z0) -> nil
fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
len(nil) -> 0'
len(cons(z0, z1)) -> s(len(z1))

Types:
FST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
ADD :: 0':s -> a -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LEN :: cons:nil -> c5:c6
nil :: cons:nil
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
fst :: 0':s -> cons:nil -> cons:nil
from :: 0':s -> cons:nil
add :: 0':s -> 0':s -> 0':s
len :: cons:nil -> 0':s
hole_c:c11_7 :: c:c1
hole_0':s2_7 :: 0':s
hole_cons:nil3_7 :: cons:nil
hole_c24_7 :: c2
hole_c3:c45_7 :: c3:c4
hole_a6_7 :: a
hole_c5:c67_7 :: c5:c6
gen_c:c18_7 :: Nat -> c:c1
gen_0':s9_7 :: Nat -> 0':s
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c211_7 :: Nat -> c2
gen_c3:c412_7 :: Nat -> c3:c4
gen_c5:c613_7 :: Nat -> c5:c6


Lemmas:
FST(gen_0':s9_7(n15_7), gen_cons:nil10_7(n15_7)) -> gen_c:c18_7(n15_7), rt in Omega(1 + n15_7)
ADD(gen_0':s9_7(n530_7), hole_a6_7) -> gen_c3:c412_7(n530_7), rt in Omega(1 + n530_7)


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_0':s9_7(0) <=> 0'
gen_0':s9_7(+(x, 1)) <=> s(gen_0':s9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c211_7(0) <=> hole_c24_7
gen_c211_7(+(x, 1)) <=> c2(gen_c211_7(x))
gen_c3:c412_7(0) <=> c3
gen_c3:c412_7(+(x, 1)) <=> c4(gen_c3:c412_7(x))
gen_c5:c613_7(0) <=> c5
gen_c5:c613_7(+(x, 1)) <=> c6(gen_c5:c613_7(x))


The following defined symbols remain to be analysed:
LEN, fst, from, add, len
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LEN(gen_cons:nil10_7(n953_7)) -> gen_c5:c613_7(n953_7), rt in Omega(1 + n953_7)

Induction Base:
LEN(gen_cons:nil10_7(0)) ->_R^Omega(1)
c5

Induction Step:
LEN(gen_cons:nil10_7(+(n953_7, 1))) ->_R^Omega(1)
c6(LEN(gen_cons:nil10_7(n953_7))) ->_IH
c6(gen_c5:c613_7(c954_7))

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:
FST(0', z0) -> c
FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
ADD(0', z0) -> c3
ADD(s(z0), z1) -> c4(ADD(z0, z1))
LEN(nil) -> c5
LEN(cons(z0, z1)) -> c6(LEN(z1))
fst(0', z0) -> nil
fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
len(nil) -> 0'
len(cons(z0, z1)) -> s(len(z1))

Types:
FST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
ADD :: 0':s -> a -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LEN :: cons:nil -> c5:c6
nil :: cons:nil
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
fst :: 0':s -> cons:nil -> cons:nil
from :: 0':s -> cons:nil
add :: 0':s -> 0':s -> 0':s
len :: cons:nil -> 0':s
hole_c:c11_7 :: c:c1
hole_0':s2_7 :: 0':s
hole_cons:nil3_7 :: cons:nil
hole_c24_7 :: c2
hole_c3:c45_7 :: c3:c4
hole_a6_7 :: a
hole_c5:c67_7 :: c5:c6
gen_c:c18_7 :: Nat -> c:c1
gen_0':s9_7 :: Nat -> 0':s
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c211_7 :: Nat -> c2
gen_c3:c412_7 :: Nat -> c3:c4
gen_c5:c613_7 :: Nat -> c5:c6


Lemmas:
FST(gen_0':s9_7(n15_7), gen_cons:nil10_7(n15_7)) -> gen_c:c18_7(n15_7), rt in Omega(1 + n15_7)
ADD(gen_0':s9_7(n530_7), hole_a6_7) -> gen_c3:c412_7(n530_7), rt in Omega(1 + n530_7)
LEN(gen_cons:nil10_7(n953_7)) -> gen_c5:c613_7(n953_7), rt in Omega(1 + n953_7)


