The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^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(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))

(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:

top(ok(X)) → top(active(X))
proper(tt) → ok(tt)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
proper(0) → ok(0)
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
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 2.

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, 6]
transitions:
ok0(0) → 0
active0(0) → 0
tt0() → 0
mark0(0) → 0
00() → 0
top0(0) → 1
proper0(0) → 2
U110(0, 0, 0) → 3
U120(0, 0, 0) → 4
s0(0) → 5
plus0(0, 0) → 6
active1(0) → 7
top1(7) → 1
tt1() → 8
ok1(8) → 2
U111(0, 0, 0) → 9
mark1(9) → 3
U121(0, 0, 0) → 10
ok1(10) → 4
U111(0, 0, 0) → 11
ok1(11) → 3
U121(0, 0, 0) → 12
mark1(12) → 4
s1(0) → 13
ok1(13) → 5
s1(0) → 14
mark1(14) → 5
plus1(0, 0) → 15
mark1(15) → 6
01() → 16
ok1(16) → 2
plus1(0, 0) → 17
ok1(17) → 6
proper1(0) → 18
top1(18) → 1
ok1(8) → 18
mark1(9) → 9
mark1(9) → 11
ok1(10) → 10
ok1(10) → 12
ok1(11) → 9
ok1(11) → 11
mark1(12) → 10
mark1(12) → 12
ok1(13) → 13
ok1(13) → 14
mark1(14) → 13
mark1(14) → 14
mark1(15) → 15
mark1(15) → 17
ok1(16) → 18
ok1(17) → 15
ok1(17) → 17
active2(8) → 19
top2(19) → 1
active2(16) → 19

(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:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(tt) → c2
PROPER(0) → c3
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
PROPER(tt) → c2
PROPER(0) → c3
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

K tuples:none
Defined Rule Symbols:

top, proper, U11, U12, s, plus

Defined Pair Symbols:

TOP, PROPER, U11', U12', S, PLUS

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

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

Removed 3 trailing nodes:

PROPER(0) → c3
PROPER(tt) → c2
TOP(ok(z0)) → c(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))

Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

S tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

K tuples:none
Defined Rule Symbols:

top, proper, U11, U12, s, plus

Defined Pair Symbols:

TOP, U11', U12', S, PLUS

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c10, c11, c12

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

top, proper, U11, U12, s, plus

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

(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.

U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]   
POL(PLUS(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = [3]x₂   
POL(U12'(x₁, x₂, x₃)) = [3]x₂ + [3]x₃   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(ok(x₁)) = [3] + x₁   
POL(proper(x₁)) = [2] + [3]x₁   
POL(tt) = [2]   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

(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.

TOP(mark(z0)) → c1(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(PLUS(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(TOP(x₁)) = x₁   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1]   
POL(ok(x₁)) = 0   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

(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.

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(PLUS(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₁   
POL(U12'(x₁, x₂, x₃)) = x₃   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = [2]   
POL(tt) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

(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.

PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(PLUS(x₁, x₂)) = x₁   
POL(S(x₁)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = [2]x₂ + [2]x₃   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = [2] + [2]x₁   
POL(tt) = [1]   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

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

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

U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(mark(z0)) → c9(S(z0))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(PLUS(x₁, x₂)) = x₁   
POL(S(x₁)) = x₁   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₁ + x₂   
POL(U12'(x₁, x₂, x₃)) = x₁ + x₃   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

S(ok(z0)) → c8(S(z0))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(mark(z0)) → c9(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

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

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

S(ok(z0)) → c8(S(z0))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(PLUS(x₁, x₂)) = 0   
POL(S(x₁)) = x₁   
POL(TOP(x₁)) = x₁   
POL(U11'(x₁, x₂, x₃)) = [2]x₂   
POL(U12'(x₁, x₂, x₃)) = [2]x₃   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [2] + x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = [1] + x₁   
POL(tt) = 0   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(mark(z0)) → c9(S(z0))
S(ok(z0)) → c8(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

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

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

PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(PLUS(x₁, x₂)) = x₂   
POL(S(x₁)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(ok(z0)) → c8(S(z0))
S(mark(z0)) → c9(S(z0))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:none
K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c5(U11'(z0, z1, z2))
U12'(ok(z0), ok(z1), ok(z2)) → c6(U12'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U11'(mark(z0), z1, z2) → c4(U11'(z0, z1, z2))
PLUS(mark(z0), z1) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
U12'(mark(z0), z1, z2) → c7(U12'(z0, z1, z2))
S(mark(z0)) → c9(S(z0))
S(ok(z0)) → c8(S(z0))
PLUS(z0, mark(z1)) → c12(PLUS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', S, PLUS, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c1

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

The set S is empty

(28) BOUNDS(1, 1)