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

0 CpxTRS
    ↳1 CpxTrsMatchBoundsTAProof (⇔, 3 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 22 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)), 114 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 4 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(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)

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
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

(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(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, comp_f_g#1, main

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c, c1, c2, c3, c4, c5

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

Removed 3 trailing nodes:

MAIN(Nil) → c4
COMP_F_G#1(walk_xs, walk_xs_3(z0), z1) → c3
WALK#1(Nil) → c

(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(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)))
MAIN(Cons(z0, z1)) → c5(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil), WALK#1(z1))
S tuples:

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)))
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, comp_f_g#1, main

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c5

(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(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(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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
MAIN(Cons(z0, z1)) → c(WALK#1(z1))
S tuples:

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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
MAIN(Cons(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, c

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

MAIN(Cons(z0, z1)) → c(WALK#1(z1))

(10) 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(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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
S tuples:

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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), 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, c

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

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

MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))

(12) 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(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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
S tuples:

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)))
K tuples:

MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), 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, c

(13) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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)

(14) 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))
Tuples:

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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
S tuples:

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)))
K tuples:

MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
Defined Rule Symbols:

walk#1

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, 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(Cons(z0, z1)) → c1(WALK#1(z1))
We considered the (Usable) Rules:none
And the Tuples:

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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), 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)) = [1] + x2   
POL(MAIN(x1)) = 0   
POL(Nil) = 0   
POL(WALK#1(x1)) = x1   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(comp_f_g(x1, x2)) = 0   
POL(walk#1(x1)) = 0   
POL(walk_xs) = 0   
POL(walk_xs_3(x1)) = 0   

(16) 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))
Tuples:

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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
S tuples:

COMP_F_G#1(comp_f_g(z0, z1), walk_xs_3(z2), z3) → c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
K tuples:

MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
WALK#1(Cons(z0, z1)) → c1(WALK#1(z1))
Defined Rule Symbols:

walk#1

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, 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), walk_xs_3(z2), z3) → c2(COMP_F_G#1(z0, z1, Cons(z2, z3)))
We considered the (Usable) Rules:

walk#1(Nil) → walk_xs
walk#1(Cons(z0, z1)) → comp_f_g(walk#1(z1), walk_xs_3(z0))
And the Tuples:

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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), 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)) = [1] + x1 + x2   
POL(MAIN(x1)) = [1] + x1   
POL(Nil) = [1]   
POL(WALK#1(x1)) = x1   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(comp_f_g(x1, x2)) = x1 + x2   
POL(walk#1(x1)) = x1   
POL(walk_xs) = 0   
POL(walk_xs_3(x1)) = [1]   

(18) 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))
Tuples:

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)))
MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
S tuples:none
K tuples:

MAIN(Cons(z0, z1)) → c(COMP_F_G#1(walk#1(z1), walk_xs_3(z0), Nil))
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)))
Defined Rule Symbols:

walk#1

Defined Pair Symbols:

WALK#1, COMP_F_G#1, MAIN

Compound Symbols:

c1, c2, c

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

The set S is empty

(20) BOUNDS(1, 1)