YES

We show the termination of the TRS R:

  |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
  |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
  |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: |:|#(|:|(x,y),z) -> |:|#(x,|:|(y,z))
p2: |:|#(|:|(x,y),z) -> |:|#(y,z)
p3: |:|#(+(x,y),z) -> |:|#(x,z)
p4: |:|#(+(x,y),z) -> |:|#(y,z)
p5: |:|#(z,+(x,f(y))) -> |:|#(g(z,y),+(x,a()))

and R consists of:

r1: |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
r2: |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
r3: |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: |:|#(|:|(x,y),z) -> |:|#(x,|:|(y,z))
p2: |:|#(+(x,y),z) -> |:|#(y,z)
p3: |:|#(+(x,y),z) -> |:|#(x,z)
p4: |:|#(|:|(x,y),z) -> |:|#(y,z)

and R consists of:

r1: |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
r2: |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
r3: |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      |:|#_A(x1,x2) = x1 + 7
      |:|_A(x1,x2) = max{5, x1 + 3, x2}
      +_A(x1,x2) = max{x1, x2}
      f_A(x1) = max{7, x1 + 6}
      g_A(x1,x2) = max{x1 - 1, x2 + 3}
      a_A = 7
    
    2. max/plus interpretations on natural numbers:
    
      |:|#_A(x1,x2) = 0
      |:|_A(x1,x2) = 1
      +_A(x1,x2) = 0
      f_A(x1) = 0
      g_A(x1,x2) = 0
      a_A = 0
    

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: |:|#(+(x,y),z) -> |:|#(y,z)
p2: |:|#(+(x,y),z) -> |:|#(x,z)
p3: |:|#(|:|(x,y),z) -> |:|#(y,z)

and R consists of:

r1: |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
r2: |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
r3: |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: |:|#(+(x,y),z) -> |:|#(y,z)
p2: |:|#(|:|(x,y),z) -> |:|#(y,z)
p3: |:|#(+(x,y),z) -> |:|#(x,z)

and R consists of:

r1: |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
r2: |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
r3: |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      |:|#_A(x1,x2) = max{1, x1}
      +_A(x1,x2) = max{x1 + 1, x2}
      |:|_A(x1,x2) = max{x1 + 2, x2 + 2}
    
    2. max/plus interpretations on natural numbers:
    
      |:|#_A(x1,x2) = 0
      +_A(x1,x2) = 0
      |:|_A(x1,x2) = 0
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: |:|#(+(x,y),z) -> |:|#(y,z)
p2: |:|#(+(x,y),z) -> |:|#(x,z)

and R consists of:

r1: |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
r2: |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
r3: |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: |:|#(+(x,y),z) -> |:|#(y,z)
p2: |:|#(+(x,y),z) -> |:|#(x,z)

and R consists of:

r1: |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
r2: |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
r3: |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      |:|#_A(x1,x2) = x1 + 1
      +_A(x1,x2) = max{x1 + 1, x2}
    
    2. max/plus interpretations on natural numbers:
    
      |:|#_A(x1,x2) = 0
      +_A(x1,x2) = 0
    

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: |:|#(+(x,y),z) -> |:|#(y,z)

and R consists of:

r1: |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
r2: |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
r3: |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: |:|#(+(x,y),z) -> |:|#(y,z)

and R consists of:

r1: |:|(|:|(x,y),z) -> |:|(x,|:|(y,z))
r2: |:|(+(x,y),z) -> +(|:|(x,z),|:|(y,z))
r3: |:|(z,+(x,f(y))) -> |:|(g(z,y),+(x,a()))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. max/plus interpretations on natural numbers:
    
      |:|#_A(x1,x2) = x1
      +_A(x1,x2) = max{x1 + 1, x2 + 1}
    
    2. max/plus interpretations on natural numbers:
    
      |:|#_A(x1,x2) = 0
      +_A(x1,x2) = 0
    

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.