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

0 CpxTRS
    ↳1 CpxTrsMatchBoundsTAProof (⇔, 12 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 24 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)), 115 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 21 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:

revApp#2(Nil, x16) → x16
revApp#2(Cons(x6, x4), x2) → revApp#2(x4, Cons(x6, x2))
dfsAcc#3(Leaf(x8), x16) → Cons(x8, x16)
dfsAcc#3(Node(x6, x4), x2) → dfsAcc#3(x4, dfsAcc#3(x6, x2))
main(x1) → revApp#2(dfsAcc#3(x1, Nil), 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:
Nil0() → 0
Cons0(0, 0) → 0
Leaf0(0) → 0
Node0(0, 0) → 0
revApp#20(0, 0) → 1
dfsAcc#30(0, 0) → 2
main0(0) → 3
Cons1(0, 0) → 4
revApp#21(0, 4) → 1
Cons1(0, 0) → 2
dfsAcc#31(0, 0) → 5
dfsAcc#31(0, 5) → 2
Nil1() → 7
dfsAcc#31(0, 7) → 6
Nil1() → 8
revApp#21(6, 8) → 3
Cons1(0, 4) → 4
Cons1(0, 5) → 2
Cons1(0, 0) → 5
Cons1(0, 7) → 6
dfsAcc#31(0, 5) → 5
dfsAcc#31(0, 7) → 5
dfsAcc#31(0, 5) → 6
Cons2(0, 8) → 9
revApp#22(7, 9) → 3
Cons1(0, 5) → 5
Cons1(0, 7) → 5
Cons1(0, 5) → 6
revApp#22(5, 9) → 3
Cons2(0, 9) → 9
revApp#22(0, 9) → 3
Cons1(0, 9) → 4
revApp#21(0, 4) → 3
0 → 1
4 → 1
4 → 3
9 → 3

(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:

revApp#2(Nil, z0) → z0
revApp#2(Cons(z0, z1), z2) → revApp#2(z1, Cons(z0, z2))
dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
main(z0) → revApp#2(dfsAcc#3(z0, Nil), Nil)
Tuples:

REVAPP#2(Nil, z0) → c
REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Leaf(z0), z1) → c2
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c4(REVAPP#2(dfsAcc#3(z0, Nil), Nil), DFSACC#3(z0, Nil))
S tuples:

REVAPP#2(Nil, z0) → c
REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Leaf(z0), z1) → c2
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c4(REVAPP#2(dfsAcc#3(z0, Nil), Nil), DFSACC#3(z0, Nil))
K tuples:none
Defined Rule Symbols:

revApp#2, dfsAcc#3, main

Defined Pair Symbols:

REVAPP#2, DFSACC#3, MAIN

Compound Symbols:

c, c1, c2, c3, c4

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

Removed 2 trailing nodes:

REVAPP#2(Nil, z0) → c
DFSACC#3(Leaf(z0), z1) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

revApp#2(Nil, z0) → z0
revApp#2(Cons(z0, z1), z2) → revApp#2(z1, Cons(z0, z2))
dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
main(z0) → revApp#2(dfsAcc#3(z0, Nil), Nil)
Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c4(REVAPP#2(dfsAcc#3(z0, Nil), Nil), DFSACC#3(z0, Nil))
S tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c4(REVAPP#2(dfsAcc#3(z0, Nil), Nil), DFSACC#3(z0, Nil))
K tuples:none
Defined Rule Symbols:

revApp#2, dfsAcc#3, main

Defined Pair Symbols:

REVAPP#2, DFSACC#3, MAIN

Compound Symbols:

c1, c3, c4

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

revApp#2(Nil, z0) → z0
revApp#2(Cons(z0, z1), z2) → revApp#2(z1, Cons(z0, z2))
dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
main(z0) → revApp#2(dfsAcc#3(z0, Nil), Nil)
Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
MAIN(z0) → c(DFSACC#3(z0, Nil))
S tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
MAIN(z0) → c(DFSACC#3(z0, Nil))
K tuples:none
Defined Rule Symbols:

revApp#2, dfsAcc#3, main

Defined Pair Symbols:

