WORST_CASE(Omega(n^1),?)
proof of input_2fEXgAFATg.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 8 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 302 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), 36.7 s]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 80 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 464 ms]
        (22) BOUNDS(1, INF)


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(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:

   int(0, 0) -> .(0, nil)
   int(0, s(y)) -> .(0, int(s(0), s(y)))
   int(s(x), 0) -> nil
   int(s(x), s(y)) -> int_list(int(x, y))
   int_list(nil) -> nil
   int_list(.(x, y)) -> .(s(x), int_list(y))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))
Tuples:
   INT(0, 0) -> c
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), 0) -> c2
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(nil) -> c4
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
S tuples:
   INT(0, 0) -> c
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), 0) -> c2
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(nil) -> c4
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
K tuples:none
Defined Rule Symbols:   int_2, int_list_1

Defined Pair Symbols:   INT_2, INT_LIST_1

Compound Symbols:   c, c1_1, c2, c3_2, c4, c5_1


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(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:

   INT(0, 0) -> c
   INT(0, s(z0)) -> c1(INT(s(0), s(z0)))
   INT(s(z0), 0) -> c2
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(nil) -> c4
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))

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

   int(0, 0) -> .(0, nil)
   int(0, s(z0)) -> .(0, int(s(0), s(z0)))
   int(s(z0), 0) -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))

Rewrite Strategy: INNERMOST
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(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:

   INT(0', 0') -> c
   INT(0', s(z0)) -> c1(INT(s(0'), s(z0)))
   INT(s(z0), 0') -> c2
   INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
   INT_LIST(nil) -> c4
   INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))

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

   int(0', 0') -> .(0', nil)
   int(0', s(z0)) -> .(0', int(s(0'), s(z0)))
   int(s(z0), 0') -> nil
   int(s(z0), s(z1)) -> int_list(int(z0, z1))
   int_list(nil) -> nil
   int_list(.(z0, z1)) -> .(s(z0), int_list(z1))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(8)
Obligation:
Innermost TRS:
Rules:
INT(0', 0') -> c
INT(0', s(z0)) -> c1(INT(s(0'), s(z0)))
INT(s(z0), 0') -> c2
INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
INT_LIST(nil) -> c4
INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
int(0', 0') -> .(0', nil)
int(0', s(z0)) -> .(0', int(s(0'), s(z0)))
int(s(z0), 0') -> nil
int(s(z0), s(z1)) -> int_list(int(z0, z1))
int_list(nil) -> nil
int_list(.(z0, z1)) -> .(s(z0), int_list(z1))

Types:
INT :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
INT_LIST :: nil:. -> c4:c5
int :: 0':s -> 0':s -> nil:.
nil :: nil:.
c4 :: c4:c5
. :: 0':s -> nil:. -> nil:.
c5 :: c4:c5 -> c4:c5
int_list :: nil:. -> nil:.
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_nil:.4_6 :: nil:.
gen_c:c1:c2:c35_6 :: Nat -> c:c1:c2:c3
gen_0':s6_6 :: Nat -> 0':s
gen_c4:c57_6 :: Nat -> c4:c5
gen_nil:.8_6 :: Nat -> nil:.

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
INT, INT_LIST, int, int_list

They will be analysed ascendingly in the following order:
INT_LIST < INT
int < INT
int_list < int

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(10)
Obligation:
Innermost TRS:
Rules:
INT(0', 0') -> c
INT(0', s(z0)) -> c1(INT(s(0'), s(z0)))
INT(s(z0), 0') -> c2
INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
INT_LIST(nil) -> c4
INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
int(0', 0') -> .(0', nil)
int(0', s(z0)) -> .(0', int(s(0'), s(z0)))
int(s(z0), 0') -> nil
int(s(z0), s(z1)) -> int_list(int(z0, z1))
int_list(nil) -> nil
int_list(.(z0, z1)) -> .(s(z0), int_list(z1))

Types:
INT :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
INT_LIST :: nil:. -> c4:c5
int :: 0':s -> 0':s -> nil:.
nil :: nil:.
c4 :: c4:c5
. :: 0':s -> nil:. -> nil:.
c5 :: c4:c5 -> c4:c5
int_list :: nil:. -> nil:.
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_nil:.4_6 :: nil:.
gen_c:c1:c2:c35_6 :: Nat -> c:c1:c2:c3
gen_0':s6_6 :: Nat -> 0':s
gen_c4:c57_6 :: Nat -> c4:c5
gen_nil:.8_6 :: Nat -> nil:.