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_0':s9_7(0) <=> 0'
gen_0':s9_7(+(x, 1)) <=> s(gen_0':s9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c211_7(0) <=> hole_c24_7
gen_c211_7(+(x, 1)) <=> c2(gen_c211_7(x))
gen_c3:c412_7(0) <=> c3
gen_c3:c412_7(+(x, 1)) <=> c4(gen_c3:c412_7(x))
gen_c5:c613_7(0) <=> c5
gen_c5:c613_7(+(x, 1)) <=> c6(gen_c5:c613_7(x))


The following defined symbols remain to be analysed:
fst, from, add, len
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fst(gen_0':s9_7(n1402_7), gen_cons:nil10_7(n1402_7)) -> gen_cons:nil10_7(n1402_7), rt in Omega(0)

Induction Base:
fst(gen_0':s9_7(0), gen_cons:nil10_7(0)) ->_R^Omega(0)
nil

Induction Step:
fst(gen_0':s9_7(+(n1402_7, 1)), gen_cons:nil10_7(+(n1402_7, 1))) ->_R^Omega(0)
cons(0', fst(gen_0':s9_7(n1402_7), gen_cons:nil10_7(n1402_7))) ->_IH
cons(0', gen_cons:nil10_7(c1403_7))

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:
FST(0', z0) -> c
FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
ADD(0', z0) -> c3
ADD(s(z0), z1) -> c4(ADD(z0, z1))
LEN(nil) -> c5
LEN(cons(z0, z1)) -> c6(LEN(z1))
fst(0', z0) -> nil
fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
len(nil) -> 0'
len(cons(z0, z1)) -> s(len(z1))

Types:
FST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
ADD :: 0':s -> a -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LEN :: cons:nil -> c5:c6
nil :: cons:nil
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
fst :: 0':s -> cons:nil -> cons:nil
from :: 0':s -> cons:nil
add :: 0':s -> 0':s -> 0':s
len :: cons:nil -> 0':s
hole_c:c11_7 :: c:c1
hole_0':s2_7 :: 0':s
hole_cons:nil3_7 :: cons:nil
hole_c24_7 :: c2
hole_c3:c45_7 :: c3:c4
hole_a6_7 :: a
hole_c5:c67_7 :: c5:c6
gen_c:c18_7 :: Nat -> c:c1
gen_0':s9_7 :: Nat -> 0':s
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c211_7 :: Nat -> c2
gen_c3:c412_7 :: Nat -> c3:c4
gen_c5:c613_7 :: Nat -> c5:c6


Lemmas:
FST(gen_0':s9_7(n15_7), gen_cons:nil10_7(n15_7)) -> gen_c:c18_7(n15_7), rt in Omega(1 + n15_7)
ADD(gen_0':s9_7(n530_7), hole_a6_7) -> gen_c3:c412_7(n530_7), rt in Omega(1 + n530_7)
LEN(gen_cons:nil10_7(n953_7)) -> gen_c5:c613_7(n953_7), rt in Omega(1 + n953_7)
fst(gen_0':s9_7(n1402_7), gen_cons:nil10_7(n1402_7)) -> gen_cons:nil10_7(n1402_7), rt in Omega(0)


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_0':s9_7(0) <=> 0'
gen_0':s9_7(+(x, 1)) <=> s(gen_0':s9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c211_7(0) <=> hole_c24_7
gen_c211_7(+(x, 1)) <=> c2(gen_c211_7(x))
gen_c3:c412_7(0) <=> c3
gen_c3:c412_7(+(x, 1)) <=> c4(gen_c3:c412_7(x))
gen_c5:c613_7(0) <=> c5
gen_c5:c613_7(+(x, 1)) <=> c6(gen_c5:c613_7(x))


The following defined symbols remain to be analysed:
from, add, len
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s9_7(n2043_7), gen_0':s9_7(b)) -> gen_0':s9_7(+(n2043_7, b)), rt in Omega(0)

Induction Base:
add(gen_0':s9_7(0), gen_0':s9_7(b)) ->_R^Omega(0)
gen_0':s9_7(b)

