WORST_CASE(Omega(n^1),O(n^1))
proof of input_L3ZHqmhByL.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 97 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), 518 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 3926 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (24) 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(Leaf(x2)) -> cons_x(x2)
   walk#1(Node(x5, x3)) -> comp_f_g(walk#1(x5), walk#1(x3))
   comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) -> comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
   comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) -> comp_f_g#1(x7, x9, Cons(x2, x4))
   comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) -> Cons(x2, comp_f_g#1(x5, x7, x3))
   comp_f_g#1(cons_x(x5), cons_x(x2), x4) -> Cons(x5, Cons(x2, x4))
   main(Leaf(x4)) -> Cons(x4, Nil)
   main(Node(x9, x5)) -> comp_f_g#1(walk#1(x9), walk#1(x5), 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(Leaf(x2)) -> cons_x(x2)
   walk#1(Node(x5, x3)) -> comp_f_g(walk#1(x5), walk#1(x3))
   comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) -> comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
   comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) -> comp_f_g#1(x7, x9, Cons(x2, x4))
   comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) -> Cons(x2, comp_f_g#1(x5, x7, x3))
   comp_f_g#1(cons_x(x5), cons_x(x2), x4) -> Cons(x5, Cons(x2, x4))
   main(Leaf(x4)) -> Cons(x4, Nil)
   main(Node(x9, x5)) -> comp_f_g#1(walk#1(x9), walk#1(x5), 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: 
Leaf0(0) -> 0
cons_x0(0) -> 0
Node0(0, 0) -> 0
comp_f_g0(0, 0) -> 0
Cons0(0, 0) -> 0
Nil0() -> 0
walk#10(0) -> 1
comp_f_g#10(0, 0, 0) -> 2
main0(0) -> 3
cons_x1(0) -> 1
walk#11(0) -> 4
walk#11(0) -> 5
comp_f_g1(4, 5) -> 1
comp_f_g#11(0, 0, 0) -> 6
comp_f_g#11(0, 0, 6) -> 2
Cons1(0, 0) -> 7
comp_f_g#11(0, 0, 7) -> 2
comp_f_g#11(0, 0, 0) -> 8
Cons1(0, 8) -> 2
Cons1(0, 0) -> 9
Cons1(0, 9) -> 2
Nil1() -> 10
Cons1(0, 10) -> 3
walk#11(0) -> 11
walk#11(0) -> 12
Nil1() -> 13
comp_f_g#11(11, 12, 13) -> 3
cons_x1(0) -> 4
cons_x1(0) -> 5
cons_x1(0) -> 11
cons_x1(0) -> 12
comp_f_g1(4, 5) -> 4
comp_f_g1(4, 5) -> 5
comp_f_g1(4, 5) -> 11
comp_f_g1(4, 5) -> 12
comp_f_g#11(0, 0, 6) -> 6
comp_f_g#11(0, 0, 7) -> 6
comp_f_g#11(0, 0, 6) -> 8
Cons1(0, 6) -> 7
Cons1(0, 7) -> 7
comp_f_g#11(0, 0, 7) -> 8
Cons1(0, 8) -> 6
Cons1(0, 8) -> 8
Cons1(0, 6) -> 9
Cons1(0, 7) -> 9
Cons1(0, 9) -> 6
Cons1(0, 9) -> 8
comp_f_g#12(4, 5, 13) -> 14
comp_f_g#12(4, 5, 14) -> 3
Cons2(0, 13) -> 15
comp_f_g#12(4, 5, 15) -> 3
comp_f_g#12(4, 5, 13) -> 16
Cons2(0, 16) -> 3
Cons2(0, 13) -> 17
Cons2(0, 17) -> 3
comp_f_g#12(4, 5, 14) -> 14
comp_f_g#12(4, 5, 15) -> 14
comp_f_g#12(4, 5, 14) -> 16
Cons2(0, 14) -> 15
Cons2(0, 15) -> 15
comp_f_g#12(4, 5, 15) -> 16
Cons2(0, 16) -> 14
Cons2(0, 16) -> 16
Cons2(0, 14) -> 17
Cons2(0, 15) -> 17
Cons2(0, 17) -> 14
Cons2(0, 17) -> 16

