### (0) Obligation:

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

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
function(p, s(s(x)), dummy, dummy2) →+ s(function(p, s(x), x, x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x)].
The result substitution is [dummy / x, dummy2 / x].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

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

function(iszero, 0', dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0', dummy, dummy2) → 0'
function(p, s(0'), dummy, dummy2) → 0'
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
function(iszero, 0', dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0', dummy, dummy2) → 0'
function(p, s(0'), dummy, dummy2) → 0'
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

Types:
function :: iszero:p:plus:if:third → 0':true:s:false → 0':true:s:false → 0':true:s:false → 0':true:s:false
iszero :: iszero:p:plus:if:third
0' :: 0':true:s:false
true :: 0':true:s:false
s :: 0':true:s:false → 0':true:s:false
false :: 0':true:s:false
p :: iszero:p:plus:if:third
plus :: iszero:p:plus:if:third
if :: iszero:p:plus:if:third
third :: iszero:p:plus:if:third
hole_0':true:s:false1_0 :: 0':true:s:false
hole_iszero:p:plus:if:third2_0 :: iszero:p:plus:if:third
gen_0':true:s:false3_0 :: Nat → 0':true:s:false

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
function

### (8) Obligation:

TRS:
Rules:
function(iszero, 0', dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0', dummy, dummy2) → 0'
function(p, s(0'), dummy, dummy2) → 0'
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

Types:
function :: iszero:p:plus:if:third → 0':true:s:false → 0':true:s:false → 0':true:s:false → 0':true:s:false
iszero :: iszero:p:plus:if:third
0' :: 0':true:s:false
true :: 0':true:s:false
s :: 0':true:s:false → 0':true:s:false
false :: 0':true:s:false
p :: iszero:p:plus:if:third
plus :: iszero:p:plus:if:third
if :: iszero:p:plus:if:third
third :: iszero:p:plus:if:third
hole_0':true:s:false1_0 :: 0':true:s:false
hole_iszero:p:plus:if:third2_0 :: iszero:p:plus:if:third
gen_0':true:s:false3_0 :: Nat → 0':true:s:false

Generator Equations:
gen_0':true:s:false3_0(0) ⇔ 0'
gen_0':true:s:false3_0(+(x, 1)) ⇔ s(gen_0':true:s:false3_0(x))

The following defined symbols remain to be analysed:
function

### (9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol function.

### (10) Obligation:

TRS:
Rules:
function(iszero, 0', dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0', dummy, dummy2) → 0'
function(p, s(0'), dummy, dummy2) → 0'
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

Types:
function :: iszero:p:plus:if:third → 0':true:s:false → 0':true:s:false → 0':true:s:false → 0':true:s:false
iszero :: iszero:p:plus:if:third
0' :: 0':true:s:false
true :: 0':true:s:false
s :: 0':true:s:false → 0':true:s:false
false :: 0':true:s:false
p :: iszero:p:plus:if:third
plus :: iszero:p:plus:if:third
if :: iszero:p:plus:if:third
third :: iszero:p:plus:if:third
hole_0':true:s:false1_0 :: 0':true:s:false
hole_iszero:p:plus:if:third2_0 :: iszero:p:plus:if:third
gen_0':true:s:false3_0 :: Nat → 0':true:s:false

Generator Equations:
gen_0':true:s:false3_0(0) ⇔ 0'
gen_0':true:s:false3_0(+(x, 1)) ⇔ s(gen_0':true:s:false3_0(x))

No more defined symbols left to analyse.