Term Rewriting System R:
[dummy, dummy2, x, y, z]
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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

FUNCTION(p, s(s(x)), dummy, dummy2) -> FUNCTION(p, s(x), x, x)
FUNCTION(plus, dummy, x, y) -> FUNCTION(if, function(iszero, x, x, x), x, y)
FUNCTION(plus, dummy, x, y) -> FUNCTION(iszero, x, x, x)
FUNCTION(if, false, x, y) -> FUNCTION(plus, function(third, x, y, y), function(p, x, x, y), s(y))
FUNCTION(if, false, x, y) -> FUNCTION(third, x, y, y)
FUNCTION(if, false, x, y) -> FUNCTION(p, x, x, y)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳Remaining


Dependency Pair:

FUNCTION(p, s(s(x)), dummy, dummy2) -> FUNCTION(p, s(x), x, x)


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





The following dependency pair can be strictly oriented:

FUNCTION(p, s(s(x)), dummy, dummy2) -> FUNCTION(p, s(x), x, x)


There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
FUNCTION(x1, x2, x3, x4) -> FUNCTION(x1, x2, x3, x4)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳Remaining


Dependency Pair:


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





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

FUNCTION(if, false, x, y) -> FUNCTION(plus, function(third, x, y, y), function(p, x, x, y), s(y))
FUNCTION(plus, dummy, x, y) -> FUNCTION(if, function(iszero, x, x, x), x, y)


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




Termination of R could not be shown.
Duration:
0:05 minutes