REVAPP#2, DFSACC#3, MAIN

Compound Symbols:

c1, c3, c

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

MAIN(z0) → c(DFSACC#3(z0, Nil))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

revApp#2(Nil, z0) → z0
revApp#2(Cons(z0, z1), z2) → revApp#2(z1, Cons(z0, z2))
dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
main(z0) → revApp#2(dfsAcc#3(z0, Nil), Nil)
Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
S tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
K tuples:none
Defined Rule Symbols:

revApp#2, dfsAcc#3, main

Defined Pair Symbols:

REVAPP#2, DFSACC#3, MAIN

Compound Symbols:

c1, c3, c

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

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

MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

revApp#2(Nil, z0) → z0
revApp#2(Cons(z0, z1), z2) → revApp#2(z1, Cons(z0, z2))
dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
main(z0) → revApp#2(dfsAcc#3(z0, Nil), Nil)
Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
S tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
K tuples:

MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
Defined Rule Symbols:

revApp#2, dfsAcc#3, main

Defined Pair Symbols:

REVAPP#2, DFSACC#3, MAIN

Compound Symbols:

c1, c3, c

(13) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

revApp#2(Nil, z0) → z0
revApp#2(Cons(z0, z1), z2) → revApp#2(z1, Cons(z0, z2))
main(z0) → revApp#2(dfsAcc#3(z0, Nil), Nil)

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
S tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
K tuples:

MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
Defined Rule Symbols:

dfsAcc#3

Defined Pair Symbols:

REVAPP#2, DFSACC#3, MAIN

Compound Symbols:

c1, c3, 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.

DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(Cons(x1, x2)) = 0   
POL(DFSACC#3(x1, x2)) = x1   
POL(Leaf(x1)) = [3] + x1   
POL(MAIN(x1)) = [2] + [3]x1   
POL(Nil) = 0   
POL(Node(x1, x2)) = [3] + x1 + x2   
POL(REVAPP#2(x1, x2)) = [1]   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   
POL(dfsAcc#3(x1, x2)) = [2] + [3]x1 + x2   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
S tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
K tuples:

MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
Defined Rule Symbols:

dfsAcc#3

Defined Pair Symbols:

REVAPP#2, DFSACC#3, MAIN

Compound Symbols:

c1, c3, 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.

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
We considered the (Usable) Rules:

dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
And the Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(Cons(x1, x2)) = [1] + x2   
POL(DFSACC#3(x1, x2)) = x1   
POL(Leaf(x1)) = [1] + x1   
POL(MAIN(x1)) = [1] + x1   
POL(Nil) = 0   
POL(Node(x1, x2)) = [1] + x1 + x2   
POL(REVAPP#2(x1, x2)) = x1   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c3(x1, x2)) = x1 + x2   
POL(dfsAcc#3(x1, x2)) = x1 + x2   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

dfsAcc#3(Leaf(z0), z1) → Cons(z0, z1)
dfsAcc#3(Node(z0, z1), z2) → dfsAcc#3(z1, dfsAcc#3(z0, z2))
Tuples:

REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
S tuples:none
K tuples:

MAIN(z0) → c(REVAPP#2(dfsAcc#3(z0, Nil), Nil))
DFSACC#3(Node(z0, z1), z2) → c3(DFSACC#3(z1, dfsAcc#3(z0, z2)), DFSACC#3(z0, z2))
REVAPP#2(Cons(z0, z1), z2) → c1(REVAPP#2(z1, Cons(z0, z2)))
Defined Rule Symbols:

dfsAcc#3

Defined Pair Symbols:

REVAPP#2, DFSACC#3, MAIN

Compound Symbols:

c1, c3, c

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

The set S is empty

(20) BOUNDS(1, 1)