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), 0 ms)
    ↳4 CdtProblem
        ↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳6 CdtProblem
        ↳7 CdtUsableRulesProof (⇔, 0 ms)
        ↳8 CdtProblem
        ↳9 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 101 ms)
        ↳10 CdtProblem
        ↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳12 CdtProblem
        ↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳14 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:

norm(nil) → 0
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0) → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

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 1.

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
00() → 0
g0(0, 0) → 0
s0(0) → 0
norm0(0) → 1
f0(0, 0) → 2
rem0(0, 0) → 3
01() → 1
norm1(0) → 4
s1(4) → 1
nil1() → 5
g1(5, 0) → 2
f1(0, 0) → 6
g1(6, 0) → 2
nil1() → 3
g1(0, 0) → 3
rem1(0, 0) → 3
01() → 4
s1(4) → 4
g1(5, 0) → 6
g1(6, 0) → 6

(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:

norm(nil) → 0
norm(g(z0, z1)) → s(norm(z0))
f(z0, nil) → g(nil, z0)
f(z0, g(z1, z2)) → g(f(z0, z1), z2)
rem(nil, z0) → nil
rem(g(z0, z1), 0) → g(z0, z1)
rem(g(z0, z1), s(z2)) → rem(z0, z2)
Tuples:

NORM(nil) → c
NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, nil) → c2
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(nil, z0) → c4
REM(g(z0, z1), 0) → c5
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:

NORM(nil) → c
NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, nil) → c2
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(nil, z0) → c4
REM(g(z0, z1), 0) → c5
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
K tuples:none
Defined Rule Symbols:

norm, f, rem

Defined Pair Symbols:

NORM, F, REM

Compound Symbols:

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

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

Removed 4 trailing nodes:

NORM(nil) → c
F(z0, nil) → c2
REM(g(z0, z1), 0) → c5
REM(nil, z0) → c4

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

norm(nil) → 0
norm(g(z0, z1)) → s(norm(z0))
f(z0, nil) → g(nil, z0)
f(z0, g(z1, z2)) → g(f(z0, z1), z2)
rem(nil, z0) → nil
rem(g(z0, z1), 0) → g(z0, z1)
rem(g(z0, z1), s(z2)) → rem(z0, z2)
Tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
K tuples:none
Defined Rule Symbols:

norm, f, rem

Defined Pair Symbols:

NORM, F, REM

Compound Symbols:

c1, c3, c6

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

norm(nil) → 0
norm(g(z0, z1)) → s(norm(z0))
f(z0, nil) → g(nil, z0)
f(z0, g(z1, z2)) → g(f(z0, z1), z2)
rem(nil, z0) → nil
rem(g(z0, z1), 0) → g(z0, z1)
rem(g(z0, z1), s(z2)) → rem(z0, z2)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

NORM, F, REM

Compound Symbols:

c1, c3, c6

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

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

REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = 0   
POL(NORM(x1)) = 0   
POL(REM(x1, x2)) = x2   
POL(c1(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(g(x1, x2)) = 0   
POL(s(x1)) = [1] + x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
K tuples:

REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

NORM, F, REM

Compound Symbols:

c1, c3, c6

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

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

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = [2]x2   
POL(NORM(x1)) = x1   
POL(REM(x1, x2)) = 0   
POL(c1(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(g(x1, x2)) = [1] + x1   
POL(s(x1)) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:none
K tuples:

REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
NORM(g(z0, z1)) → c1(NORM(z0))
F(z0, g(z1, z2)) → c3(F(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

NORM, F, REM

Compound Symbols:

c1, c3, c6

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

The set S is empty

(14) BOUNDS(1, 1)