Generator Equations:
gen_c:c1:c2:c35_6(0) <=> c
gen_c:c1:c2:c35_6(+(x, 1)) <=> c1(gen_c:c1:c2:c35_6(x))
gen_0':s6_6(0) <=> 0'
gen_0':s6_6(+(x, 1)) <=> s(gen_0':s6_6(x))
gen_c4:c57_6(0) <=> c4
gen_c4:c57_6(+(x, 1)) <=> c5(gen_c4:c57_6(x))
gen_nil:.8_6(0) <=> nil
gen_nil:.8_6(+(x, 1)) <=> .(0', gen_nil:.8_6(x))


The following defined symbols remain to be analysed:
INT_LIST, INT, int, int_list

They will be analysed ascendingly in the following order:
INT_LIST < INT
int < INT
int_list < int

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INT_LIST(gen_nil:.8_6(n10_6)) -> gen_c4:c57_6(n10_6), rt in Omega(1 + n10_6)

Induction Base:
INT_LIST(gen_nil:.8_6(0)) ->_R^Omega(1)
c4

Induction Step:
INT_LIST(gen_nil:.8_6(+(n10_6, 1))) ->_R^Omega(1)
c5(INT_LIST(gen_nil:.8_6(n10_6))) ->_IH
c5(gen_c4:c57_6(c11_6))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(12)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
INT(0', 0') -> c
INT(0', s(z0)) -> c1(INT(s(0'), s(z0)))
INT(s(z0), 0') -> c2
INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
INT_LIST(nil) -> c4
INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
int(0', 0') -> .(0', nil)
int(0', s(z0)) -> .(0', int(s(0'), s(z0)))
int(s(z0), 0') -> nil
int(s(z0), s(z1)) -> int_list(int(z0, z1))
int_list(nil) -> nil
int_list(.(z0, z1)) -> .(s(z0), int_list(z1))

Types:
INT :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
INT_LIST :: nil:. -> c4:c5
int :: 0':s -> 0':s -> nil:.
nil :: nil:.
c4 :: c4:c5
. :: 0':s -> nil:. -> nil:.
c5 :: c4:c5 -> c4:c5
int_list :: nil:. -> nil:.
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_nil:.4_6 :: nil:.
gen_c:c1:c2:c35_6 :: Nat -> c:c1:c2:c3
gen_0':s6_6 :: Nat -> 0':s
gen_c4:c57_6 :: Nat -> c4:c5
gen_nil:.8_6 :: Nat -> nil:.


Generator Equations:
gen_c:c1:c2:c35_6(0) <=> c
gen_c:c1:c2:c35_6(+(x, 1)) <=> c1(gen_c:c1:c2:c35_6(x))
gen_0':s6_6(0) <=> 0'
gen_0':s6_6(+(x, 1)) <=> s(gen_0':s6_6(x))
gen_c4:c57_6(0) <=> c4
gen_c4:c57_6(+(x, 1)) <=> c5(gen_c4:c57_6(x))
gen_nil:.8_6(0) <=> nil
gen_nil:.8_6(+(x, 1)) <=> .(0', gen_nil:.8_6(x))


The following defined symbols remain to be analysed:
INT_LIST, INT, int, int_list

They will be analysed ascendingly in the following order:
INT_LIST < INT
int < INT
int_list < int

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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
INT(0', 0') -> c
INT(0', s(z0)) -> c1(INT(s(0'), s(z0)))
INT(s(z0), 0') -> c2
INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
INT_LIST(nil) -> c4
INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
int(0', 0') -> .(0', nil)
int(0', s(z0)) -> .(0', int(s(0'), s(z0)))
int(s(z0), 0') -> nil
int(s(z0), s(z1)) -> int_list(int(z0, z1))
int_list(nil) -> nil
int_list(.(z0, z1)) -> .(s(z0), int_list(z1))

