### (0) Obligation:

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

The TRS R consists of the following rules:

f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
++(x, nil) → x
++(x, g(y, z)) → g(++(x, y), z)
null(nil) → true
null(g(x, y)) → false
mem(nil, y) → false
mem(g(x, y), z) → or(=(y, z), mem(x, z))
mem(x, max(x)) → not(null(x))
max(g(g(nil, x), y)) → max'(x, y)
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)

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:
mem(x, max(x)) → not(null(x))

### (2) Obligation:

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

The TRS R consists of the following rules:

f(x, nil) → g(nil, x)
max(g(g(nil, x), y)) → max'(x, y)
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)
++(x, g(y, z)) → g(++(x, y), z)
null(nil) → true
mem(g(x, y), z) → or(=(y, z), mem(x, z))
mem(nil, y) → false
++(x, nil) → x
null(g(x, y)) → false
f(x, g(y, z)) → g(f(x, y), z)

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, n^1).

The TRS R consists of the following rules:

f(x, nil) → g(nil, x)
max(g(g(nil, x), y)) → max'(x, y)
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)
++(x, g(y, z)) → g(++(x, y), z)
null(nil) → true
mem(g(x, y), z) → or(=(y, z), mem(x, z))
mem(nil, y) → false
++(x, nil) → x
null(g(x, y)) → false
f(x, g(y, z)) → g(f(x, y), z)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, nil) → g(nil, z0)
f(z0, g(z1, z2)) → g(f(z0, z1), z2)
max(g(g(nil, z0), z1)) → max'(z0, z1)
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
++(z0, g(z1, z2)) → g(++(z0, z1), z2)
++(z0, nil) → z0
null(nil) → true
null(g(z0, z1)) → false
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2))
mem(nil, z0) → false
Tuples:

F(z0, nil) → c
F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(nil, z0), z1)) → c2
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
++'(z0, nil) → c5
NULL(nil) → c6
NULL(g(z0, z1)) → c7
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
MEM(nil, z0) → c9
S tuples:

F(z0, nil) → c
F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(nil, z0), z1)) → c2
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
++'(z0, nil) → c5
NULL(nil) → c6
NULL(g(z0, z1)) → c7
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
MEM(nil, z0) → c9
K tuples:none
Defined Rule Symbols:

f, max, ++, null, mem

Defined Pair Symbols:

F, MAX, ++', NULL, MEM

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

Removed 6 trailing nodes:

++'(z0, nil) → c5
MEM(nil, z0) → c9
F(z0, nil) → c
NULL(g(z0, z1)) → c7
NULL(nil) → c6
MAX(g(g(nil, z0), z1)) → c2

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, nil) → g(nil, z0)
f(z0, g(z1, z2)) → g(f(z0, z1), z2)
max(g(g(nil, z0), z1)) → max'(z0, z1)
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
++(z0, g(z1, z2)) → g(++(z0, z1), z2)
++(z0, nil) → z0
null(nil) → true
null(g(z0, z1)) → false
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2))
mem(nil, z0) → false
Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
S tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
K tuples:none
Defined Rule Symbols:

f, max, ++, null, mem

Defined Pair Symbols:

F, MAX, ++', MEM

Compound Symbols:

c1, c3, c4, c8

### (9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(z0, nil) → g(nil, z0)
f(z0, g(z1, z2)) → g(f(z0, z1), z2)
max(g(g(nil, z0), z1)) → max'(z0, z1)
max(g(g(g(z0, z1), z2), u)) → max'(max(g(g(z0, z1), z2)), u)
++(z0, g(z1, z2)) → g(++(z0, z1), z2)
++(z0, nil) → z0
null(nil) → true
null(g(z0, z1)) → false
mem(g(z0, z1), z2) → or(=(z1, z2), mem(z0, z2))
mem(nil, z0) → false

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
S tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F, MAX, ++', MEM

Compound Symbols:

c1, c3, c4, c8

### (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.

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(++'(x1, x2)) = 0
POL(F(x1, x2)) = x2
POL(MAX(x1)) = x1
POL(MEM(x1, x2)) = x1
POL(c1(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c8(x1)) = x1
POL(g(x1, x2)) = [1] + x1
POL(u) = 0

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
S tuples:

++'(z0, g(z1, z2)) → c4(++'(z0, z1))
K tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

F, MAX, ++', MEM

Compound Symbols:

c1, c3, c4, c8

### (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.

++'(z0, g(z1, z2)) → c4(++'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(++'(x1, x2)) = x2
POL(F(x1, x2)) = 0
POL(MAX(x1)) = 0
POL(MEM(x1, x2)) = 0
POL(c1(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c8(x1)) = x1
POL(g(x1, x2)) = [1] + x1
POL(u) = 0

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
S tuples:none
K tuples:

F(z0, g(z1, z2)) → c1(F(z0, z1))
MAX(g(g(g(z0, z1), z2), u)) → c3(MAX(g(g(z0, z1), z2)))
MEM(g(z0, z1), z2) → c8(MEM(z0, z2))
++'(z0, g(z1, z2)) → c4(++'(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

F, MAX, ++', MEM

Compound Symbols:

c1, c3, c4, c8

### (15) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty