YES

We show the termination of the TRS R:

  not(not(x)) -> x
  not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
  not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(x))
p3: not#(or(x,y)) -> not#(x)
p4: not#(or(x,y)) -> not#(not(not(y)))
p5: not#(or(x,y)) -> not#(not(y))
p6: not#(or(x,y)) -> not#(y)
p7: not#(and(x,y)) -> not#(not(not(x)))
p8: not#(and(x,y)) -> not#(not(x))
p9: not#(and(x,y)) -> not#(x)
p10: not#(and(x,y)) -> not#(not(not(y)))
p11: not#(and(x,y)) -> not#(not(y))
p12: not#(and(x,y)) -> not#(y)

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(y)
p3: not#(and(x,y)) -> not#(not(y))
p4: not#(and(x,y)) -> not#(not(not(y)))
p5: not#(and(x,y)) -> not#(x)
p6: not#(and(x,y)) -> not#(not(x))
p7: not#(and(x,y)) -> not#(not(not(x)))
p8: not#(or(x,y)) -> not#(y)
p9: not#(or(x,y)) -> not#(not(y))
p10: not#(or(x,y)) -> not#(not(not(y)))
p11: not#(or(x,y)) -> not#(x)
p12: not#(or(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,0),(1,1)) x1 + (1,0)
          or_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,0)
          not_A(x1) = x1
          and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,0)
  
    precedence:
    
      or = not > and > not#
    
    partial status:
  
      pi(not#) = [1]
      pi(or) = []
      pi(not) = []
      pi(and) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(y)
p3: not#(and(x,y)) -> not#(not(y))
p4: not#(and(x,y)) -> not#(x)
p5: not#(and(x,y)) -> not#(not(x))
p6: not#(and(x,y)) -> not#(not(not(x)))
p7: not#(or(x,y)) -> not#(y)
p8: not#(or(x,y)) -> not#(not(y))
p9: not#(or(x,y)) -> not#(not(not(y)))
p10: not#(or(x,y)) -> not#(x)
p11: not#(or(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(x))
p3: not#(or(x,y)) -> not#(x)
p4: not#(or(x,y)) -> not#(not(not(y)))
p5: not#(or(x,y)) -> not#(not(y))
p6: not#(or(x,y)) -> not#(y)
p7: not#(and(x,y)) -> not#(not(not(x)))
p8: not#(and(x,y)) -> not#(not(x))
p9: not#(and(x,y)) -> not#(x)
p10: not#(and(x,y)) -> not#(not(y))
p11: not#(and(x,y)) -> not#(y)

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,1),(0,0)) x1 + (1,1)
          or_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,1),(0,0)) x2 + (2,3)
          not_A(x1) = ((0,1),(1,0)) x1
          and_A(x1,x2) = ((0,1),(1,0)) x1 + ((0,0),(1,1)) x2 + (3,2)
  
    precedence:
    
      not > and > not# = or
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = []
      pi(and) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(x))
p3: not#(or(x,y)) -> not#(x)
p4: not#(or(x,y)) -> not#(not(not(y)))
p5: not#(or(x,y)) -> not#(y)
p6: not#(and(x,y)) -> not#(not(not(x)))
p7: not#(and(x,y)) -> not#(not(x))
p8: not#(and(x,y)) -> not#(x)
p9: not#(and(x,y)) -> not#(not(y))
p10: not#(and(x,y)) -> not#(y)

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(y)
p3: not#(and(x,y)) -> not#(not(y))
p4: not#(and(x,y)) -> not#(x)
p5: not#(and(x,y)) -> not#(not(x))
p6: not#(and(x,y)) -> not#(not(not(x)))
p7: not#(or(x,y)) -> not#(y)
p8: not#(or(x,y)) -> not#(not(not(y)))
p9: not#(or(x,y)) -> not#(x)
p10: not#(or(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,0),(1,0)) x1 + (0,3)
          or_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (1,2)
          not_A(x1) = ((1,0),(1,1)) x1 + (0,5)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,1)
  
    precedence:
    
      not > not# = or = and
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = []
      pi(and) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(y)
p3: not#(and(x,y)) -> not#(x)
p4: not#(and(x,y)) -> not#(not(x))
p5: not#(and(x,y)) -> not#(not(not(x)))
p6: not#(or(x,y)) -> not#(y)
p7: not#(or(x,y)) -> not#(not(not(y)))
p8: not#(or(x,y)) -> not#(x)
p9: not#(or(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(x))
p3: not#(or(x,y)) -> not#(x)
p4: not#(or(x,y)) -> not#(not(not(y)))
p5: not#(or(x,y)) -> not#(y)
p6: not#(and(x,y)) -> not#(not(not(x)))
p7: not#(and(x,y)) -> not#(not(x))
p8: not#(and(x,y)) -> not#(x)
p9: not#(and(x,y)) -> not#(y)

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,1),(0,1)) x1 + (1,1)
          or_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(0,1)) x2 + (2,1)
          not_A(x1) = x1
          and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(0,1)) x2 + (2,1)
  
    precedence:
    
      not > not# = and > or
    
    partial status:
  
      pi(not#) = [1]
      pi(or) = [1, 2]
      pi(not) = []
      pi(and) = [2]

The next rules are strictly ordered:

  p9

We remove them from the problem.

