WORST_CASE(?,O(n^1))
proof of input_5FkrvyjjiF.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) CpxTrsMatchBoundsTAProof [FINISHED, 35 ms]
(4) BOUNDS(1, n^1)


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(0)
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
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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
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(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

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(4)
BOUNDS(1, n^1)