WORST_CASE(Omega(n^1),O(n^1))
proof of input_WXHn60Q62n.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 46 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), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 265 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), 58 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 26 ms]
            (26) 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:

   walk#1(Nil) -> walk_xs
   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
   main(Nil) -> Nil
   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

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:

   walk#1(Nil) -> walk_xs
   walk#1(Cons(x4, x3)) -> comp_f_g(walk#1(x3), walk_xs_3(x4))
   comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) -> comp_f_g#1(x7, x9, Cons(x8, x12))
   comp_f_g#1(walk_xs, walk_xs_3(x8), x12) -> Cons(x8, x12)
   main(Nil) -> Nil
   main(Cons(x4, x5)) -> comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

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 2. 

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
walk_xs0() -> 0
Cons0(0, 0) -> 0
comp_f_g0(0, 0) -> 0
walk_xs_30(0) -> 0
walk#10(0) -> 1
comp_f_g#10(0, 0, 0) -> 2
main0(0) -> 3
walk_xs1() -> 1
walk#11(0) -> 4
walk_xs_31(0) -> 5
comp_f_g1(4, 5) -> 1
Cons1(0, 0) -> 6
comp_f_g#11(0, 0, 6) -> 2
Cons1(0, 0) -> 2
Nil1() -> 3
walk#11(0) -> 7
walk_xs_31(0) -> 8
Nil1() -> 9
comp_f_g#11(7, 8, 9) -> 3
walk_xs1() -> 4
walk_xs1() -> 7
comp_f_g1(4, 5) -> 4
comp_f_g1(4, 5) -> 7
Cons1(0, 6) -> 6
Cons1(0, 6) -> 2
Cons2(0, 9) -> 10
comp_f_g#12(4, 5, 10) -> 3
Cons2(0, 9) -> 3
Cons2(0, 10) -> 10
Cons2(0, 10) -> 3

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

(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:
   walk#1(Nil) -> walk_xs
   walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
   comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
   comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
   main(Nil) -> Nil
   main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)
Tuples:
   WALK#1(Nil) -> c
   WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
   COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
   COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
   MAIN(Nil) -> c4
   MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
S tuples:
   WALK#1(Nil) -> c
   WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
   COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
   COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
   MAIN(Nil) -> c4
   MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
K tuples:none
Defined Rule Symbols:   walk#1_1, comp_f_g#1_3, main_1

Defined Pair Symbols:   WALK#1_1, COMP_F_G#1_3, MAIN_1

Compound Symbols:   c, c1_1, c2_1, c3, c4, c5_2


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

(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:

   WALK#1(Nil) -> c
   WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
   COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
   COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
   MAIN(Nil) -> c4
   MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))

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

   walk#1(Nil) -> walk_xs
   walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
   comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
   comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
   main(Nil) -> Nil
   main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)

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:

   WALK#1(Nil) -> c
   WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
   COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
   COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
   MAIN(Nil) -> c4
   MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))

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

   walk#1(Nil) -> walk_xs
   walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
   comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
   comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
   main(Nil) -> Nil
   main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
WALK#1(Nil) -> c
WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
MAIN(Nil) -> c4
MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
walk#1(Nil) -> walk_xs
walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
main(Nil) -> Nil
main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)

Types:
WALK#1 :: Nil:Cons -> c:c1
Nil :: Nil:Cons
c :: c:c1
Cons :: a -> Nil:Cons -> Nil:Cons
c1 :: c:c1 -> c:c1
COMP_F_G#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> c2:c3
comp_f_g :: comp_f_g:walk_xs -> walk_xs_3 -> comp_f_g:walk_xs
walk_xs_3 :: a -> walk_xs_3
c2 :: c2:c3 -> c2:c3
walk_xs :: comp_f_g:walk_xs
c3 :: c2:c3
MAIN :: Nil:Cons -> c4:c5
c4 :: c4:c5
c5 :: c2:c3 -> c:c1 -> c4:c5
walk#1 :: Nil:Cons -> comp_f_g:walk_xs
comp_f_g#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> Nil:Cons
hole_c:c11_6 :: c:c1
hole_Nil:Cons2_6 :: Nil:Cons
hole_a3_6 :: a
hole_c2:c34_6 :: c2:c3
hole_comp_f_g:walk_xs5_6 :: comp_f_g:walk_xs
hole_walk_xs_36_6 :: walk_xs_3
hole_c4:c57_6 :: c4:c5
gen_c:c18_6 :: Nat -> c:c1
gen_Nil:Cons9_6 :: Nat -> Nil:Cons
gen_c2:c310_6 :: Nat -> c2:c3
gen_comp_f_g:walk_xs11_6 :: Nat -> comp_f_g:walk_xs

