WORST_CASE(Omega(n^1),O(n^1))
proof of input_UmDrCyfhTH.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, n^1).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 35 ms]
    (4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 7 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 281 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 49 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 121 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 22 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
            (30) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   norm(nil) -> 0
   norm(g(x, y)) -> s(norm(x))
   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   rem(nil, y) -> nil
   rem(g(x, y), 0) -> g(x, y)
   rem(g(x, y), s(z)) -> rem(x, z)

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

(3) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
nil0() -> 0
00() -> 0
g0(0, 0) -> 0
s0(0) -> 0
norm0(0) -> 1
f0(0, 0) -> 2
rem0(0, 0) -> 3
01() -> 1
norm1(0) -> 4
s1(4) -> 1
nil1() -> 5
g1(5, 0) -> 2
f1(0, 0) -> 6
g1(6, 0) -> 2
nil1() -> 3
g1(0, 0) -> 3
rem1(0, 0) -> 3
01() -> 4
s1(4) -> 4
g1(5, 0) -> 6
g1(6, 0) -> 6

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

(4)
BOUNDS(1, n^1)

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

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   norm(nil) -> 0
   norm(g(z0, z1)) -> s(norm(z0))
   f(z0, nil) -> g(nil, z0)
   f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
   rem(nil, z0) -> nil
   rem(g(z0, z1), 0) -> g(z0, z1)
   rem(g(z0, z1), s(z2)) -> rem(z0, z2)
Tuples:
   NORM(nil) -> c
   NORM(g(z0, z1)) -> c1(NORM(z0))
   F(z0, nil) -> c2
   F(z0, g(z1, z2)) -> c3(F(z0, z1))
   REM(nil, z0) -> c4
   REM(g(z0, z1), 0) -> c5
   REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
S tuples:
   NORM(nil) -> c
   NORM(g(z0, z1)) -> c1(NORM(z0))
   F(z0, nil) -> c2
   F(z0, g(z1, z2)) -> c3(F(z0, z1))
   REM(nil, z0) -> c4
   REM(g(z0, z1), 0) -> c5
   REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
K tuples:none
Defined Rule Symbols:   norm_1, f_2, rem_2

Defined Pair Symbols:   NORM_1, F_2, REM_2

Compound Symbols:   c, c1_1, c2, c3_1, c4, c5, c6_1


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
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:

   NORM(nil) -> c
   NORM(g(z0, z1)) -> c1(NORM(z0))
   F(z0, nil) -> c2
   F(z0, g(z1, z2)) -> c3(F(z0, z1))
   REM(nil, z0) -> c4
   REM(g(z0, z1), 0) -> c5
   REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))

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

   norm(nil) -> 0
   norm(g(z0, z1)) -> s(norm(z0))
   f(z0, nil) -> g(nil, z0)
   f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
   rem(nil, z0) -> nil
   rem(g(z0, z1), 0) -> g(z0, z1)
   rem(g(z0, z1), s(z2)) -> rem(z0, z2)

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

(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(10)
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:

   NORM(nil) -> c
   NORM(g(z0, z1)) -> c1(NORM(z0))
   F(z0, nil) -> c2
   F(z0, g(z1, z2)) -> c3(F(z0, z1))
   REM(nil, z0) -> c4
   REM(g(z0, z1), 0') -> c5
   REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))

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

   norm(nil) -> 0'
   norm(g(z0, z1)) -> s(norm(z0))
   f(z0, nil) -> g(nil, z0)
   f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
   rem(nil, z0) -> nil
   rem(g(z0, z1), 0') -> g(z0, z1)
   rem(g(z0, z1), s(z2)) -> rem(z0, z2)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
NORM(nil) -> c
NORM(g(z0, z1)) -> c1(NORM(z0))
F(z0, nil) -> c2
F(z0, g(z1, z2)) -> c3(F(z0, z1))
REM(nil, z0) -> c4
REM(g(z0, z1), 0') -> c5
REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
norm(nil) -> 0'
norm(g(z0, z1)) -> s(norm(z0))
f(z0, nil) -> g(nil, z0)
f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
rem(nil, z0) -> nil
rem(g(z0, z1), 0') -> g(z0, z1)
rem(g(z0, z1), s(z2)) -> rem(z0, z2)

