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

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 14 ms)
↳2 CpxTRS
↳3 RcToIrcProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxTRS
↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 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(a, x) → g1(x, x)
f1(x, a) → g2(x, x)
f2(a, x) → g1(x, x)
f2(x, a) → g2(x, x)
g1(a, x) → h1(x)
g1(x, a) → h2(x)
g2(a, x) → h1(x)
g2(x, a) → h2(x)
h1(a) → i
h2(a) → i
e1(h1(w), h2(w), x, y, z, w) → e2(x, x, y, z, z, w)
e1(x1, x1, x, y, z, a) → e5(x1, x, y, z)
e2(f1(w, w), x, y, z, f2(w, w), w) → e3(x, y, x, y, y, z, y, z, x, y, z, w)
e2(x, x, y, z, z, a) → e6(x, y, z)
e2(i, x, y, z, i, a) → e6(x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w) → e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w)
e3(x, y, x, y, y, z, y, z, x, y, z, a) → e6(x, y, z)
e4(g1(w, w), x1, g2(w, w), x1, g1(w, w), x1, g2(w, w), x1, x, y, z, w) → e1(x1, x1, x, y, z, w)
e4(i, x1, i, x1, i, x1, i, x1, x, y, z, a) → e5(x1, x, y, z)
e4(x, x, x, x, x, x, x, x, x, x, x, a) → 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(w), h2(w), x, y, z, w) → e2(x, x, y, z, z, w)
e2(f1(w, w), x, y, z, f2(w, w), w) → e3(x, y, x, y, y, z, y, z, x, y, z, w)
e4(g1(w, w), x1, g2(w, w), x1, g1(w, w), x1, g2(w, w), x1, x, y, z, w) → e1(x1, x1, x, y, z, w)

(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:

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

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:

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

e1(z0, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3)
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → e6(z0, z1, z2)
g1(a, z0) → h1(z0)
g1(z0, a) → h2(z0)
f1(a, z0) → g1(z0, z0)
f1(z0, a) → g2(z0, z0)
f2(a, z0) → g1(z0, z0)
f2(z0, a) → g2(z0, z0)
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → e6(z0, z0, z0)
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3)
e5(i, z0, z1, z2) → e6(z0, z1, z2)
h1(a) → i
h2(a) → i
g2(a, z0) → h1(z0)
g2(z0, a) → h2(z0)
e2(i, z0, z1, z2, i, a) → e6(z0, z1, z2)
e2(z0, z0, z1, z2, z2, a) → e6(z0, z1, z2)
Tuples:

E1(z0, z0, z1, z2, z3, a) → c(E5(z0, z1, z2, z3))
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c1(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7))
E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → c2
G1(a, z0) → c3(H1(z0))
G1(z0, a) → c4(H2(z0))
F1(a, z0) → c5(G1(z0, z0))
F1(z0, a) → c6(G2(z0, z0))
F2(a, z0) → c7(G1(z0, z0))
F2(z0, a) → c8(G2(z0, z0))
E4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → c9
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → c10(E5(z0, z1, z2, z3))
E5(i, z0, z1, z2) → c11
H1(a) → c12
H2(a) → c13
G2(a, z0) → c14(H1(z0))
G2(z0, a) → c15(H2(z0))
E2(i, z0, z1, z2, i, a) → c16
E2(z0, z0, z1, z2, z2, a) → c17
S tuples:

E1(z0, z0, z1, z2, z3, a) → c(E5(z0, z1, z2, z3))
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c1(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7))
E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → c2
G1(a, z0) → c3(H1(z0))
G1(z0, a) → c4(H2(z0))
F1(a, z0) → c5(G1(z0, z0))
F1(z0, a) → c6(G2(z0, z0))
F2(a, z0) → c7(G1(z0, z0))
F2(z0, a) → c8(G2(z0, z0))
E4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → c9
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → c10(E5(z0, z1, z2, z3))
E5(i, z0, z1, z2) → c11
H1(a) → c12
H2(a) → c13
G2(a, z0) → c14(H1(z0))
G2(z0, a) → c15(H2(z0))
E2(i, z0, z1, z2, i, a) → c16
E2(z0, z0, z1, z2, z2, a) → c17
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

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:

F1(z0, a) → c6(G2(z0, z0))
E2(z0, z0, z1, z2, z2, a) → c17
H2(a) → c13
G2(a, z0) → c14(H1(z0))
E3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → c2
F1(a, z0) → c5(G1(z0, z0))
G1(z0, a) → c4(H2(z0))
E3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → c1(E4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7))
E5(i, z0, z1, z2) → c11
E1(z0, z0, z1, z2, z3, a) → c(E5(z0, z1, z2, z3))
G2(z0, a) → c15(H2(z0))
H1(a) → c12
F2(z0, a) → c8(G2(z0, z0))
E4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → c10(E5(z0, z1, z2, z3))
E4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → c9
G1(a, z0) → c3(H1(z0))
F2(a, z0) → c7(G1(z0, z0))
E2(i, z0, z1, z2, i, a) → c16

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

e1(z0, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3)
e3(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7) → e4(z0, z0, z1, z1, z2, z2, z3, z3, z4, z5, z6, z7)
e3(z0, z1, z0, z1, z1, z2, z1, z2, z0, z1, z2, a) → e6(z0, z1, z2)
g1(a, z0) → h1(z0)
g1(z0, a) → h2(z0)
f1(a, z0) → g1(z0, z0)
f1(z0, a) → g2(z0, z0)
f2(a, z0) → g1(z0, z0)
f2(z0, a) → g2(z0, z0)
e4(z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, z0, a) → e6(z0, z0, z0)
e4(i, z0, i, z0, i, z0, i, z0, z1, z2, z3, a) → e5(z0, z1, z2, z3)
e5(i, z0, z1, z2) → e6(z0, z1, z2)
h1(a) → i
h2(a) → i
g2(a, z0) → h1(z0)
g2(z0, a) → h2(z0)
e2(i, z0, z1, z2, i, a) → e6(z0, z1, z2)
e2(z0, z0, z1, z2, z2, a) → e6(z0, z1, z2)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(10) BOUNDS(1, 1)