Types:
INT :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
INT_LIST :: nil:. -> c4:c5
int :: 0':s -> 0':s -> nil:.
nil :: nil:.
c4 :: c4:c5
. :: 0':s -> nil:. -> nil:.
c5 :: c4:c5 -> c4:c5
int_list :: nil:. -> nil:.
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_nil:.4_6 :: nil:.
gen_c:c1:c2:c35_6 :: Nat -> c:c1:c2:c3
gen_0':s6_6 :: Nat -> 0':s
gen_c4:c57_6 :: Nat -> c4:c5
gen_nil:.8_6 :: Nat -> nil:.


Lemmas:
INT_LIST(gen_nil:.8_6(n10_6)) -> gen_c4:c57_6(n10_6), rt in Omega(1 + n10_6)


Generator Equations:
gen_c:c1:c2:c35_6(0) <=> c
gen_c:c1:c2:c35_6(+(x, 1)) <=> c1(gen_c:c1:c2:c35_6(x))
gen_0':s6_6(0) <=> 0'
gen_0':s6_6(+(x, 1)) <=> s(gen_0':s6_6(x))
gen_c4:c57_6(0) <=> c4
gen_c4:c57_6(+(x, 1)) <=> c5(gen_c4:c57_6(x))
gen_nil:.8_6(0) <=> nil
gen_nil:.8_6(+(x, 1)) <=> .(0', gen_nil:.8_6(x))


The following defined symbols remain to be analysed:
int_list, INT, int

They will be analysed ascendingly in the following order:
int < INT
int_list < int

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
int_list(gen_nil:.8_6(+(1, n334_6))) -> *9_6, rt in Omega(0)

Induction Base:
int_list(gen_nil:.8_6(+(1, 0)))

Induction Step:
int_list(gen_nil:.8_6(+(1, +(n334_6, 1)))) ->_R^Omega(0)
.(s(0'), int_list(gen_nil:.8_6(+(1, n334_6)))) ->_IH
.(s(0'), *9_6)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
Obligation:
Innermost TRS:
Rules:
INT(0', 0') -> c
INT(0', s(z0)) -> c1(INT(s(0'), s(z0)))
INT(s(z0), 0') -> c2
INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
INT_LIST(nil) -> c4
INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
int(0', 0') -> .(0', nil)
int(0', s(z0)) -> .(0', int(s(0'), s(z0)))
int(s(z0), 0') -> nil
int(s(z0), s(z1)) -> int_list(int(z0, z1))
int_list(nil) -> nil
int_list(.(z0, z1)) -> .(s(z0), int_list(z1))

Types:
INT :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
INT_LIST :: nil:. -> c4:c5
int :: 0':s -> 0':s -> nil:.
nil :: nil:.
c4 :: c4:c5
. :: 0':s -> nil:. -> nil:.
c5 :: c4:c5 -> c4:c5
int_list :: nil:. -> nil:.
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_nil:.4_6 :: nil:.
gen_c:c1:c2:c35_6 :: Nat -> c:c1:c2:c3
gen_0':s6_6 :: Nat -> 0':s
gen_c4:c57_6 :: Nat -> c4:c5
gen_nil:.8_6 :: Nat -> nil:.


Lemmas:
INT_LIST(gen_nil:.8_6(n10_6)) -> gen_c4:c57_6(n10_6), rt in Omega(1 + n10_6)
int_list(gen_nil:.8_6(+(1, n334_6))) -> *9_6, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c35_6(0) <=> c
gen_c:c1:c2:c35_6(+(x, 1)) <=> c1(gen_c:c1:c2:c35_6(x))
gen_0':s6_6(0) <=> 0'
gen_0':s6_6(+(x, 1)) <=> s(gen_0':s6_6(x))
gen_c4:c57_6(0) <=> c4
gen_c4:c57_6(+(x, 1)) <=> c5(gen_c4:c57_6(x))
gen_nil:.8_6(0) <=> nil
gen_nil:.8_6(+(x, 1)) <=> .(0', gen_nil:.8_6(x))