Types:
NORM :: nil:g -> c:c1
nil :: nil:g
c :: c:c1
g :: nil:g -> a -> nil:g
c1 :: c:c1 -> c:c1
F :: b -> nil:g -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
REM :: nil:g -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
0' :: 0':s
c5 :: c4:c5:c6
s :: 0':s -> 0':s
c6 :: c4:c5:c6 -> c4:c5:c6
norm :: nil:g -> 0':s
f :: a -> nil:g -> nil:g
rem :: nil:g -> 0':s -> nil:g
hole_c:c11_7 :: c:c1
hole_nil:g2_7 :: nil:g
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_b5_7 :: b
hole_c4:c5:c66_7 :: c4:c5:c6
hole_0':s7_7 :: 0':s
gen_c:c18_7 :: Nat -> c:c1
gen_nil:g9_7 :: Nat -> nil:g
gen_c2:c310_7 :: Nat -> c2:c3
gen_c4:c5:c611_7 :: Nat -> c4:c5:c6
gen_0':s12_7 :: Nat -> 0':s

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
NORM, F, REM, norm, f, rem
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
NORM(nil) -> c
NORM(g(z0, z1)) -> c1(NORM(z0))
F(z0, nil) -> c2
F(z0, g(z1, z2)) -> c3(F(z0, z1))
REM(nil, z0) -> c4
REM(g(z0, z1), 0') -> c5
REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
norm(nil) -> 0'
norm(g(z0, z1)) -> s(norm(z0))
f(z0, nil) -> g(nil, z0)
f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
rem(nil, z0) -> nil
rem(g(z0, z1), 0') -> g(z0, z1)
rem(g(z0, z1), s(z2)) -> rem(z0, z2)

Types:
NORM :: nil:g -> c:c1
nil :: nil:g
c :: c:c1
g :: nil:g -> a -> nil:g
c1 :: c:c1 -> c:c1
F :: b -> nil:g -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
REM :: nil:g -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
0' :: 0':s
c5 :: c4:c5:c6
s :: 0':s -> 0':s
c6 :: c4:c5:c6 -> c4:c5:c6
norm :: nil:g -> 0':s
f :: a -> nil:g -> nil:g
rem :: nil:g -> 0':s -> nil:g
hole_c:c11_7 :: c:c1
hole_nil:g2_7 :: nil:g
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_b5_7 :: b
hole_c4:c5:c66_7 :: c4:c5:c6
hole_0':s7_7 :: 0':s
gen_c:c18_7 :: Nat -> c:c1
gen_nil:g9_7 :: Nat -> nil:g
gen_c2:c310_7 :: Nat -> c2:c3
gen_c4:c5:c611_7 :: Nat -> c4:c5:c6
gen_0':s12_7 :: Nat -> 0':s


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_nil:g9_7(0) <=> nil
gen_nil:g9_7(+(x, 1)) <=> g(gen_nil:g9_7(x), hole_a3_7)
gen_c2:c310_7(0) <=> c2
gen_c2:c310_7(+(x, 1)) <=> c3(gen_c2:c310_7(x))
gen_c4:c5:c611_7(0) <=> c4
gen_c4:c5:c611_7(+(x, 1)) <=> c6(gen_c4:c5:c611_7(x))
gen_0':s12_7(0) <=> 0'
gen_0':s12_7(+(x, 1)) <=> s(gen_0':s12_7(x))


The following defined symbols remain to be analysed:
NORM, F, REM, norm, f, rem
----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
NORM(gen_nil:g9_7(n14_7)) -> gen_c:c18_7(n14_7), rt in Omega(1 + n14_7)

