YES

We show the termination of the TRS R:

  eq(|0|(),|0|()) -> true()
  eq(|0|(),s(x)) -> false()
  eq(s(x),|0|()) -> false()
  eq(s(x),s(y)) -> eq(x,y)
  or(true(),y) -> true()
  or(false(),y) -> y
  union(empty(),h) -> h
  union(edge(x,y,i),h) -> edge(x,y,union(i,h))
  reach(x,y,empty(),h) -> false()
  reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
  if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
  if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
  if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
  if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

-- SCC decomposition.

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

p1: eq#(s(x),s(y)) -> eq#(x,y)
p2: union#(edge(x,y,i),h) -> union#(i,h)
p3: reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
p4: reach#(x,y,edge(u,v,i),h) -> eq#(x,u)
p5: if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
p6: if_reach_1#(true(),x,y,edge(u,v,i),h) -> eq#(y,v)
p7: if_reach_2#(false(),x,y,edge(u,v,i),h) -> or#(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
p8: if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)
p9: if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty())
p10: if_reach_2#(false(),x,y,edge(u,v,i),h) -> union#(i,h)
p11: if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

The estimated dependency graph contains the following SCCs:

  {p3, p5, p8, p9, p11}
  {p1}
  {p2}


-- Reduction pair.

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

p1: if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(v,y,union(i,h),empty())
p2: reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
p3: if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
p4: if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
p5: if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

The set of usable rules consists of

  r1, r2, r3, r4, r7, r8

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          if_reach_2#_A(x1,x2,x3,x4,x5) = ((1,0),(1,0)) x4 + ((1,0),(0,0)) x5 + (1,18)
          false_A() = (2,20)
          edge_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (8,21)
          reach#_A(x1,x2,x3,x4) = ((1,0),(1,0)) x3 + ((1,0),(0,0)) x4 + (3,22)
          union_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,22)
          empty_A() = (1,19)
          if_reach_1#_A(x1,x2,x3,x4,x5) = ((1,0),(1,0)) x4 + ((1,0),(0,0)) x5 + (3,19)
          eq_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (12,1)
          true_A() = (10,2)
          |0|_A() = (11,3)
          s_A(x1) = ((1,0),(0,0)) x1 + (3,2)
  
    precedence:
    
      if_reach_2# = reach# = union = if_reach_1# > empty > |0| > false = edge = eq = true = s
    
    partial status:
  
      pi(if_reach_2#) = []
      pi(false) = []
      pi(edge) = []
      pi(reach#) = []
      pi(union) = []
      pi(empty) = []
      pi(if_reach_1#) = []
      pi(eq) = [2]
      pi(true) = []
      pi(|0|) = []
      pi(s) = []

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: reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
p2: if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))
p3: if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
p4: if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

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: reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
p2: if_reach_1#(true(),x,y,edge(u,v,i),h) -> if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)
p3: if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)
p4: if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

The set of usable rules consists of

  r1, r2, r3, r4

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          reach#_A(x1,x2,x3,x4) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (3,7)
          edge_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + x3 + (5,1)
          if_reach_1#_A(x1,x2,x3,x4,x5) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + ((1,0),(1,1)) x4 + ((0,0),(1,0)) x5 + (2,0)
          eq_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (6,10)
          true_A() = (4,9)
          if_reach_2#_A(x1,x2,x3,x4,x5) = ((1,0),(1,0)) x2 + x3 + ((1,0),(0,0)) x4 + ((0,0),(1,0)) x5 + (0,8)
          false_A() = (4,8)
          |0|_A() = (5,10)
          s_A(x1) = x1 + (5,11)
  
    precedence:
    
      |0| > if_reach_1# > if_reach_2# > edge > true > reach# = eq = false = s
    
    partial status:
  
      pi(reach#) = []
      pi(edge) = [2]
      pi(if_reach_1#) = []
      pi(eq) = [1, 2]
      pi(true) = []
      pi(if_reach_2#) = [3]
      pi(false) = []
      pi(|0|) = []
      pi(s) = []

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: reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
p2: if_reach_2#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,h)
p3: if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

The estimated dependency graph contains the following SCCs:

  {p1, p3}


-- Reduction pair.

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

p1: reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)
p2: if_reach_1#(false(),x,y,edge(u,v,i),h) -> reach#(x,y,i,edge(u,v,h))

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

The set of usable rules consists of

  r1, r2, r3, r4

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          reach#_A(x1,x2,x3,x4) = ((1,0),(1,0)) x3 + (2,1)
          edge_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x3 + (3,2)
          if_reach_1#_A(x1,x2,x3,x4,x5) = x4 + (1,1)
          eq_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (6,5)
          false_A() = (8,3)
          |0|_A() = (9,4)
          true_A() = (0,5)
          s_A(x1) = ((1,0),(0,0)) x1 + (7,1)
  
    precedence:
    
      reach# = edge > if_reach_1# = eq = false = |0| = true = s
    
    partial status:
  
      pi(reach#) = []
      pi(edge) = []
      pi(if_reach_1#) = [4]
      pi(eq) = []
      pi(false) = []
      pi(|0|) = []
      pi(true) = []
      pi(s) = []

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: reach#(x,y,edge(u,v,i),h) -> if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: eq#(s(x),s(y)) -> eq#(x,y)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          eq#_A(x1,x2) = x1
          s_A(x1) = x1 + (1,1)
  
    precedence:
    
      s > eq#
    
    partial status:
  
      pi(eq#) = []
      pi(s) = [1]

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: union#(edge(x,y,i),h) -> union#(i,h)

and R consists of:

r1: eq(|0|(),|0|()) -> true()
r2: eq(|0|(),s(x)) -> false()
r3: eq(s(x),|0|()) -> false()
r4: eq(s(x),s(y)) -> eq(x,y)
r5: or(true(),y) -> true()
r6: or(false(),y) -> y
r7: union(empty(),h) -> h
r8: union(edge(x,y,i),h) -> edge(x,y,union(i,h))
r9: reach(x,y,empty(),h) -> false()
r10: reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
r11: if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
r12: if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
r13: if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
r14: if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          union#_A(x1,x2) = x1 + (1,1)
          edge_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + x3 + (2,0)
  
    precedence:
    
      union# = edge
    
    partial status:
  
      pi(union#) = [1]
      pi(edge) = [2]

The next rules are strictly ordered:

  p1

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