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
WALK#1, COMP_F_G#1, walk#1, comp_f_g#1
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
WALK#1(Nil) -> c
WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
MAIN(Nil) -> c4
MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
walk#1(Nil) -> walk_xs
walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
main(Nil) -> Nil
main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)

Types:
WALK#1 :: Nil:Cons -> c:c1
Nil :: Nil:Cons
c :: c:c1
Cons :: a -> Nil:Cons -> Nil:Cons
c1 :: c:c1 -> c:c1
COMP_F_G#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> c2:c3
comp_f_g :: comp_f_g:walk_xs -> walk_xs_3 -> comp_f_g:walk_xs
walk_xs_3 :: a -> walk_xs_3
c2 :: c2:c3 -> c2:c3
walk_xs :: comp_f_g:walk_xs
c3 :: c2:c3
MAIN :: Nil:Cons -> c4:c5
c4 :: c4:c5
c5 :: c2:c3 -> c:c1 -> c4:c5
walk#1 :: Nil:Cons -> comp_f_g:walk_xs
comp_f_g#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> Nil:Cons
hole_c:c11_6 :: c:c1
hole_Nil:Cons2_6 :: Nil:Cons
hole_a3_6 :: a
hole_c2:c34_6 :: c2:c3
hole_comp_f_g:walk_xs5_6 :: comp_f_g:walk_xs
hole_walk_xs_36_6 :: walk_xs_3
hole_c4:c57_6 :: c4:c5
gen_c:c18_6 :: Nat -> c:c1
gen_Nil:Cons9_6 :: Nat -> Nil:Cons
gen_c2:c310_6 :: Nat -> c2:c3
gen_comp_f_g:walk_xs11_6 :: Nat -> comp_f_g:walk_xs


Generator Equations:
gen_c:c18_6(0) <=> c
gen_c:c18_6(+(x, 1)) <=> c1(gen_c:c18_6(x))
gen_Nil:Cons9_6(0) <=> Nil
gen_Nil:Cons9_6(+(x, 1)) <=> Cons(hole_a3_6, gen_Nil:Cons9_6(x))
gen_c2:c310_6(0) <=> c3
gen_c2:c310_6(+(x, 1)) <=> c2(gen_c2:c310_6(x))
gen_comp_f_g:walk_xs11_6(0) <=> walk_xs
gen_comp_f_g:walk_xs11_6(+(x, 1)) <=> comp_f_g(gen_comp_f_g:walk_xs11_6(x), walk_xs_3(hole_a3_6))


The following defined symbols remain to be analysed:
WALK#1, COMP_F_G#1, walk#1, comp_f_g#1
----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
WALK#1(gen_Nil:Cons9_6(n13_6)) -> gen_c:c18_6(n13_6), rt in Omega(1 + n13_6)

Induction Base:
WALK#1(gen_Nil:Cons9_6(0)) ->_R^Omega(1)
c

Induction Step:
WALK#1(gen_Nil:Cons9_6(+(n13_6, 1))) ->_R^Omega(1)
c1(WALK#1(gen_Nil:Cons9_6(n13_6))) ->_IH
c1(gen_c:c18_6(c14_6))

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:
WALK#1(Nil) -> c
WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
MAIN(Nil) -> c4
MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
walk#1(Nil) -> walk_xs
walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
main(Nil) -> Nil
main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)

Types:
WALK#1 :: Nil:Cons -> c:c1
Nil :: Nil:Cons
c :: c:c1
Cons :: a -> Nil:Cons -> Nil:Cons
c1 :: c:c1 -> c:c1
COMP_F_G#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> c2:c3
comp_f_g :: comp_f_g:walk_xs -> walk_xs_3 -> comp_f_g:walk_xs
walk_xs_3 :: a -> walk_xs_3
c2 :: c2:c3 -> c2:c3
walk_xs :: comp_f_g:walk_xs
c3 :: c2:c3
MAIN :: Nil:Cons -> c4:c5
c4 :: c4:c5
c5 :: c2:c3 -> c:c1 -> c4:c5
walk#1 :: Nil:Cons -> comp_f_g:walk_xs
comp_f_g#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> Nil:Cons
hole_c:c11_6 :: c:c1
hole_Nil:Cons2_6 :: Nil:Cons
hole_a3_6 :: a
hole_c2:c34_6 :: c2:c3
hole_comp_f_g:walk_xs5_6 :: comp_f_g:walk_xs
hole_walk_xs_36_6 :: walk_xs_3
hole_c4:c57_6 :: c4:c5
gen_c:c18_6 :: Nat -> c:c1
gen_Nil:Cons9_6 :: Nat -> Nil:Cons
gen_c2:c310_6 :: Nat -> c2:c3
gen_comp_f_g:walk_xs11_6 :: Nat -> comp_f_g:walk_xs