Induction Base:
NORM(gen_nil:g9_7(0)) ->_R^Omega(1)
c

Induction Step:
NORM(gen_nil:g9_7(+(n14_7, 1))) ->_R^Omega(1)
c1(NORM(gen_nil:g9_7(n14_7))) ->_IH
c1(gen_c:c18_7(c15_7))

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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
NORM(nil) -> c
NORM(g(z0, z1)) -> c1(NORM(z0))
F(z0, nil) -> c2
F(z0, g(z1, z2)) -> c3(F(z0, z1))
REM(nil, z0) -> c4
REM(g(z0, z1), 0') -> c5
REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
norm(nil) -> 0'
norm(g(z0, z1)) -> s(norm(z0))
f(z0, nil) -> g(nil, z0)
f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
rem(nil, z0) -> nil
rem(g(z0, z1), 0') -> g(z0, z1)
rem(g(z0, z1), s(z2)) -> rem(z0, z2)

Types:
NORM :: nil:g -> c:c1
nil :: nil:g
c :: c:c1
g :: nil:g -> a -> nil:g
c1 :: c:c1 -> c:c1
F :: b -> nil:g -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
REM :: nil:g -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
0' :: 0':s
c5 :: c4:c5:c6
s :: 0':s -> 0':s
c6 :: c4:c5:c6 -> c4:c5:c6
norm :: nil:g -> 0':s
f :: a -> nil:g -> nil:g
rem :: nil:g -> 0':s -> nil:g
hole_c:c11_7 :: c:c1
hole_nil:g2_7 :: nil:g
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_b5_7 :: b
hole_c4:c5:c66_7 :: c4:c5:c6
hole_0':s7_7 :: 0':s
gen_c:c18_7 :: Nat -> c:c1
gen_nil:g9_7 :: Nat -> nil:g
gen_c2:c310_7 :: Nat -> c2:c3
gen_c4:c5:c611_7 :: Nat -> c4:c5:c6
gen_0':s12_7 :: Nat -> 0':s


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_nil:g9_7(0) <=> nil
gen_nil:g9_7(+(x, 1)) <=> g(gen_nil:g9_7(x), hole_a3_7)
gen_c2:c310_7(0) <=> c2
gen_c2:c310_7(+(x, 1)) <=> c3(gen_c2:c310_7(x))
gen_c4:c5:c611_7(0) <=> c4
gen_c4:c5:c611_7(+(x, 1)) <=> c6(gen_c4:c5:c611_7(x))
gen_0':s12_7(0) <=> 0'
gen_0':s12_7(+(x, 1)) <=> s(gen_0':s12_7(x))


The following defined symbols remain to be analysed:
NORM, F, REM, norm, f, rem
----------------------------------------

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
NORM(nil) -> c
NORM(g(z0, z1)) -> c1(NORM(z0))
F(z0, nil) -> c2
F(z0, g(z1, z2)) -> c3(F(z0, z1))
REM(nil, z0) -> c4
REM(g(z0, z1), 0') -> c5
REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
norm(nil) -> 0'
norm(g(z0, z1)) -> s(norm(z0))
f(z0, nil) -> g(nil, z0)
f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
rem(nil, z0) -> nil
rem(g(z0, z1), 0') -> g(z0, z1)
rem(g(z0, z1), s(z2)) -> rem(z0, z2)

Types:
NORM :: nil:g -> c:c1
nil :: nil:g
c :: c:c1
g :: nil:g -> a -> nil:g
c1 :: c:c1 -> c:c1
F :: b -> nil:g -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
REM :: nil:g -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
0' :: 0':s
c5 :: c4:c5:c6
s :: 0':s -> 0':s
c6 :: c4:c5:c6 -> c4:c5:c6
norm :: nil:g -> 0':s
f :: a -> nil:g -> nil:g
rem :: nil:g -> 0':s -> nil:g
hole_c:c11_7 :: c:c1
hole_nil:g2_7 :: nil:g
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_b5_7 :: b
hole_c4:c5:c66_7 :: c4:c5:c6
hole_0':s7_7 :: 0':s
gen_c:c18_7 :: Nat -> c:c1
gen_nil:g9_7 :: Nat -> nil:g
gen_c2:c310_7 :: Nat -> c2:c3
gen_c4:c5:c611_7 :: Nat -> c4:c5:c6
gen_0':s12_7 :: Nat -> 0':s