Induction Step:
add(gen_0':s9_7(+(n2043_7, 1)), gen_0':s9_7(b)) ->_R^Omega(0)
s(add(gen_0':s9_7(n2043_7), gen_0':s9_7(b))) ->_IH
s(gen_0':s9_7(+(b, c2044_7)))

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:
FST(0', z0) -> c
FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2))
FROM(z0) -> c2(FROM(s(z0)))
ADD(0', z0) -> c3
ADD(s(z0), z1) -> c4(ADD(z0, z1))
LEN(nil) -> c5
LEN(cons(z0, z1)) -> c6(LEN(z1))
fst(0', z0) -> nil
fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2))
from(z0) -> cons(z0, from(s(z0)))
add(0', z0) -> z0
add(s(z0), z1) -> s(add(z0, z1))
len(nil) -> 0'
len(cons(z0, z1)) -> s(len(z1))

Types:
FST :: 0':s -> cons:nil -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
cons :: 0':s -> cons:nil -> cons:nil
c1 :: c:c1 -> c:c1
FROM :: 0':s -> c2
c2 :: c2 -> c2
ADD :: 0':s -> a -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
LEN :: cons:nil -> c5:c6
nil :: cons:nil
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
fst :: 0':s -> cons:nil -> cons:nil
from :: 0':s -> cons:nil
add :: 0':s -> 0':s -> 0':s
len :: cons:nil -> 0':s
hole_c:c11_7 :: c:c1
hole_0':s2_7 :: 0':s
hole_cons:nil3_7 :: cons:nil
hole_c24_7 :: c2
hole_c3:c45_7 :: c3:c4
hole_a6_7 :: a
hole_c5:c67_7 :: c5:c6
gen_c:c18_7 :: Nat -> c:c1
gen_0':s9_7 :: Nat -> 0':s
gen_cons:nil10_7 :: Nat -> cons:nil
gen_c211_7 :: Nat -> c2
gen_c3:c412_7 :: Nat -> c3:c4
gen_c5:c613_7 :: Nat -> c5:c6


Lemmas:
FST(gen_0':s9_7(n15_7), gen_cons:nil10_7(n15_7)) -> gen_c:c18_7(n15_7), rt in Omega(1 + n15_7)
ADD(gen_0':s9_7(n530_7), hole_a6_7) -> gen_c3:c412_7(n530_7), rt in Omega(1 + n530_7)
LEN(gen_cons:nil10_7(n953_7)) -> gen_c5:c613_7(n953_7), rt in Omega(1 + n953_7)
fst(gen_0':s9_7(n1402_7), gen_cons:nil10_7(n1402_7)) -> gen_cons:nil10_7(n1402_7), rt in Omega(0)
add(gen_0':s9_7(n2043_7), gen_0':s9_7(b)) -> gen_0':s9_7(+(n2043_7, b)), rt in Omega(0)


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_0':s9_7(0) <=> 0'
gen_0':s9_7(+(x, 1)) <=> s(gen_0':s9_7(x))
gen_cons:nil10_7(0) <=> nil
gen_cons:nil10_7(+(x, 1)) <=> cons(0', gen_cons:nil10_7(x))
gen_c211_7(0) <=> hole_c24_7
gen_c211_7(+(x, 1)) <=> c2(gen_c211_7(x))
gen_c3:c412_7(0) <=> c3
gen_c3:c412_7(+(x, 1)) <=> c4(gen_c3:c412_7(x))
gen_c5:c613_7(0) <=> c5
gen_c5:c613_7(+(x, 1)) <=> c6(gen_c5:c613_7(x))


The following defined symbols remain to be analysed:
len
----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
len(gen_cons:nil10_7(n3336_7)) -> gen_0':s9_7(n3336_7), rt in Omega(0)

Induction Base:
len(gen_cons:nil10_7(0)) ->_R^Omega(0)
0'

Induction Step:
len(gen_cons:nil10_7(+(n3336_7, 1))) ->_R^Omega(0)
s(len(gen_cons:nil10_7(n3336_7))) ->_IH
s(gen_0':s9_7(c3337_7))

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

(26)
BOUNDS(1, INF)