Generator Equations:
gen_c:c18_6(0) <=> c
gen_c:c18_6(+(x, 1)) <=> c1(gen_c:c18_6(x))
gen_Nil:Cons9_6(0) <=> Nil
gen_Nil:Cons9_6(+(x, 1)) <=> Cons(hole_a3_6, gen_Nil:Cons9_6(x))
gen_c2:c310_6(0) <=> c3
gen_c2:c310_6(+(x, 1)) <=> c2(gen_c2:c310_6(x))
gen_comp_f_g:walk_xs11_6(0) <=> walk_xs
gen_comp_f_g:walk_xs11_6(+(x, 1)) <=> comp_f_g(gen_comp_f_g:walk_xs11_6(x), walk_xs_3(hole_a3_6))


The following defined symbols remain to be analysed:
WALK#1, COMP_F_G#1, walk#1, comp_f_g#1
----------------------------------------

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
WALK#1(Nil) -> c
WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
MAIN(Nil) -> c4
MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
walk#1(Nil) -> walk_xs
walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
main(Nil) -> Nil
main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)

Types:
WALK#1 :: Nil:Cons -> c:c1
Nil :: Nil:Cons
c :: c:c1
Cons :: a -> Nil:Cons -> Nil:Cons
c1 :: c:c1 -> c:c1
COMP_F_G#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> c2:c3
comp_f_g :: comp_f_g:walk_xs -> walk_xs_3 -> comp_f_g:walk_xs
walk_xs_3 :: a -> walk_xs_3
c2 :: c2:c3 -> c2:c3
walk_xs :: comp_f_g:walk_xs
c3 :: c2:c3
MAIN :: Nil:Cons -> c4:c5
c4 :: c4:c5
c5 :: c2:c3 -> c:c1 -> c4:c5
walk#1 :: Nil:Cons -> comp_f_g:walk_xs
comp_f_g#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> Nil:Cons
hole_c:c11_6 :: c:c1
hole_Nil:Cons2_6 :: Nil:Cons
hole_a3_6 :: a
hole_c2:c34_6 :: c2:c3
hole_comp_f_g:walk_xs5_6 :: comp_f_g:walk_xs
hole_walk_xs_36_6 :: walk_xs_3
hole_c4:c57_6 :: c4:c5
gen_c:c18_6 :: Nat -> c:c1
gen_Nil:Cons9_6 :: Nat -> Nil:Cons
gen_c2:c310_6 :: Nat -> c2:c3
gen_comp_f_g:walk_xs11_6 :: Nat -> comp_f_g:walk_xs


Lemmas:
WALK#1(gen_Nil:Cons9_6(n13_6)) -> gen_c:c18_6(n13_6), rt in Omega(1 + n13_6)


Generator Equations:
gen_c:c18_6(0) <=> c
gen_c:c18_6(+(x, 1)) <=> c1(gen_c:c18_6(x))
gen_Nil:Cons9_6(0) <=> Nil
gen_Nil:Cons9_6(+(x, 1)) <=> Cons(hole_a3_6, gen_Nil:Cons9_6(x))
gen_c2:c310_6(0) <=> c3
gen_c2:c310_6(+(x, 1)) <=> c2(gen_c2:c310_6(x))
gen_comp_f_g:walk_xs11_6(0) <=> walk_xs
gen_comp_f_g:walk_xs11_6(+(x, 1)) <=> comp_f_g(gen_comp_f_g:walk_xs11_6(x), walk_xs_3(hole_a3_6))


The following defined symbols remain to be analysed:
COMP_F_G#1, walk#1, comp_f_g#1
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
COMP_F_G#1(gen_comp_f_g:walk_xs11_6(n263_6), walk_xs_3(hole_a3_6), gen_Nil:Cons9_6(b)) -> gen_c2:c310_6(n263_6), rt in Omega(1 + n263_6)