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

(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(Leaf(z0)) -> cons_x(z0)
   walk#1(Node(z0, z1)) -> comp_f_g(walk#1(z0), walk#1(z1))
   comp_f_g#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> comp_f_g#1(z0, z1, comp_f_g#1(z2, z3, z4))
   comp_f_g#1(comp_f_g(z0, z1), cons_x(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
   comp_f_g#1(cons_x(z0), comp_f_g(z1, z2), z3) -> Cons(z0, comp_f_g#1(z1, z2, z3))
   comp_f_g#1(cons_x(z0), cons_x(z1), z2) -> Cons(z0, Cons(z1, z2))
   main(Leaf(z0)) -> Cons(z0, Nil)
   main(Node(z0, z1)) -> comp_f_g#1(walk#1(z0), walk#1(z1), Nil)
Tuples:
   WALK#1(Leaf(z0)) -> c
   WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
   WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
   COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
   COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
   COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
   COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
   MAIN(Leaf(z0)) -> c7
   MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
   MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z1))
S tuples:
   WALK#1(Leaf(z0)) -> c
   WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
   WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
   COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
   COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
   COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
   COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
   MAIN(Leaf(z0)) -> c7
   MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
   MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), 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_2, c4_1, c5_1, c6, c7, c8_2, c9_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(Leaf(z0)) -> c
   WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
   WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
   COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
   COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
   COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
   COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
   MAIN(Leaf(z0)) -> c7
   MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
   MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z1))

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

   walk#1(Leaf(z0)) -> cons_x(z0)
   walk#1(Node(z0, z1)) -> comp_f_g(walk#1(z0), walk#1(z1))
   comp_f_g#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> comp_f_g#1(z0, z1, comp_f_g#1(z2, z3, z4))
   comp_f_g#1(comp_f_g(z0, z1), cons_x(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
   comp_f_g#1(cons_x(z0), comp_f_g(z1, z2), z3) -> Cons(z0, comp_f_g#1(z1, z2, z3))
   comp_f_g#1(cons_x(z0), cons_x(z1), z2) -> Cons(z0, Cons(z1, z2))
   main(Leaf(z0)) -> Cons(z0, Nil)
   main(Node(z0, z1)) -> comp_f_g#1(walk#1(z0), walk#1(z1), 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(Leaf(z0)) -> c
   WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
   WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
   COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
   COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
   COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
   COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
   MAIN(Leaf(z0)) -> c7
   MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
   MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z1))

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

   walk#1(Leaf(z0)) -> cons_x(z0)
   walk#1(Node(z0, z1)) -> comp_f_g(walk#1(z0), walk#1(z1))
   comp_f_g#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> comp_f_g#1(z0, z1, comp_f_g#1(z2, z3, z4))
   comp_f_g#1(comp_f_g(z0, z1), cons_x(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
   comp_f_g#1(cons_x(z0), comp_f_g(z1, z2), z3) -> Cons(z0, comp_f_g#1(z1, z2, z3))
   comp_f_g#1(cons_x(z0), cons_x(z1), z2) -> Cons(z0, Cons(z1, z2))
   main(Leaf(z0)) -> Cons(z0, Nil)
   main(Node(z0, z1)) -> comp_f_g#1(walk#1(z0), walk#1(z1), Nil)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
WALK#1(Leaf(z0)) -> c
WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
MAIN(Leaf(z0)) -> c7
MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z1))
walk#1(Leaf(z0)) -> cons_x(z0)
walk#1(Node(z0, z1)) -> comp_f_g(walk#1(z0), walk#1(z1))
comp_f_g#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> comp_f_g#1(z0, z1, comp_f_g#1(z2, z3, z4))
comp_f_g#1(comp_f_g(z0, z1), cons_x(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(cons_x(z0), comp_f_g(z1, z2), z3) -> Cons(z0, comp_f_g#1(z1, z2, z3))
comp_f_g#1(cons_x(z0), cons_x(z1), z2) -> Cons(z0, Cons(z1, z2))
main(Leaf(z0)) -> Cons(z0, Nil)
main(Node(z0, z1)) -> comp_f_g#1(walk#1(z0), walk#1(z1), Nil)

