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, r5, r6, r7, r8, r9, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          if_reach_2#_A(x1,x2,x3,x4,x5) = x4 + x5 + 6
          false_A() = 2
          edge_A(x1,x2,x3) = x1 + x2 + x3 + 6
          reach#_A(x1,x2,x3,x4) = x3 + x4 + 7
          union_A(x1,x2) = x1 + x2 + 3
          empty_A() = 1
          if_reach_1#_A(x1,x2,x3,x4,x5) = x4 + x5 + 7
          eq_A(x1,x2) = 5
          true_A() = 4
          |0|_A() = 5
          s_A(x1) = 6
          or_A(x1,x2) = x1 + x2 + 1
          reach_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 8
          if_reach_2_A(x1,x2,x3,x4,x5) = x4 + 1
  
    base order 2:
  
      precedence:
      
        reach# = if_reach_1# > if_reach_2# = union = or = reach > if_reach_2 > edge = s > eq = |0| > false = true > empty
      
  
    partial status:
  
      pi(if_reach_2#) = [5]
      pi(false) = []
      pi(edge) = [1]
      pi(reach#) = []
      pi(union) = []
      pi(empty) = []
      pi(if_reach_1#) = []
      pi(eq) = []
      pi(true) = []
      pi(|0|) = []
      pi(s) = []
      pi(or) = []
      pi(reach) = []
      pi(if_reach_2) = [4]

The following rules are removed:

  p4



-- SCC decomposition.

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_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:

  {p2, 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, r5, r6, r9, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          reach#_A(x1,x2,x3,x4) = x3 + x4 + 1
          edge_A(x1,x2,x3) = x1 + x2 + x3 + 3
          if_reach_1#_A(x1,x2,x3,x4,x5) = x4 + x5 + 1
          eq_A(x1,x2) = 6
          false_A() = 5
          |0|_A() = 6
          true_A() = 1
          s_A(x1) = 7
          or_A(x1,x2) = x1 + x2 + 1
          reach_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 6
          empty_A() = 1
          if_reach_2_A(x1,x2,x3,x4,x5) = 2
  
    base order 2:
  
      precedence:
      
        if_reach_2 > |0| = s = reach > eq = true = empty > false > reach# = edge = if_reach_1# = or
      
  
    partial status:
  
      pi(reach#) = [3]
      pi(edge) = [1, 2]
      pi(if_reach_1#) = [4]
      pi(eq) = []
      pi(false) = []
      pi(|0|) = []
      pi(true) = []
      pi(s) = []
      pi(or) = []
      pi(reach) = [1, 3]
      pi(empty) = []
      pi(if_reach_2) = []

The following rules are removed:

  p2



-- 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

  r1, r2, r3, r4, r5, r6, r9, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          eq#_A(x1,x2) = x1 + x2 + 7
          s_A(x1) = x1 + 6
          eq_A(x1,x2) = x1 + x2 + 2
          |0|_A() = 6
          true_A() = 1
          false_A() = 5
          or_A(x1,x2) = x1 + x2 + 1
          reach_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 6
          empty_A() = 4
          if_reach_2_A(x1,x2,x3,x4,x5) = x4 + 2
          edge_A(x1,x2,x3) = x2 + x3 + 3
  
    base order 2:
  
      precedence:
      
        eq > eq# = reach > s = |0| = or > false = if_reach_2 > empty > edge > true
      
  
    partial status:
  
      pi(eq#) = [1]
      pi(s) = [1]
      pi(eq) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(or) = []
      pi(reach) = []
      pi(empty) = []
      pi(if_reach_2) = []
      pi(edge) = [3]

The following rules are removed:

  p1

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

  r1, r2, r3, r4, r5, r6, r9, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          union#_A(x1,x2) = x1 + x2 + 5
          edge_A(x1,x2,x3) = x1 + x2 + x3 + 4
          eq_A(x1,x2) = 3
          |0|_A() = 3
          true_A() = 2
          s_A(x1) = 4
          false_A() = 2
          or_A(x1,x2) = x1 + x2 + 1
          reach_A(x1,x2,x3,x4) = x1 + x2 + x3 + x4 + 6
          empty_A() = 1
          if_reach_2_A(x1,x2,x3,x4,x5) = x4 + 1
  
    base order 2:
  
      precedence:
      
        reach > or > |0| = if_reach_2 > true = s > union# = eq > edge > false > empty
      
  
    partial status:
  
      pi(union#) = [1, 2]
      pi(edge) = []
      pi(eq) = []
      pi(|0|) = []
      pi(true) = []
      pi(s) = []
      pi(false) = []
      pi(or) = [1, 2]
      pi(reach) = [1, 2, 3]
      pi(empty) = []
      pi(if_reach_2) = []

The following rules are removed:

  p1

Then no dependency pair remains.