Induction Base:
COMP_F_G#1(gen_comp_f_g:walk_xs11_6(0), walk_xs_3(hole_a3_6), gen_Nil:Cons9_6(b)) ->_R^Omega(1)
c3

Induction Step:
COMP_F_G#1(gen_comp_f_g:walk_xs11_6(+(n263_6, 1)), walk_xs_3(hole_a3_6), gen_Nil:Cons9_6(b)) ->_R^Omega(1)
c2(COMP_F_G#1(gen_comp_f_g:walk_xs11_6(n263_6), walk_xs_3(hole_a3_6), Cons(hole_a3_6, gen_Nil:Cons9_6(b)))) ->_IH
c2(gen_c2:c310_6(c264_6))

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:
WALK#1(Nil) -> c
WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
MAIN(Nil) -> c4
MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
walk#1(Nil) -> walk_xs
walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
main(Nil) -> Nil
main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)

Types:
WALK#1 :: Nil:Cons -> c:c1
Nil :: Nil:Cons
c :: c:c1
Cons :: a -> Nil:Cons -> Nil:Cons
c1 :: c:c1 -> c:c1
COMP_F_G#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> c2:c3
comp_f_g :: comp_f_g:walk_xs -> walk_xs_3 -> comp_f_g:walk_xs
walk_xs_3 :: a -> walk_xs_3
c2 :: c2:c3 -> c2:c3
walk_xs :: comp_f_g:walk_xs
c3 :: c2:c3
MAIN :: Nil:Cons -> c4:c5
c4 :: c4:c5
c5 :: c2:c3 -> c:c1 -> c4:c5
walk#1 :: Nil:Cons -> comp_f_g:walk_xs
comp_f_g#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> Nil:Cons
hole_c:c11_6 :: c:c1
hole_Nil:Cons2_6 :: Nil:Cons
hole_a3_6 :: a
hole_c2:c34_6 :: c2:c3
hole_comp_f_g:walk_xs5_6 :: comp_f_g:walk_xs
hole_walk_xs_36_6 :: walk_xs_3
hole_c4:c57_6 :: c4:c5
gen_c:c18_6 :: Nat -> c:c1
gen_Nil:Cons9_6 :: Nat -> Nil:Cons
gen_c2:c310_6 :: Nat -> c2:c3
gen_comp_f_g:walk_xs11_6 :: Nat -> comp_f_g:walk_xs


Lemmas:
WALK#1(gen_Nil:Cons9_6(n13_6)) -> gen_c:c18_6(n13_6), rt in Omega(1 + n13_6)
COMP_F_G#1(gen_comp_f_g:walk_xs11_6(n263_6), walk_xs_3(hole_a3_6), gen_Nil:Cons9_6(b)) -> gen_c2:c310_6(n263_6), rt in Omega(1 + n263_6)


Generator Equations:
gen_c:c18_6(0) <=> c
gen_c:c18_6(+(x, 1)) <=> c1(gen_c:c18_6(x))
gen_Nil:Cons9_6(0) <=> Nil
gen_Nil:Cons9_6(+(x, 1)) <=> Cons(hole_a3_6, gen_Nil:Cons9_6(x))
gen_c2:c310_6(0) <=> c3
gen_c2:c310_6(+(x, 1)) <=> c2(gen_c2:c310_6(x))
gen_comp_f_g:walk_xs11_6(0) <=> walk_xs
gen_comp_f_g:walk_xs11_6(+(x, 1)) <=> comp_f_g(gen_comp_f_g:walk_xs11_6(x), walk_xs_3(hole_a3_6))


The following defined symbols remain to be analysed:
walk#1, comp_f_g#1
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
walk#1(gen_Nil:Cons9_6(n797_6)) -> gen_comp_f_g:walk_xs11_6(n797_6), rt in Omega(0)

Induction Base:
walk#1(gen_Nil:Cons9_6(0)) ->_R^Omega(0)
walk_xs

Induction Step:
walk#1(gen_Nil:Cons9_6(+(n797_6, 1))) ->_R^Omega(0)
comp_f_g(walk#1(gen_Nil:Cons9_6(n797_6)), walk_xs_3(hole_a3_6)) ->_IH
comp_f_g(gen_comp_f_g:walk_xs11_6(c798_6), walk_xs_3(hole_a3_6))