Types:
WALK#1 :: Leaf:Node -> c:c1:c2
Leaf :: a -> Leaf:Node
c :: c:c1:c2
Node :: Leaf:Node -> Leaf:Node -> Leaf:Node
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
COMP_F_G#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> c3:c4:c5:c6
comp_f_g :: comp_f_g:cons_x -> comp_f_g:cons_x -> comp_f_g:cons_x
c3 :: c3:c4:c5:c6 -> c3:c4:c5:c6 -> c3:c4:c5:c6
comp_f_g#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> Cons:Nil
cons_x :: a -> comp_f_g:cons_x
c4 :: c3:c4:c5:c6 -> c3:c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6
MAIN :: Leaf:Node -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
walk#1 :: Leaf:Node -> comp_f_g:cons_x
Nil :: Cons:Nil
c9 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
main :: Leaf:Node -> Cons:Nil
hole_c:c1:c21_10 :: c:c1:c2
hole_Leaf:Node2_10 :: Leaf:Node
hole_a3_10 :: a
hole_c3:c4:c5:c64_10 :: c3:c4:c5:c6
hole_comp_f_g:cons_x5_10 :: comp_f_g:cons_x
hole_Cons:Nil6_10 :: Cons:Nil
hole_c7:c8:c97_10 :: c7:c8:c9
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_Leaf:Node9_10 :: Nat -> Leaf:Node
gen_c3:c4:c5:c610_10 :: Nat -> c3:c4:c5:c6
gen_comp_f_g:cons_x11_10 :: Nat -> comp_f_g:cons_x
gen_Cons:Nil12_10 :: Nat -> Cons:Nil

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

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

They will be analysed ascendingly in the following order:
comp_f_g#1 < COMP_F_G#1

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

(14)
Obligation:
Innermost TRS:
Rules:
WALK#1(Leaf(z0)) -> c
WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
MAIN(Leaf(z0)) -> c7
MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z1))
walk#1(Leaf(z0)) -> cons_x(z0)
walk#1(Node(z0, z1)) -> comp_f_g(walk#1(z0), walk#1(z1))
comp_f_g#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> comp_f_g#1(z0, z1, comp_f_g#1(z2, z3, z4))
comp_f_g#1(comp_f_g(z0, z1), cons_x(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(cons_x(z0), comp_f_g(z1, z2), z3) -> Cons(z0, comp_f_g#1(z1, z2, z3))
comp_f_g#1(cons_x(z0), cons_x(z1), z2) -> Cons(z0, Cons(z1, z2))
main(Leaf(z0)) -> Cons(z0, Nil)
main(Node(z0, z1)) -> comp_f_g#1(walk#1(z0), walk#1(z1), Nil)

Types:
WALK#1 :: Leaf:Node -> c:c1:c2
Leaf :: a -> Leaf:Node
c :: c:c1:c2
Node :: Leaf:Node -> Leaf:Node -> Leaf:Node
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
COMP_F_G#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> c3:c4:c5:c6
comp_f_g :: comp_f_g:cons_x -> comp_f_g:cons_x -> comp_f_g:cons_x
c3 :: c3:c4:c5:c6 -> c3:c4:c5:c6 -> c3:c4:c5:c6
comp_f_g#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> Cons:Nil
cons_x :: a -> comp_f_g:cons_x
c4 :: c3:c4:c5:c6 -> c3:c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6
MAIN :: Leaf:Node -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
walk#1 :: Leaf:Node -> comp_f_g:cons_x
Nil :: Cons:Nil
c9 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
main :: Leaf:Node -> Cons:Nil
hole_c:c1:c21_10 :: c:c1:c2
hole_Leaf:Node2_10 :: Leaf:Node
hole_a3_10 :: a
hole_c3:c4:c5:c64_10 :: c3:c4:c5:c6
hole_comp_f_g:cons_x5_10 :: comp_f_g:cons_x
hole_Cons:Nil6_10 :: Cons:Nil
hole_c7:c8:c97_10 :: c7:c8:c9
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_Leaf:Node9_10 :: Nat -> Leaf:Node
gen_c3:c4:c5:c610_10 :: Nat -> c3:c4:c5:c6
gen_comp_f_g:cons_x11_10 :: Nat -> comp_f_g:cons_x
gen_Cons:Nil12_10 :: Nat -> Cons:Nil