Lemmas:
NORM(gen_nil:g9_7(n14_7)) -> gen_c:c18_7(n14_7), rt in Omega(1 + n14_7)


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_nil:g9_7(0) <=> nil
gen_nil:g9_7(+(x, 1)) <=> g(gen_nil:g9_7(x), hole_a3_7)
gen_c2:c310_7(0) <=> c2
gen_c2:c310_7(+(x, 1)) <=> c3(gen_c2:c310_7(x))
gen_c4:c5:c611_7(0) <=> c4
gen_c4:c5:c611_7(+(x, 1)) <=> c6(gen_c4:c5:c611_7(x))
gen_0':s12_7(0) <=> 0'
gen_0':s12_7(+(x, 1)) <=> s(gen_0':s12_7(x))


The following defined symbols remain to be analysed:
F, REM, norm, f, rem
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
F(hole_b5_7, gen_nil:g9_7(n283_7)) -> gen_c2:c310_7(n283_7), rt in Omega(1 + n283_7)

Induction Base:
F(hole_b5_7, gen_nil:g9_7(0)) ->_R^Omega(1)
c2

Induction Step:
F(hole_b5_7, gen_nil:g9_7(+(n283_7, 1))) ->_R^Omega(1)
c3(F(hole_b5_7, gen_nil:g9_7(n283_7))) ->_IH
c3(gen_c2:c310_7(c284_7))

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

(22)
Obligation:
Innermost TRS:
Rules:
NORM(nil) -> c
NORM(g(z0, z1)) -> c1(NORM(z0))
F(z0, nil) -> c2
F(z0, g(z1, z2)) -> c3(F(z0, z1))
REM(nil, z0) -> c4
REM(g(z0, z1), 0') -> c5
REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
norm(nil) -> 0'
norm(g(z0, z1)) -> s(norm(z0))
f(z0, nil) -> g(nil, z0)
f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
rem(nil, z0) -> nil
rem(g(z0, z1), 0') -> g(z0, z1)
rem(g(z0, z1), s(z2)) -> rem(z0, z2)

Types:
NORM :: nil:g -> c:c1
nil :: nil:g
c :: c:c1
g :: nil:g -> a -> nil:g
c1 :: c:c1 -> c:c1
F :: b -> nil:g -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
REM :: nil:g -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
0' :: 0':s
c5 :: c4:c5:c6
s :: 0':s -> 0':s
c6 :: c4:c5:c6 -> c4:c5:c6
norm :: nil:g -> 0':s
f :: a -> nil:g -> nil:g
rem :: nil:g -> 0':s -> nil:g
hole_c:c11_7 :: c:c1
hole_nil:g2_7 :: nil:g
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_b5_7 :: b
hole_c4:c5:c66_7 :: c4:c5:c6
hole_0':s7_7 :: 0':s
gen_c:c18_7 :: Nat -> c:c1
gen_nil:g9_7 :: Nat -> nil:g
gen_c2:c310_7 :: Nat -> c2:c3
gen_c4:c5:c611_7 :: Nat -> c4:c5:c6
gen_0':s12_7 :: Nat -> 0':s


Lemmas:
NORM(gen_nil:g9_7(n14_7)) -> gen_c:c18_7(n14_7), rt in Omega(1 + n14_7)
F(hole_b5_7, gen_nil:g9_7(n283_7)) -> gen_c2:c310_7(n283_7), rt in Omega(1 + n283_7)


