The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

map(Cons(x, xs)) → Cons(f(x), map(xs))
map(Nil) → Nil
goal(xs) → map(xs)
f(x) → *(x, x)
+Full(S(x), y) → +Full(x, S(y))
+Full(0, y) → y

The (relative) TRS S consists of the following rules:

*(x, S(S(y))) → +(x, *(x, S(y)))
*(x, S(0)) → x
*(x, 0) → 0
*(0, y) → 0

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (UPPER BOUND(ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, S(S(z1))) → +(z0, *(z0, S(z1)))
*(z0, S(0)) → z0
*(z0, 0) → 0
*(0, z0) → 0
map(Cons(z0, z1)) → Cons(f(z0), map(z1))
map(Nil) → Nil
goal(z0) → map(z0)
f(z0) → *(z0, z0)
+Full(S(z0), z1) → +Full(z0, S(z1))
+Full(0, z0) → z0

Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1)))
*'(z0, S(0)) → c1
*'(z0, 0) → c2
*'(0, z0) → c3
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
MAP(Nil) → c5
GOAL(z0) → c6(MAP(z0))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
+FULL(0, z0) → c9

S tuples:

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
MAP(Nil) → c5
GOAL(z0) → c6(MAP(z0))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
+FULL(0, z0) → c9

K tuples:none
Defined Rule Symbols:

map, goal, f, +Full, *

Defined Pair Symbols:

*', MAP, GOAL, F, +FULL

Compound Symbols:

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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0) → c6(MAP(z0))

Removed 5 trailing nodes:

*'(z0, 0) → c2
+FULL(0, z0) → c9
*'(z0, S(0)) → c1
*'(0, z0) → c3
MAP(Nil) → c5

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, S(S(z1))) → +(z0, *(z0, S(z1)))
*(z0, S(0)) → z0
*(z0, 0) → 0
*(0, z0) → 0
map(Cons(z0, z1)) → Cons(f(z0), map(z1))
map(Nil) → Nil
goal(z0) → map(z0)
f(z0) → *(z0, z0)
+Full(S(z0), z1) → +Full(z0, S(z1))
+Full(0, z0) → z0

Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1)))
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

S tuples:

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

K tuples:none
Defined Rule Symbols:

map, goal, f, +Full, *

Defined Pair Symbols:

*', MAP, F, +FULL

Compound Symbols:

c, c4, c7, c8

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

*(z0, S(S(z1))) → +(z0, *(z0, S(z1)))
*(z0, S(0)) → z0
*(z0, 0) → 0
*(0, z0) → 0
map(Cons(z0, z1)) → Cons(f(z0), map(z1))
map(Nil) → Nil
goal(z0) → map(z0)
f(z0) → *(z0, z0)
+Full(S(z0), z1) → +Full(z0, S(z1))
+Full(0, z0) → z0

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1)))
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

S tuples:

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

*', MAP, F, +FULL

Compound Symbols:

c, c4, c7, c8

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

F(z0) → c7(*'(z0, z0))

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

*'(z0, S(S(z1))) → c(*'(z0, S(z1)))
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

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

POL(*'(x₁, x₂)) = 0   
POL(+FULL(x₁, x₂)) = [2]x₂²   
POL(Cons(x₁, x₂)) = [2] + x₁ + x₂   
POL(F(x₁)) = [2] + x₁   
POL(MAP(x₁)) = x₁   
POL(S(x₁)) = 0   
POL(c(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1)))
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

S tuples:

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

K tuples:

F(z0) → c7(*'(z0, z0))

Defined Rule Symbols:none

Defined Pair Symbols:

*', MAP, F, +FULL

Compound Symbols:

c, c4, c7, c8

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

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

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

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

*'(z0, S(S(z1))) → c(*'(z0, S(z1)))
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

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

POL(*'(x₁, x₂)) = [3]   
POL(+FULL(x₁, x₂)) = [4]x₁   
POL(Cons(x₁, x₂)) = [4] + x₂   
POL(F(x₁)) = [4]   
POL(MAP(x₁)) = [4]x₁   
POL(S(x₁)) = [4] + x₁   
POL(c(x₁)) = x₁   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1)))
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
F(z0) → c7(*'(z0, z0))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

S tuples:none
K tuples:

F(z0) → c7(*'(z0, z0))
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))

Defined Rule Symbols:none

Defined Pair Symbols:

*', MAP, F, +FULL

Compound Symbols:

c, c4, c7, c8

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

The set S is empty

(12) BOUNDS(1, 1)