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c1(gen_c:c1:c28_10(x))
gen_Leaf:Node9_10(0) <=> Leaf(hole_a3_10)
gen_Leaf:Node9_10(+(x, 1)) <=> Node(Leaf(hole_a3_10), gen_Leaf:Node9_10(x))
gen_c3:c4:c5:c610_10(0) <=> c6
gen_c3:c4:c5:c610_10(+(x, 1)) <=> c3(c6, gen_c3:c4:c5:c610_10(x))
gen_comp_f_g:cons_x11_10(0) <=> cons_x(hole_a3_10)
gen_comp_f_g:cons_x11_10(+(x, 1)) <=> comp_f_g(cons_x(hole_a3_10), gen_comp_f_g:cons_x11_10(x))
gen_Cons:Nil12_10(0) <=> Nil
gen_Cons:Nil12_10(+(x, 1)) <=> Cons(hole_a3_10, gen_Cons:Nil12_10(x))


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

They will be analysed ascendingly in the following order:
comp_f_g#1 < COMP_F_G#1

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
comp_f_g#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(n65_10), gen_Cons:Nil12_10(c)) -> gen_Cons:Nil12_10(+(+(2, n65_10), c)), rt in Omega(0)

Induction Base:
comp_f_g#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(0), gen_Cons:Nil12_10(c)) ->_R^Omega(0)
Cons(hole_a3_10, Cons(hole_a3_10, gen_Cons:Nil12_10(c)))

Induction Step:
comp_f_g#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(+(n65_10, 1)), gen_Cons:Nil12_10(c)) ->_R^Omega(0)
Cons(hole_a3_10, comp_f_g#1(cons_x(hole_a3_10), gen_comp_f_g:cons_x11_10(n65_10), gen_Cons:Nil12_10(c))) ->_IH
Cons(hole_a3_10, gen_Cons:Nil12_10(+(+(2, c), c66_10)))

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

(16)
Obligation:
Innermost TRS:
Rules:
WALK#1(Leaf(z0)) -> c
WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
MAIN(Leaf(z0)) -> c7
MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z1))
walk#1(Leaf(z0)) -> cons_x(z0)
walk#1(Node(z0, z1)) -> comp_f_g(walk#1(z0), walk#1(z1))
comp_f_g#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> comp_f_g#1(z0, z1, comp_f_g#1(z2, z3, z4))
comp_f_g#1(comp_f_g(z0, z1), cons_x(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(cons_x(z0), comp_f_g(z1, z2), z3) -> Cons(z0, comp_f_g#1(z1, z2, z3))
comp_f_g#1(cons_x(z0), cons_x(z1), z2) -> Cons(z0, Cons(z1, z2))
main(Leaf(z0)) -> Cons(z0, Nil)
main(Node(z0, z1)) -> comp_f_g#1(walk#1(z0), walk#1(z1), Nil)

Types:
WALK#1 :: Leaf:Node -> c:c1:c2
Leaf :: a -> Leaf:Node
c :: c:c1:c2
Node :: Leaf:Node -> Leaf:Node -> Leaf:Node
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
COMP_F_G#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> c3:c4:c5:c6
comp_f_g :: comp_f_g:cons_x -> comp_f_g:cons_x -> comp_f_g:cons_x
c3 :: c3:c4:c5:c6 -> c3:c4:c5:c6 -> c3:c4:c5:c6
comp_f_g#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> Cons:Nil
cons_x :: a -> comp_f_g:cons_x
c4 :: c3:c4:c5:c6 -> c3:c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6
MAIN :: Leaf:Node -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
walk#1 :: Leaf:Node -> comp_f_g:cons_x
Nil :: Cons:Nil
c9 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
main :: Leaf:Node -> Cons:Nil
hole_c:c1:c21_10 :: c:c1:c2
hole_Leaf:Node2_10 :: Leaf:Node
hole_a3_10 :: a
hole_c3:c4:c5:c64_10 :: c3:c4:c5:c6
hole_comp_f_g:cons_x5_10 :: comp_f_g:cons_x
hole_Cons:Nil6_10 :: Cons:Nil
hole_c7:c8:c97_10 :: c7:c8:c9
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_Leaf:Node9_10 :: Nat -> Leaf:Node
gen_c3:c4:c5:c610_10 :: Nat -> c3:c4:c5:c6
gen_comp_f_g:cons_x11_10 :: Nat -> comp_f_g:cons_x
gen_Cons:Nil12_10 :: Nat -> Cons:Nil


Lemmas:
comp_f_g#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(n65_10), gen_Cons:Nil12_10(c)) -> gen_Cons:Nil12_10(+(+(2, n65_10), c)), rt in Omega(0)


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c1(gen_c:c1:c28_10(x))
gen_Leaf:Node9_10(0) <=> Leaf(hole_a3_10)
gen_Leaf:Node9_10(+(x, 1)) <=> Node(Leaf(hole_a3_10), gen_Leaf:Node9_10(x))
gen_c3:c4:c5:c610_10(0) <=> c6
gen_c3:c4:c5:c610_10(+(x, 1)) <=> c3(c6, gen_c3:c4:c5:c610_10(x))
gen_comp_f_g:cons_x11_10(0) <=> cons_x(hole_a3_10)
gen_comp_f_g:cons_x11_10(+(x, 1)) <=> comp_f_g(cons_x(hole_a3_10), gen_comp_f_g:cons_x11_10(x))
gen_Cons:Nil12_10(0) <=> Nil
gen_Cons:Nil12_10(+(x, 1)) <=> Cons(hole_a3_10, gen_Cons:Nil12_10(x))


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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
COMP_F_G#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(+(1, n7341_10)), gen_Cons:Nil12_10(c)) -> *13_10, rt in Omega(n7341_10)

Induction Base:
COMP_F_G#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(+(1, 0)), gen_Cons:Nil12_10(c))

Induction Step:
COMP_F_G#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(+(1, +(n7341_10, 1))), gen_Cons:Nil12_10(c)) ->_R^Omega(1)
c5(COMP_F_G#1(cons_x(hole_a3_10), gen_comp_f_g:cons_x11_10(+(1, n7341_10)), gen_Cons:Nil12_10(c))) ->_IH
c5(*13_10)

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

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
WALK#1(Leaf(z0)) -> c
WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
MAIN(Leaf(z0)) -> c7
MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z1))
walk#1(Leaf(z0)) -> cons_x(z0)
walk#1(Node(z0, z1)) -> comp_f_g(walk#1(z0), walk#1(z1))
comp_f_g#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> comp_f_g#1(z0, z1, comp_f_g#1(z2, z3, z4))
comp_f_g#1(comp_f_g(z0, z1), cons_x(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(cons_x(z0), comp_f_g(z1, z2), z3) -> Cons(z0, comp_f_g#1(z1, z2, z3))
comp_f_g#1(cons_x(z0), cons_x(z1), z2) -> Cons(z0, Cons(z1, z2))
main(Leaf(z0)) -> Cons(z0, Nil)
main(Node(z0, z1)) -> comp_f_g#1(walk#1(z0), walk#1(z1), Nil)

Types:
WALK#1 :: Leaf:Node -> c:c1:c2
Leaf :: a -> Leaf:Node
c :: c:c1:c2
Node :: Leaf:Node -> Leaf:Node -> Leaf:Node
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
COMP_F_G#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> c3:c4:c5:c6
comp_f_g :: comp_f_g:cons_x -> comp_f_g:cons_x -> comp_f_g:cons_x
c3 :: c3:c4:c5:c6 -> c3:c4:c5:c6 -> c3:c4:c5:c6
comp_f_g#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> Cons:Nil
cons_x :: a -> comp_f_g:cons_x
c4 :: c3:c4:c5:c6 -> c3:c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6
MAIN :: Leaf:Node -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
walk#1 :: Leaf:Node -> comp_f_g:cons_x
Nil :: Cons:Nil
c9 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
main :: Leaf:Node -> Cons:Nil
hole_c:c1:c21_10 :: c:c1:c2
hole_Leaf:Node2_10 :: Leaf:Node
hole_a3_10 :: a
hole_c3:c4:c5:c64_10 :: c3:c4:c5:c6
hole_comp_f_g:cons_x5_10 :: comp_f_g:cons_x
hole_Cons:Nil6_10 :: Cons:Nil
hole_c7:c8:c97_10 :: c7:c8:c9
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_Leaf:Node9_10 :: Nat -> Leaf:Node
gen_c3:c4:c5:c610_10 :: Nat -> c3:c4:c5:c6
gen_comp_f_g:cons_x11_10 :: Nat -> comp_f_g:cons_x
gen_Cons:Nil12_10 :: Nat -> Cons:Nil


Lemmas:
comp_f_g#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(n65_10), gen_Cons:Nil12_10(c)) -> gen_Cons:Nil12_10(+(+(2, n65_10), c)), rt in Omega(0)


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c1(gen_c:c1:c28_10(x))
gen_Leaf:Node9_10(0) <=> Leaf(hole_a3_10)
gen_Leaf:Node9_10(+(x, 1)) <=> Node(Leaf(hole_a3_10), gen_Leaf:Node9_10(x))
gen_c3:c4:c5:c610_10(0) <=> c6
gen_c3:c4:c5:c610_10(+(x, 1)) <=> c3(c6, gen_c3:c4:c5:c610_10(x))
gen_comp_f_g:cons_x11_10(0) <=> cons_x(hole_a3_10)
gen_comp_f_g:cons_x11_10(+(x, 1)) <=> comp_f_g(cons_x(hole_a3_10), gen_comp_f_g:cons_x11_10(x))
gen_Cons:Nil12_10(0) <=> Nil
gen_Cons:Nil12_10(+(x, 1)) <=> Cons(hole_a3_10, gen_Cons:Nil12_10(x))


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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
WALK#1(Leaf(z0)) -> c
WALK#1(Node(z0, z1)) -> c1(WALK#1(z0))
WALK#1(Node(z0, z1)) -> c2(WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> c3(COMP_F_G#1(z0, z1, comp_f_g#1(z2, z3, z4)), COMP_F_G#1(z2, z3, z4))
COMP_F_G#1(comp_f_g(z0, z1), cons_x(z2), z3) -> c4(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) -> c5(COMP_F_G#1(z1, z2, z3))
COMP_F_G#1(cons_x(z0), cons_x(z1), z2) -> c6
MAIN(Leaf(z0)) -> c7
MAIN(Node(z0, z1)) -> c8(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0))
MAIN(Node(z0, z1)) -> c9(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z1))
walk#1(Leaf(z0)) -> cons_x(z0)
walk#1(Node(z0, z1)) -> comp_f_g(walk#1(z0), walk#1(z1))
comp_f_g#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) -> comp_f_g#1(z0, z1, comp_f_g#1(z2, z3, z4))
comp_f_g#1(comp_f_g(z0, z1), cons_x(z2), z3) -> comp_f_g#1(z0, z1, Cons(z2, z3))
comp_f_g#1(cons_x(z0), comp_f_g(z1, z2), z3) -> Cons(z0, comp_f_g#1(z1, z2, z3))
comp_f_g#1(cons_x(z0), cons_x(z1), z2) -> Cons(z0, Cons(z1, z2))
main(Leaf(z0)) -> Cons(z0, Nil)
main(Node(z0, z1)) -> comp_f_g#1(walk#1(z0), walk#1(z1), Nil)