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_nil:g9_7(0) <=> nil
gen_nil:g9_7(+(x, 1)) <=> g(gen_nil:g9_7(x), hole_a3_7)
gen_c2:c310_7(0) <=> c2
gen_c2:c310_7(+(x, 1)) <=> c3(gen_c2:c310_7(x))
gen_c4:c5:c611_7(0) <=> c4
gen_c4:c5:c611_7(+(x, 1)) <=> c6(gen_c4:c5:c611_7(x))
gen_0':s12_7(0) <=> 0'
gen_0':s12_7(+(x, 1)) <=> s(gen_0':s12_7(x))


The following defined symbols remain to be analysed:
REM, norm, f, rem
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
REM(gen_nil:g9_7(n651_7), gen_0':s12_7(n651_7)) -> gen_c4:c5:c611_7(n651_7), rt in Omega(1 + n651_7)

Induction Base:
REM(gen_nil:g9_7(0), gen_0':s12_7(0)) ->_R^Omega(1)
c4

Induction Step:
REM(gen_nil:g9_7(+(n651_7, 1)), gen_0':s12_7(+(n651_7, 1))) ->_R^Omega(1)
c6(REM(gen_nil:g9_7(n651_7), gen_0':s12_7(n651_7))) ->_IH
c6(gen_c4:c5:c611_7(c652_7))

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

(24)
Obligation:
Innermost TRS:
Rules:
NORM(nil) -> c
NORM(g(z0, z1)) -> c1(NORM(z0))
F(z0, nil) -> c2
F(z0, g(z1, z2)) -> c3(F(z0, z1))
REM(nil, z0) -> c4
REM(g(z0, z1), 0') -> c5
REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
norm(nil) -> 0'
norm(g(z0, z1)) -> s(norm(z0))
f(z0, nil) -> g(nil, z0)
f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
rem(nil, z0) -> nil
rem(g(z0, z1), 0') -> g(z0, z1)
rem(g(z0, z1), s(z2)) -> rem(z0, z2)

Types:
NORM :: nil:g -> c:c1
nil :: nil:g
c :: c:c1
g :: nil:g -> a -> nil:g
c1 :: c:c1 -> c:c1
F :: b -> nil:g -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
REM :: nil:g -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
0' :: 0':s
c5 :: c4:c5:c6
s :: 0':s -> 0':s
c6 :: c4:c5:c6 -> c4:c5:c6
norm :: nil:g -> 0':s
f :: a -> nil:g -> nil:g
rem :: nil:g -> 0':s -> nil:g
hole_c:c11_7 :: c:c1
hole_nil:g2_7 :: nil:g
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_b5_7 :: b
hole_c4:c5:c66_7 :: c4:c5:c6
hole_0':s7_7 :: 0':s
gen_c:c18_7 :: Nat -> c:c1
gen_nil:g9_7 :: Nat -> nil:g
gen_c2:c310_7 :: Nat -> c2:c3
gen_c4:c5:c611_7 :: Nat -> c4:c5:c6
gen_0':s12_7 :: Nat -> 0':s


Lemmas:
NORM(gen_nil:g9_7(n14_7)) -> gen_c:c18_7(n14_7), rt in Omega(1 + n14_7)
F(hole_b5_7, gen_nil:g9_7(n283_7)) -> gen_c2:c310_7(n283_7), rt in Omega(1 + n283_7)
REM(gen_nil:g9_7(n651_7), gen_0':s12_7(n651_7)) -> gen_c4:c5:c611_7(n651_7), rt in Omega(1 + n651_7)


Generator Equations:
gen_c:c18_7(0) <=> c
gen_c:c18_7(+(x, 1)) <=> c1(gen_c:c18_7(x))
gen_nil:g9_7(0) <=> nil
gen_nil:g9_7(+(x, 1)) <=> g(gen_nil:g9_7(x), hole_a3_7)
gen_c2:c310_7(0) <=> c2
gen_c2:c310_7(+(x, 1)) <=> c3(gen_c2:c310_7(x))
gen_c4:c5:c611_7(0) <=> c4
gen_c4:c5:c611_7(+(x, 1)) <=> c6(gen_c4:c5:c611_7(x))
gen_0':s12_7(0) <=> 0'
gen_0':s12_7(+(x, 1)) <=> s(gen_0':s12_7(x))


