(0) Obligation:

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

admit(x, nil) → nil
admit(x, .(u, .(v, .(w, z)))) → cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) → y

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

admit(z0, nil) → nil
admit(z0, .(z1, .(z2, .(w, z3)))) → cond(=(sum(z0, z1, z2), w), .(z1, .(z2, .(w, admit(carry(z0, z1, z2), z3)))))
cond(true, z0) → z0
Tuples:

ADMIT(z0, nil) → c
ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(COND(=(sum(z0, z1, z2), w), .(z1, .(z2, .(w, admit(carry(z0, z1, z2), z3))))), ADMIT(carry(z0, z1, z2), z3))
COND(true, z0) → c2
S tuples:

ADMIT(z0, nil) → c
ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(COND(=(sum(z0, z1, z2), w), .(z1, .(z2, .(w, admit(carry(z0, z1, z2), z3))))), ADMIT(carry(z0, z1, z2), z3))
COND(true, z0) → c2
K tuples:none
Defined Rule Symbols:

admit, cond

Defined Pair Symbols:

ADMIT, COND

Compound Symbols:

c, c1, c2

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

Removed 2 trailing nodes:

ADMIT(z0, nil) → c
COND(true, z0) → c2

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

admit(z0, nil) → nil
admit(z0, .(z1, .(z2, .(w, z3)))) → cond(=(sum(z0, z1, z2), w), .(z1, .(z2, .(w, admit(carry(z0, z1, z2), z3)))))
cond(true, z0) → z0
Tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(COND(=(sum(z0, z1, z2), w), .(z1, .(z2, .(w, admit(carry(z0, z1, z2), z3))))), ADMIT(carry(z0, z1, z2), z3))
S tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(COND(=(sum(z0, z1, z2), w), .(z1, .(z2, .(w, admit(carry(z0, z1, z2), z3))))), ADMIT(carry(z0, z1, z2), z3))
K tuples:none
Defined Rule Symbols:

admit, cond

Defined Pair Symbols:

ADMIT

Compound Symbols:

c1

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

admit(z0, nil) → nil
admit(z0, .(z1, .(z2, .(w, z3)))) → cond(=(sum(z0, z1, z2), w), .(z1, .(z2, .(w, admit(carry(z0, z1, z2), z3)))))
cond(true, z0) → z0
Tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(ADMIT(carry(z0, z1, z2), z3))
S tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(ADMIT(carry(z0, z1, z2), z3))
K tuples:none
Defined Rule Symbols:

admit, cond

Defined Pair Symbols:

ADMIT

Compound Symbols:

c1

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

admit(z0, nil) → nil
admit(z0, .(z1, .(z2, .(w, z3)))) → cond(=(sum(z0, z1, z2), w), .(z1, .(z2, .(w, admit(carry(z0, z1, z2), z3)))))
cond(true, z0) → z0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(ADMIT(carry(z0, z1, z2), z3))
S tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(ADMIT(carry(z0, z1, z2), z3))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ADMIT

Compound Symbols:

c1

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

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

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(ADMIT(carry(z0, z1, z2), z3))
We considered the (Usable) Rules:none
And the Tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(ADMIT(carry(z0, z1, z2), z3))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x1, x2)) = [4] + x1 + x2
POL(ADMIT(x1, x2)) = [2]x1 + [2]x2
POL(c1(x1)) = x1
POL(carry(x1, x2, x3)) = [3]
POL(w) = [4]

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(ADMIT(carry(z0, z1, z2), z3))
S tuples:none
K tuples:

ADMIT(z0, .(z1, .(z2, .(w, z3)))) → c1(ADMIT(carry(z0, z1, z2), z3))
Defined Rule Symbols:none

Defined Pair Symbols:

ADMIT

Compound Symbols:

c1

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

The set S is empty