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(x))
p3: not#(or(x,y)) -> not#(x)
p4: not#(or(x,y)) -> not#(not(not(y)))
p5: not#(or(x,y)) -> not#(y)
p6: not#(and(x,y)) -> not#(not(not(x)))
p7: not#(and(x,y)) -> not#(not(x))
p8: not#(and(x,y)) -> not#(x)

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(x)
p3: not#(and(x,y)) -> not#(not(x))
p4: not#(and(x,y)) -> not#(not(not(x)))
p5: not#(or(x,y)) -> not#(y)
p6: not#(or(x,y)) -> not#(not(not(y)))
p7: not#(or(x,y)) -> not#(x)
p8: not#(or(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,0),(1,0)) x1 + (1,2)
          or_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
          not_A(x1) = ((1,0),(1,1)) x1
          and_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
  
    precedence:
    
      not = and > not# > or
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = []
      pi(and) = []

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: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(not(x))
p3: not#(and(x,y)) -> not#(not(not(x)))
p4: not#(or(x,y)) -> not#(y)
p5: not#(or(x,y)) -> not#(not(not(y)))
p6: not#(or(x,y)) -> not#(x)
p7: not#(or(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(x))
p3: not#(or(x,y)) -> not#(x)
p4: not#(or(x,y)) -> not#(not(not(y)))
p5: not#(or(x,y)) -> not#(y)
p6: not#(and(x,y)) -> not#(not(not(x)))
p7: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          or_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
          not_A(x1) = ((1,0),(1,1)) x1
          and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
  
    precedence:
    
      not > or = and > not#
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = []
      pi(and) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(x))
p3: not#(or(x,y)) -> not#(not(not(y)))
p4: not#(or(x,y)) -> not#(y)
p5: not#(and(x,y)) -> not#(not(not(x)))
p6: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(not(x))
p3: not#(and(x,y)) -> not#(not(not(x)))
p4: not#(or(x,y)) -> not#(y)
p5: not#(or(x,y)) -> not#(not(not(y)))
p6: not#(or(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((0,1),(0,0)) x1 + (1,2)
          or_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,0),(0,1)) x2 + (2,3)
          not_A(x1) = ((0,1),(1,0)) x1
          and_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(0,0)) x2 + (3,2)
  
    precedence:
    
      or = not = and > not#
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = []
      pi(and) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(not(x))
p3: not#(and(x,y)) -> not#(not(not(x)))
p4: not#(or(x,y)) -> not#(y)
p5: not#(or(x,y)) -> not#(not(not(y)))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(not(y)))
p3: not#(or(x,y)) -> not#(y)
p4: not#(and(x,y)) -> not#(not(not(x)))
p5: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,1),(0,0)) x1 + (2,1)
          or_A(x1,x2) = ((0,0),(1,1)) x1 + ((1,1),(0,0)) x2 + (1,3)
          not_A(x1) = ((0,1),(1,0)) x1
          and_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,0),(1,1)) x2 + (3,1)
  
    precedence:
    
      not# = not = and > or
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = []
      pi(and) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(not(y)))
p3: not#(and(x,y)) -> not#(not(not(x)))
p4: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

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: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(not(x))
p3: not#(and(x,y)) -> not#(not(not(x)))
p4: not#(or(x,y)) -> not#(not(not(y)))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,0),(1,0)) x1 + (1,2)
          or_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (2,2)
          not_A(x1) = x1 + (0,4)
          and_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
  
    precedence:
    
      or = not > not# > and
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = [1]
      pi(and) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(and(x,y)) -> not#(not(x))
p3: not#(or(x,y)) -> not#(not(not(y)))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

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: not#(or(x,y)) -> not#(not(not(x)))
p2: not#(or(x,y)) -> not#(not(not(y)))
p3: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,0),(1,0)) x1 + (2,4)
          or_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (3,1)
          not_A(x1) = ((0,1),(1,0)) x1
          and_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (1,3)
  
    precedence:
    
      not# = or = not = and
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = []
      pi(and) = []

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: not#(or(x,y)) -> not#(not(not(y)))
p2: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: not#(or(x,y)) -> not#(not(not(y)))
p2: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((1,0),(0,0)) x1 + (2,3)
          or_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (1,1)
          not_A(x1) = ((1,0),(1,1)) x1
          and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,2)
  
    precedence:
    
      not = and > or > not#
    
    partial status:
  
      pi(not#) = []
      pi(or) = []
      pi(not) = []
      pi(and) = []

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: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: not#(and(x,y)) -> not#(not(x))

and R consists of:

r1: not(not(x)) -> x
r2: not(or(x,y)) -> and(not(not(not(x))),not(not(not(y))))
r3: not(and(x,y)) -> or(not(not(not(x))),not(not(not(y))))

The set of usable rules consists of

  r1, r2, r3

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          not#_A(x1) = ((0,1),(0,0)) x1 + (3,2)
          and_A(x1,x2) = ((0,0),(0,1)) x1 + (2,1)
          not_A(x1) = ((1,1),(0,1)) x1 + (5,0)
          or_A(x1,x2) = ((0,0),(0,1)) x1 + (1,1)
  
    precedence:
    
      not = or > and > not#
    
    partial status:
  
      pi(not#) = []
      pi(and) = []
      pi(not) = []
      pi(or) = []

The next rules are strictly ordered:

  p1

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