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:
WALK#1(Nil) -> c
WALK#1(Cons(z0, z1)) -> c1(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) -> c3
MAIN(Nil) -> c4
MAIN(Cons(z0, z1)) -> c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
walk#1(Nil) -> walk_xs
walk#1(Cons(z0, z1)) -> comp_f_g(walk#1(z1), walk_xs_3(z0))
comp_f_g#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(walk_xs, walk_xs_3(z0), z1) -> Cons(z0, z1)
main(Nil) -> Nil
main(Cons(z0, z1)) -> comp_f_g#1(walk#1(z1), walk_xs_3(z0), Nil)

Types:
WALK#1 :: Nil:Cons -> c:c1
Nil :: Nil:Cons
c :: c:c1
Cons :: a -> Nil:Cons -> Nil:Cons
c1 :: c:c1 -> c:c1
COMP_F_G#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> c2:c3
comp_f_g :: comp_f_g:walk_xs -> walk_xs_3 -> comp_f_g:walk_xs
walk_xs_3 :: a -> walk_xs_3
c2 :: c2:c3 -> c2:c3
walk_xs :: comp_f_g:walk_xs
c3 :: c2:c3
MAIN :: Nil:Cons -> c4:c5
c4 :: c4:c5
c5 :: c2:c3 -> c:c1 -> c4:c5
walk#1 :: Nil:Cons -> comp_f_g:walk_xs
comp_f_g#1 :: comp_f_g:walk_xs -> walk_xs_3 -> Nil:Cons -> Nil:Cons
main :: Nil:Cons -> Nil:Cons
hole_c:c11_6 :: c:c1
hole_Nil:Cons2_6 :: Nil:Cons
hole_a3_6 :: a
hole_c2:c34_6 :: c2:c3
hole_comp_f_g:walk_xs5_6 :: comp_f_g:walk_xs
hole_walk_xs_36_6 :: walk_xs_3
hole_c4:c57_6 :: c4:c5
gen_c:c18_6 :: Nat -> c:c1
gen_Nil:Cons9_6 :: Nat -> Nil:Cons
gen_c2:c310_6 :: Nat -> c2:c3
gen_comp_f_g:walk_xs11_6 :: Nat -> comp_f_g:walk_xs


Lemmas:
WALK#1(gen_Nil:Cons9_6(n13_6)) -> gen_c:c18_6(n13_6), rt in Omega(1 + n13_6)
COMP_F_G#1(gen_comp_f_g:walk_xs11_6(n263_6), walk_xs_3(hole_a3_6), gen_Nil:Cons9_6(b)) -> gen_c2:c310_6(n263_6), rt in Omega(1 + n263_6)
walk#1(gen_Nil:Cons9_6(n797_6)) -> gen_comp_f_g:walk_xs11_6(n797_6), rt in Omega(0)


Generator Equations:
gen_c:c18_6(0) <=> c
gen_c:c18_6(+(x, 1)) <=> c1(gen_c:c18_6(x))
gen_Nil:Cons9_6(0) <=> Nil
gen_Nil:Cons9_6(+(x, 1)) <=> Cons(hole_a3_6, gen_Nil:Cons9_6(x))
gen_c2:c310_6(0) <=> c3
gen_c2:c310_6(+(x, 1)) <=> c2(gen_c2:c310_6(x))
gen_comp_f_g:walk_xs11_6(0) <=> walk_xs
gen_comp_f_g:walk_xs11_6(+(x, 1)) <=> comp_f_g(gen_comp_f_g:walk_xs11_6(x), walk_xs_3(hole_a3_6))


The following defined symbols remain to be analysed:
comp_f_g#1
----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
comp_f_g#1(gen_comp_f_g:walk_xs11_6(n1157_6), walk_xs_3(hole_a3_6), gen_Nil:Cons9_6(b)) -> gen_Nil:Cons9_6(+(+(1, n1157_6), b)), rt in Omega(0)

Induction Base:
comp_f_g#1(gen_comp_f_g:walk_xs11_6(0), walk_xs_3(hole_a3_6), gen_Nil:Cons9_6(b)) ->_R^Omega(0)
Cons(hole_a3_6, gen_Nil:Cons9_6(b))

Induction Step:
comp_f_g#1(gen_comp_f_g:walk_xs11_6(+(n1157_6, 1)), walk_xs_3(hole_a3_6), gen_Nil:Cons9_6(b)) ->_R^Omega(0)
comp_f_g#1(gen_comp_f_g:walk_xs11_6(n1157_6), walk_xs_3(hole_a3_6), Cons(hole_a3_6, gen_Nil:Cons9_6(b))) ->_IH
gen_Nil:Cons9_6(+(+(1, +(b, 1)), c1158_6))

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

(26)
BOUNDS(1, INF)