Types:
WALK#1 :: Leaf:Node -> c:c1:c2
Leaf :: a -> Leaf:Node
c :: c:c1:c2
Node :: Leaf:Node -> Leaf:Node -> Leaf:Node
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
COMP_F_G#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> c3:c4:c5:c6
comp_f_g :: comp_f_g:cons_x -> comp_f_g:cons_x -> comp_f_g:cons_x
c3 :: c3:c4:c5:c6 -> c3:c4:c5:c6 -> c3:c4:c5:c6
comp_f_g#1 :: comp_f_g:cons_x -> comp_f_g:cons_x -> Cons:Nil -> Cons:Nil
cons_x :: a -> comp_f_g:cons_x
c4 :: c3:c4:c5:c6 -> c3:c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6
MAIN :: Leaf:Node -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
walk#1 :: Leaf:Node -> comp_f_g:cons_x
Nil :: Cons:Nil
c9 :: c3:c4:c5:c6 -> c:c1:c2 -> c7:c8:c9
main :: Leaf:Node -> Cons:Nil
hole_c:c1:c21_10 :: c:c1:c2
hole_Leaf:Node2_10 :: Leaf:Node
hole_a3_10 :: a
hole_c3:c4:c5:c64_10 :: c3:c4:c5:c6
hole_comp_f_g:cons_x5_10 :: comp_f_g:cons_x
hole_Cons:Nil6_10 :: Cons:Nil
hole_c7:c8:c97_10 :: c7:c8:c9
gen_c:c1:c28_10 :: Nat -> c:c1:c2
gen_Leaf:Node9_10 :: Nat -> Leaf:Node
gen_c3:c4:c5:c610_10 :: Nat -> c3:c4:c5:c6
gen_comp_f_g:cons_x11_10 :: Nat -> comp_f_g:cons_x
gen_Cons:Nil12_10 :: Nat -> Cons:Nil


Lemmas:
comp_f_g#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(n65_10), gen_Cons:Nil12_10(c)) -> gen_Cons:Nil12_10(+(+(2, n65_10), c)), rt in Omega(0)
COMP_F_G#1(gen_comp_f_g:cons_x11_10(0), gen_comp_f_g:cons_x11_10(+(1, n7341_10)), gen_Cons:Nil12_10(c)) -> *13_10, rt in Omega(n7341_10)


Generator Equations:
gen_c:c1:c28_10(0) <=> c
gen_c:c1:c28_10(+(x, 1)) <=> c1(gen_c:c1:c28_10(x))
gen_Leaf:Node9_10(0) <=> Leaf(hole_a3_10)
gen_Leaf:Node9_10(+(x, 1)) <=> Node(Leaf(hole_a3_10), gen_Leaf:Node9_10(x))
gen_c3:c4:c5:c610_10(0) <=> c6
gen_c3:c4:c5:c610_10(+(x, 1)) <=> c3(c6, gen_c3:c4:c5:c610_10(x))
gen_comp_f_g:cons_x11_10(0) <=> cons_x(hole_a3_10)
gen_comp_f_g:cons_x11_10(+(x, 1)) <=> comp_f_g(cons_x(hole_a3_10), gen_comp_f_g:cons_x11_10(x))
gen_Cons:Nil12_10(0) <=> Nil
gen_Cons:Nil12_10(+(x, 1)) <=> Cons(hole_a3_10, gen_Cons:Nil12_10(x))


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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
walk#1(gen_Leaf:Node9_10(n44526_10)) -> gen_comp_f_g:cons_x11_10(n44526_10), rt in Omega(0)

Induction Base:
walk#1(gen_Leaf:Node9_10(0)) ->_R^Omega(0)
cons_x(hole_a3_10)

Induction Step:
walk#1(gen_Leaf:Node9_10(+(n44526_10, 1))) ->_R^Omega(0)
comp_f_g(walk#1(Leaf(hole_a3_10)), walk#1(gen_Leaf:Node9_10(n44526_10))) ->_R^Omega(0)
comp_f_g(cons_x(hole_a3_10), walk#1(gen_Leaf:Node9_10(n44526_10))) ->_IH
comp_f_g(cons_x(hole_a3_10), gen_comp_f_g:cons_x11_10(c44527_10))

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

(24)
BOUNDS(1, INF)