The following defined symbols remain to be analysed:
norm, f, rem
----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
norm(gen_nil:g9_7(n1365_7)) -> gen_0':s12_7(n1365_7), rt in Omega(0)

Induction Base:
norm(gen_nil:g9_7(0)) ->_R^Omega(0)
0'

Induction Step:
norm(gen_nil:g9_7(+(n1365_7, 1))) ->_R^Omega(0)
s(norm(gen_nil:g9_7(n1365_7))) ->_IH
s(gen_0':s12_7(c1366_7))

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

(26)
Obligation:
Innermost TRS:
Rules:
NORM(nil) -> c
NORM(g(z0, z1)) -> c1(NORM(z0))
F(z0, nil) -> c2
F(z0, g(z1, z2)) -> c3(F(z0, z1))
REM(nil, z0) -> c4
REM(g(z0, z1), 0') -> c5
REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
norm(nil) -> 0'
norm(g(z0, z1)) -> s(norm(z0))
f(z0, nil) -> g(nil, z0)
f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
rem(nil, z0) -> nil
rem(g(z0, z1), 0') -> g(z0, z1)
rem(g(z0, z1), s(z2)) -> rem(z0, z2)

Types:
NORM :: nil:g -> c:c1
nil :: nil:g
c :: c:c1
g :: nil:g -> a -> nil:g
c1 :: c:c1 -> c:c1
F :: b -> nil:g -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
REM :: nil:g -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
0' :: 0':s
c5 :: c4:c5:c6
s :: 0':s -> 0':s
c6 :: c4:c5:c6 -> c4:c5:c6
norm :: nil:g -> 0':s
f :: a -> nil:g -> nil:g
rem :: nil:g -> 0':s -> nil:g
hole_c:c11_7 :: c:c1
hole_nil:g2_7 :: nil:g
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_b5_7 :: b
hole_c4:c5:c66_7 :: c4:c5:c6
hole_0':s7_7 :: 0':s
gen_c:c18_7 :: Nat -> c:c1
gen_nil:g9_7 :: Nat -> nil:g
gen_c2:c310_7 :: Nat -> c2:c3
gen_c4:c5:c611_7 :: Nat -> c4:c5:c6
gen_0':s12_7 :: Nat -> 0':s


Lemmas:
NORM(gen_nil:g9_7(n14_7)) -> gen_c:c18_7(n14_7), rt in Omega(1 + n14_7)
F(hole_b5_7, gen_nil:g9_7(n283_7)) -> gen_c2:c310_7(n283_7), rt in Omega(1 + n283_7)
REM(gen_nil:g9_7(n651_7), gen_0':s12_7(n651_7)) -> gen_c4:c5:c611_7(n651_7), rt in Omega(1 + n651_7)
norm(gen_nil:g9_7(n1365_7)) -> gen_0':s12_7(n1365_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_nil:g9_7(0) <=> nil
gen_nil:g9_7(+(x, 1)) <=> g(gen_nil:g9_7(x), hole_a3_7)
gen_c2:c310_7(0) <=> c2
gen_c2:c310_7(+(x, 1)) <=> c3(gen_c2:c310_7(x))
gen_c4:c5:c611_7(0) <=> c4
gen_c4:c5:c611_7(+(x, 1)) <=> c6(gen_c4:c5:c611_7(x))
gen_0':s12_7(0) <=> 0'
gen_0':s12_7(+(x, 1)) <=> s(gen_0':s12_7(x))


The following defined symbols remain to be analysed:
f, rem
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(hole_a3_7, gen_nil:g9_7(n1782_7)) -> gen_nil:g9_7(+(1, n1782_7)), rt in Omega(0)

Induction Base:
f(hole_a3_7, gen_nil:g9_7(0)) ->_R^Omega(0)
g(nil, hole_a3_7)

