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

0 CpxTRS
    ↳1 CpxTrsMatchBoundsTAProof (⇔, 119 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
        ↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳6 CdtProblem
        ↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳8 CdtProblem
        ↳9 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
        ↳10 CdtProblem
        ↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳12 CdtProblem
        ↳13 CdtUsableRulesProof (⇔, 0 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 148 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 53 ms)
        ↳18 CdtProblem
        ↳19 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳20 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (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)

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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

(2) BOUNDS(1, n^1)

(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(4) 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(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
COMP_F_G#1(cons_x(z0), cons_x(z1), z2) → c5
MAIN(Leaf(z0)) → c6
MAIN(Node(z0, z1)) → c7(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0), WALK#1(z1))
S tuples:

WALK#1(Leaf(z0)) → c
WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
COMP_F_G#1(cons_x(z0), cons_x(z1), z2) → c5
MAIN(Leaf(z0)) → c6
MAIN(Node(z0, z1)) → c7(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0), WALK#1(z1))
K tuples:none
Defined Rule Symbols:

walk#1, comp_f_g#1, main

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

COMP_F_G#1(cons_x(z0), cons_x(z1), z2) → c5
MAIN(Leaf(z0)) → c6
WALK#1(Leaf(z0)) → c

(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(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c7(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0), WALK#1(z1))
S tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c7(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil), WALK#1(z0), WALK#1(z1))
K tuples:none
Defined Rule Symbols:

walk#1, comp_f_g#1, main

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c3, c4, c7

(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(8) 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(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
MAIN(Node(z0, z1)) → c(WALK#1(z0))
MAIN(Node(z0, z1)) → c(WALK#1(z1))
S tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
MAIN(Node(z0, z1)) → c(WALK#1(z0))
MAIN(Node(z0, z1)) → c(WALK#1(z1))
K tuples:none
Defined Rule Symbols:

walk#1, comp_f_g#1, main

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c3, c4, c

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

MAIN(Node(z0, z1)) → c(WALK#1(z0))
MAIN(Node(z0, z1)) → c(WALK#1(z1))

(10) 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(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
S tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
K tuples:none
Defined Rule Symbols:

walk#1, comp_f_g#1, main

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c3, c4, c

(11) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))

(12) 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(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
S tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
K tuples:

MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
Defined Rule Symbols:

walk#1, comp_f_g#1, main

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c3, c4, c

(13) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

main(Leaf(z0)) → Cons(z0, Nil)
main(Node(z0, z1)) → comp_f_g#1(walk#1(z0), walk#1(z1), Nil)

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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))
walk#1(Leaf(z0)) → cons_x(z0)
walk#1(Node(z0, z1)) → comp_f_g(walk#1(z0), walk#1(z1))
Tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
S tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
K tuples:

MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
Defined Rule Symbols:

comp_f_g#1, walk#1

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c3, c4, c

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
We considered the (Usable) Rules:none
And the Tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(COMP_F_G#1(x1, x2, x3)) = 0   
POL(Cons(x1, x2)) = 0   
POL(Leaf(x1)) = 0   
POL(MAIN(x1)) = 0   
POL(Nil) = 0   
POL(Node(x1, x2)) = [1] + x1 + x2   
POL(WALK#1(x1)) = x1   
POL(c(x1)) = x1   
POL(c1(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(comp_f_g(x1, x2)) = 0   
POL(comp_f_g#1(x1, x2, x3)) = 0   
POL(cons_x(x1)) = 0   
POL(walk#1(x1)) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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))
walk#1(Leaf(z0)) → cons_x(z0)
walk#1(Node(z0, z1)) → comp_f_g(walk#1(z0), walk#1(z1))
Tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
S tuples:

COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
K tuples:

MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
Defined Rule Symbols:

comp_f_g#1, walk#1

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c3, c4, c

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
We considered the (Usable) Rules:

walk#1(Node(z0, z1)) → comp_f_g(walk#1(z0), walk#1(z1))
walk#1(Leaf(z0)) → cons_x(z0)
And the Tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(COMP_F_G#1(x1, x2, x3)) = x1 + x2   
POL(Cons(x1, x2)) = 0   
POL(Leaf(x1)) = 0   
POL(MAIN(x1)) = x1   
POL(Nil) = 0   
POL(Node(x1, x2)) = [1] + x1 + x2   
POL(WALK#1(x1)) = 0   
POL(c(x1)) = x1   
POL(c1(x1, x2)) = x1 + x2   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(comp_f_g(x1, x2)) = [1] + x1 + x2   
POL(comp_f_g#1(x1, x2, x3)) = 0   
POL(cons_x(x1)) = 0   
POL(walk#1(x1)) = x1   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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))
walk#1(Leaf(z0)) → cons_x(z0)
walk#1(Node(z0, z1)) → comp_f_g(walk#1(z0), walk#1(z1))
Tuples:

WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
S tuples:none
K tuples:

MAIN(Node(z0, z1)) → c(COMP_F_G#1(walk#1(z0), walk#1(z1), Nil))
WALK#1(Node(z0, z1)) → c1(WALK#1(z0), WALK#1(z1))
COMP_F_G#1(comp_f_g(z0, z1), comp_f_g(z2, z3), z4) → c2(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) → c3(COMP_F_G#1(z0, z1, Cons(z2, z3)))
COMP_F_G#1(cons_x(z0), comp_f_g(z1, z2), z3) → c4(COMP_F_G#1(z1, z2, z3))
Defined Rule Symbols:

comp_f_g#1, walk#1

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c3, c4, c

(19) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(20) BOUNDS(1, 1)