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

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

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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
int(gen_0':s6_6(+(1, n412259_6)), gen_0':s6_6(n412259_6)) -> gen_nil:.8_6(0), rt in Omega(0)

Induction Base:
int(gen_0':s6_6(+(1, 0)), gen_0':s6_6(0)) ->_R^Omega(0)
nil

Induction Step:
int(gen_0':s6_6(+(1, +(n412259_6, 1))), gen_0':s6_6(+(n412259_6, 1))) ->_R^Omega(0)
int_list(int(gen_0':s6_6(+(1, n412259_6)), gen_0':s6_6(n412259_6))) ->_IH
int_list(gen_nil:.8_6(0)) ->_R^Omega(0)
nil

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(20)
Obligation:
Innermost TRS:
Rules:
INT(0', 0') -> c
INT(0', s(z0)) -> c1(INT(s(0'), s(z0)))
INT(s(z0), 0') -> c2
INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1))
INT_LIST(nil) -> c4
INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1))
int(0', 0') -> .(0', nil)
int(0', s(z0)) -> .(0', int(s(0'), s(z0)))
int(s(z0), 0') -> nil
int(s(z0), s(z1)) -> int_list(int(z0, z1))
int_list(nil) -> nil
int_list(.(z0, z1)) -> .(s(z0), int_list(z1))

Types:
INT :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
INT_LIST :: nil:. -> c4:c5
int :: 0':s -> 0':s -> nil:.
nil :: nil:.
c4 :: c4:c5
. :: 0':s -> nil:. -> nil:.
c5 :: c4:c5 -> c4:c5
int_list :: nil:. -> nil:.
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s2_6 :: 0':s
hole_c4:c53_6 :: c4:c5
hole_nil:.4_6 :: nil:.
gen_c:c1:c2:c35_6 :: Nat -> c:c1:c2:c3
gen_0':s6_6 :: Nat -> 0':s
gen_c4:c57_6 :: Nat -> c4:c5
gen_nil:.8_6 :: Nat -> nil:.


Lemmas:
INT_LIST(gen_nil:.8_6(n10_6)) -> gen_c4:c57_6(n10_6), rt in Omega(1 + n10_6)
int_list(gen_nil:.8_6(+(1, n334_6))) -> *9_6, rt in Omega(0)
int(gen_0':s6_6(+(1, n412259_6)), gen_0':s6_6(n412259_6)) -> gen_nil:.8_6(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c35_6(0) <=> c
gen_c:c1:c2:c35_6(+(x, 1)) <=> c1(gen_c:c1:c2:c35_6(x))
gen_0':s6_6(0) <=> 0'
gen_0':s6_6(+(x, 1)) <=> s(gen_0':s6_6(x))
gen_c4:c57_6(0) <=> c4
gen_c4:c57_6(+(x, 1)) <=> c5(gen_c4:c57_6(x))
gen_nil:.8_6(0) <=> nil
gen_nil:.8_6(+(x, 1)) <=> .(0', gen_nil:.8_6(x))


The following defined symbols remain to be analysed:
INT
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INT(gen_0':s6_6(+(1, n414974_6)), gen_0':s6_6(n414974_6)) -> *9_6, rt in Omega(n414974_6)

Induction Base:
INT(gen_0':s6_6(+(1, 0)), gen_0':s6_6(0))

Induction Step:
INT(gen_0':s6_6(+(1, +(n414974_6, 1))), gen_0':s6_6(+(n414974_6, 1))) ->_R^Omega(1)
c3(INT_LIST(int(gen_0':s6_6(+(1, n414974_6)), gen_0':s6_6(n414974_6))), INT(gen_0':s6_6(+(1, n414974_6)), gen_0':s6_6(n414974_6))) ->_L^Omega(0)
c3(INT_LIST(gen_nil:.8_6(0)), INT(gen_0':s6_6(+(1, n414974_6)), gen_0':s6_6(n414974_6))) ->_L^Omega(1)
c3(gen_c4:c57_6(0), INT(gen_0':s6_6(+(1, n414974_6)), gen_0':s6_6(n414974_6))) ->_IH
c3(gen_c4:c57_6(0), *9_6)

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