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

(0) Obligation:

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


The TRS R consists of the following rules:

f1 → g1
f1 → g2
f2 → g1
f2 → g2
g1 → h1
g1 → h2
g2 → h1
g2 → h2
h1 → i
h2 → i
e1(h1, h2, x, y, z) → e2(x, x, y, z, z)
e1(x1, x1, x, y, z) → e5(x1, x, y, z)
e2(f1, x, y, z, f2) → e3(x, y, x, y, y, z, y, z, x, y, z)
e2(x, x, y, z, z) → e6(x, y, z)
e2(i, x, y, z, i) → e6(x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) → e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e3(x, y, x, y, y, z, y, z, x, y, z) → e6(x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) → e1(x1, x1, x, y, z)
e4(i, x1, i, x1, i, x1, i, x1, x, y, z) → e5(x1, x, y, z)
e4(x, x, x, x, x, x, x, x, x, x, x) → e6(x, x, x)
e5(i, x, y, z) → e6(x, y, z)

Rewrite Strategy: FULL

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
e1(h1, h2, x, y, z) → e2(x, x, y, z, z)
e2(f1, x, y, z, f2) → e3(x, y, x, y, y, z, y, z, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) → e1(x1, x1, x, y, z)

(2) Obligation:

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


The TRS R consists of the following rules:

h1 → i
f2 → g2
f1 → g1
e4(i, x1, i, x1, i, x1, i, x1, x, y, z) → e5(x1, x, y, z)
f1 → g2
f2 → g1
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) → e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e3(x, y, x, y, y, z, y, z, x, y, z) → e6(x, y, z)
e2(i, x, y, z, i) → e6(x, y, z)
e5(i, x, y, z) → e6(x, y, z)
g2 → h1
g1 → h1
g2 → h2
e1(x1, x1, x, y, z) → e5(x1, x, y, z)
e2(x, x, y, z, z) → e6(x, y, z)
e4(x, x, x, x, x, x, x, x, x, x, x) → e6(x, x, x)
g1 → h2
h2 → i

Rewrite Strategy: FULL

(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

(4) Obligation:

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


The TRS R consists of the following rules:

h1 → i
f2 → g2
f1 → g1
e4(i, x1, i, x1, i, x1, i, x1, x, y, z) → e5(x1, x, y, z)
f1 → g2
f2 → g1
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) → e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e3(x, y, x, y, y, z, y, z, x, y, z) → e6(x, y, z)
e2(i, x, y, z, i) → e6(x, y, z)
e5(i, x, y, z) → e6(x, y, z)
g2 → h1
g1 → h1
g2 → h2
e1(x1, x1, x, y, z) → e5(x1, x, y, z)
e2(x, x, y, z, z) → e6(x, y, z)
e4(x, x, x, x, x, x, x, x, x, x, x) → e6(x, x, x)
g1 → h2
h2 → i

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

h1 → i
f2 → g2
f2 → g1
f1 → g1
f1 → g2
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → e5(z0, z1, z2, z3)
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → e6(z0, z0, z0)
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → e6(z0, z1, z2)
e2(i, z0, z1, z2, i) → e6(z0, z1, z2)
e2(z0, z0, z1, z2, z2) → e6(z0, z1, z2)
e5(i, z0, z1, z2) → e6(z0, z1, z2)
g2 → h1
g2 → h2
g1 → h1
g1 → h2
e1(z0, z0, z1, z2, z3) → e5(z0, z1, z2, z3)
h2 → i

Tuples:

H1 → c
F2 → c1(G2)
F2 → c2(G1)
F1 → c3(G1)
F1 → c4(G2)
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c5(E5(z0, z1, z2, z3))
E4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → c6
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c7(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6))
E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → c8
E2(i, z0, z1, z2, i) → c9
E2(z0, z0, z1, z2, z2) → c10
E5(i, z0, z1, z2) → c11
G2 → c12(H1)
G2 → c13(H2)
G1 → c14(H1)
G1 → c15(H2)
E1(z0, z0, z1, z2, z3) → c16(E5(z0, z1, z2, z3))
H2 → c17

S tuples:

H1 → c
F2 → c1(G2)
F2 → c2(G1)
F1 → c3(G1)
F1 → c4(G2)
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c5(E5(z0, z1, z2, z3))
E4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → c6
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c7(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6))
E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → c8
E2(i, z0, z1, z2, i) → c9
E2(z0, z0, z1, z2, z2) → c10
E5(i, z0, z1, z2) → c11
G2 → c12(H1)
G2 → c13(H2)
G1 → c14(H1)
G1 → c15(H2)
E1(z0, z0, z1, z2, z3) → c16(E5(z0, z1, z2, z3))
H2 → c17

K tuples:none
Defined Rule Symbols:

h1, f2, f1, e4, e3, e2, e5, g2, g1, e1, h2

Defined Pair Symbols:

H1, F2, F1, E4, E3, E2, E5, G2, G1, E1, H2

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17

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

Removed 18 trailing nodes:

E2(z0, z0, z1, z2, z2) → c10
F1 → c3(G1)
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → c7(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6))
F2 → c2(G1)
E1(z0, z0, z1, z2, z3) → c16(E5(z0, z1, z2, z3))
E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → c8
G2 → c13(H2)
E4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → c6
H1 → c
G2 → c12(H1)
F1 → c4(G2)
H2 → c17
F2 → c1(G2)
E5(i, z0, z1, z2) → c11
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → c5(E5(z0, z1, z2, z3))
G1 → c15(H2)
G1 → c14(H1)
E2(i, z0, z1, z2, i) → c9

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

h1 → i
f2 → g2
f2 → g1
f1 → g1
f1 → g2
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3) → e5(z0, z1, z2, z3)
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0) → e6(z0, z0, z0)
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6)
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2) → e6(z0, z1, z2)
e2(i, z0, z1, z2, i) → e6(z0, z1, z2)
e2(z0, z0, z1, z2, z2) → e6(z0, z1, z2)
e5(i, z0, z1, z2) → e6(z0, z1, z2)
g2 → h1
g2 → h2
g1 → h1
g1 → h2
e1(z0, z0, z1, z2, z3) → e5(z0, z1, z2, z3)
h2 → i

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

h1, f2, f1, e4, e3, e2, e5, g2, g1, e1, h2

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(10) BOUNDS(1, 1)