Induction Step:
f(hole_a3_7, gen_nil:g9_7(+(n1782_7, 1))) ->_R^Omega(0)
g(f(hole_a3_7, gen_nil:g9_7(n1782_7)), hole_a3_7) ->_IH
g(gen_nil:g9_7(+(1, c1783_7)), hole_a3_7)

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

(28)
Obligation:
Innermost TRS:
Rules:
NORM(nil) -> c
NORM(g(z0, z1)) -> c1(NORM(z0))
F(z0, nil) -> c2
F(z0, g(z1, z2)) -> c3(F(z0, z1))
REM(nil, z0) -> c4
REM(g(z0, z1), 0') -> c5
REM(g(z0, z1), s(z2)) -> c6(REM(z0, z2))
norm(nil) -> 0'
norm(g(z0, z1)) -> s(norm(z0))
f(z0, nil) -> g(nil, z0)
f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
rem(nil, z0) -> nil
rem(g(z0, z1), 0') -> g(z0, z1)
rem(g(z0, z1), s(z2)) -> rem(z0, z2)

Types:
NORM :: nil:g -> c:c1
nil :: nil:g
c :: c:c1
g :: nil:g -> a -> nil:g
c1 :: c:c1 -> c:c1
F :: b -> nil:g -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
REM :: nil:g -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
0' :: 0':s
c5 :: c4:c5:c6
s :: 0':s -> 0':s
c6 :: c4:c5:c6 -> c4:c5:c6
norm :: nil:g -> 0':s
f :: a -> nil:g -> nil:g
rem :: nil:g -> 0':s -> nil:g
hole_c:c11_7 :: c:c1
hole_nil:g2_7 :: nil:g
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_b5_7 :: b
hole_c4:c5:c66_7 :: c4:c5:c6
hole_0':s7_7 :: 0':s
gen_c:c18_7 :: Nat -> c:c1
gen_nil:g9_7 :: Nat -> nil:g
gen_c2:c310_7 :: Nat -> c2:c3
gen_c4:c5:c611_7 :: Nat -> c4:c5:c6
gen_0':s12_7 :: Nat -> 0':s


Lemmas:
NORM(gen_nil:g9_7(n14_7)) -> gen_c:c18_7(n14_7), rt in Omega(1 + n14_7)
F(hole_b5_7, gen_nil:g9_7(n283_7)) -> gen_c2:c310_7(n283_7), rt in Omega(1 + n283_7)
REM(gen_nil:g9_7(n651_7), gen_0':s12_7(n651_7)) -> gen_c4:c5:c611_7(n651_7), rt in Omega(1 + n651_7)
norm(gen_nil:g9_7(n1365_7)) -> gen_0':s12_7(n1365_7), rt in Omega(0)
f(hole_a3_7, gen_nil:g9_7(n1782_7)) -> gen_nil:g9_7(+(1, n1782_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_nil:g9_7(0) <=> nil
gen_nil:g9_7(+(x, 1)) <=> g(gen_nil:g9_7(x), hole_a3_7)
gen_c2:c310_7(0) <=> c2
gen_c2:c310_7(+(x, 1)) <=> c3(gen_c2:c310_7(x))
gen_c4:c5:c611_7(0) <=> c4
gen_c4:c5:c611_7(+(x, 1)) <=> c6(gen_c4:c5:c611_7(x))
gen_0':s12_7(0) <=> 0'
gen_0':s12_7(+(x, 1)) <=> s(gen_0':s12_7(x))


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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rem(gen_nil:g9_7(n2129_7), gen_0':s12_7(n2129_7)) -> gen_nil:g9_7(0), rt in Omega(0)

Induction Base:
rem(gen_nil:g9_7(0), gen_0':s12_7(0)) ->_R^Omega(0)
nil

Induction Step:
rem(gen_nil:g9_7(+(n2129_7, 1)), gen_0':s12_7(+(n2129_7, 1))) ->_R^Omega(0)
rem(gen_nil:g9_7(n2129_7), gen_0':s12_7(n2129_7)) ->_IH
gen_nil:g9_7(0)

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

(30)
BOUNDS(1, INF)