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

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
    ↳3 CpxTrsMatchBoundsTAProof (⇔, 150 ms)
    ↳4 BOUNDS(1, n^1)
    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
        ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳8 CdtProblem
        ↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳10 CdtProblem
        ↳11 CdtUsableRulesProof (⇔, 0 ms)
        ↳12 CdtProblem
        ↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 93 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 23 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳18 CdtProblem
        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 19 ms)
        ↳20 CdtProblem
        ↳21 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳22 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:

active(f(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(f(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(g(X)) → g(active(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))

(2) 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:

g(ok(X)) → ok(g(X))
top(ok(X)) → top(active(X))
active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(mark(X)) → mark(g(X))
top(mark(X)) → top(proper(X))

Rewrite Strategy: INNERMOST

(3) 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 4.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5]
transitions:
ok0(0) → 0
b0() → 0
mark0(0) → 0
c0() → 0
g0(0) → 1
top0(0) → 2
active0(0) → 3
proper0(0) → 4
f0(0, 0, 0) → 5
g1(0) → 6
ok1(6) → 1
active1(0) → 7
top1(7) → 2
c1() → 8
mark1(8) → 3
b1() → 9
ok1(9) → 4
c1() → 10
ok1(10) → 4
f1(0, 0, 0) → 11
ok1(11) → 5
g1(0) → 12
mark1(12) → 1
proper1(0) → 13
top1(13) → 2
ok1(6) → 6
ok1(6) → 12
mark1(8) → 7
ok1(9) → 13
ok1(10) → 13
ok1(11) → 11
mark1(12) → 6
mark1(12) → 12
active2(9) → 14
top2(14) → 2
active2(10) → 14
proper2(8) → 15
top2(15) → 2
c2() → 16
mark2(16) → 14
c2() → 17
ok2(17) → 15
active3(17) → 18
top3(18) → 2
proper3(16) → 19
top3(19) → 2
c3() → 20
ok3(20) → 19
active4(20) → 21
top4(21) → 2

(4) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(ok(z0)) → ok(g(z0))
g(mark(z0)) → mark(g(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c4(TOP(proper(z0)), PROPER(z0))
ACTIVE(b) → c5
PROPER(b) → c6
PROPER(c) → c7
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c4(TOP(proper(z0)), PROPER(z0))
ACTIVE(b) → c5
PROPER(b) → c6
PROPER(c) → c7
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

g, top, active, proper, f

Defined Pair Symbols:

G, TOP, ACTIVE, PROPER, F

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8

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

Removed 3 trailing nodes:

PROPER(c) → c7
PROPER(b) → c6
ACTIVE(b) → c5

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(ok(z0)) → ok(g(z0))
g(mark(z0)) → mark(g(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c4(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
S tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c4(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

g, top, active, proper, f

Defined Pair Symbols:

G, TOP, F

Compound Symbols:

c1, c2, c3, c4, c8

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(ok(z0)) → ok(g(z0))
g(mark(z0)) → mark(g(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))
Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
S tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

g, top, active, proper, f

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c1, c2, c8, c3, c4

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

g(ok(z0)) → ok(g(z0))
g(mark(z0)) → mark(g(z0))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
f(ok(z0), ok(z1), ok(z2)) → ok(f(z0, z1, z2))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
S tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

active, proper

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c1, c2, c8, c3, c4

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

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

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = x1 + x3   
POL(G(x1)) = 0   
POL(TOP(x1)) = 0   
POL(active(x1)) = 0   
POL(b) = 0   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c8(x1)) = x1   
POL(mark(x1)) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
S tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
Defined Rule Symbols:

active, proper

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c1, c2, c8, c3, c4

(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.

G(mark(z0)) → c2(G(z0))
We considered the (Usable) Rules:none
And the Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = 0   
POL(G(x1)) = x1   
POL(TOP(x1)) = 0   
POL(active(x1)) = 0   
POL(b) = 0   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c8(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
S tuples:

G(ok(z0)) → c1(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
G(mark(z0)) → c2(G(z0))
Defined Rule Symbols:

active, proper

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c1, c2, c8, c3, c4

(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.

G(ok(z0)) → c1(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)))
We considered the (Usable) Rules:

active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
And the Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = 0   
POL(G(x1)) = x1   
POL(TOP(x1)) = x1   
POL(active(x1)) = x1   
POL(b) = [1]   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c8(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = [1] + x1   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
S tuples:

TOP(mark(z0)) → c4(TOP(proper(z0)))
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c1(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)))
Defined Rule Symbols:

active, proper

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c1, c2, c8, c3, c4

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

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

TOP(mark(z0)) → c4(TOP(proper(z0)))
We considered the (Usable) Rules:

active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
And the Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2, x3)) = 0   
POL(G(x1)) = 0   
POL(TOP(x1)) = x1   
POL(active(x1)) = x1   
POL(b) = [1]   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c8(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(b) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
Tuples:

G(ok(z0)) → c1(G(z0))
G(mark(z0)) → c2(G(z0))
F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
S tuples:none
K tuples:

F(ok(z0), ok(z1), ok(z2)) → c8(F(z0, z1, z2))
G(mark(z0)) → c2(G(z0))
G(ok(z0)) → c1(G(z0))
TOP(ok(z0)) → c3(TOP(active(z0)))
TOP(mark(z0)) → c4(TOP(proper(z0)))
Defined Rule Symbols:

active, proper

Defined Pair Symbols:

G, F, TOP

Compound Symbols:

c1, c2, c8, c3, c4

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

The set S is empty

(22) BOUNDS(1, 1)