YES

Problem:
 not(tt()) -> ff()
 not(ff()) -> tt()
 or(tt(),x) -> tt()
 or(ff(),x) -> x
 eq(0(),0()) -> tt()
 eq(s(x),0()) -> ff()
 eq(0(),s(y)) -> ff()
 eq(s(x),s(y)) -> eq(x,y)
 main(phi) -> ver(phi,nil())
 in(x,nil()) -> ff()
 in(x,cons(a,l)) -> or(eq(x,a),in(x,l))
 ver(Var(x),t()) -> in(x,t())
 ver(Or(phi,psi),t()) -> or(ver(phi,t()),ver(psi,t()))
 ver(Not(phi),t()) -> not(ver(phi,t()))
 ver(Exists(n,phi),t()) -> or(ver(phi,cons(n,t())),ver(phi,t()))

Proof:
 DP Processor:
  DPs:
   eq#(s(x),s(y)) -> eq#(x,y)
   main#(phi) -> ver#(phi,nil())
   in#(x,cons(a,l)) -> in#(x,l)
   in#(x,cons(a,l)) -> eq#(x,a)
   in#(x,cons(a,l)) -> or#(eq(x,a),in(x,l))
   ver#(Var(x),t()) -> in#(x,t())
   ver#(Or(phi,psi),t()) -> ver#(psi,t())
   ver#(Or(phi,psi),t()) -> ver#(phi,t())
   ver#(Or(phi,psi),t()) -> or#(ver(phi,t()),ver(psi,t()))
   ver#(Not(phi),t()) -> ver#(phi,t())
   ver#(Not(phi),t()) -> not#(ver(phi,t()))
   ver#(Exists(n,phi),t()) -> ver#(phi,t())
   ver#(Exists(n,phi),t()) -> ver#(phi,cons(n,t()))
   ver#(Exists(n,phi),t()) -> or#(ver(phi,cons(n,t())),ver(phi,t()))
  TRS:
   not(tt()) -> ff()
   not(ff()) -> tt()
   or(tt(),x) -> tt()
   or(ff(),x) -> x
   eq(0(),0()) -> tt()
   eq(s(x),0()) -> ff()
   eq(0(),s(y)) -> ff()
   eq(s(x),s(y)) -> eq(x,y)
   main(phi) -> ver(phi,nil())
   in(x,nil()) -> ff()
   in(x,cons(a,l)) -> or(eq(x,a),in(x,l))
   ver(Var(x),t()) -> in(x,t())
   ver(Or(phi,psi),t()) -> or(ver(phi,t()),ver(psi,t()))
   ver(Not(phi),t()) -> not(ver(phi,t()))
   ver(Exists(n,phi),t()) -> or(ver(phi,cons(n,t())),ver(phi,t()))
  Usable Rule Processor:
   DPs:
    eq#(s(x),s(y)) -> eq#(x,y)
    main#(phi) -> ver#(phi,nil())
    in#(x,cons(a,l)) -> in#(x,l)
    in#(x,cons(a,l)) -> eq#(x,a)
    in#(x,cons(a,l)) -> or#(eq(x,a),in(x,l))
    ver#(Var(x),t()) -> in#(x,t())
    ver#(Or(phi,psi),t()) -> ver#(psi,t())
    ver#(Or(phi,psi),t()) -> ver#(phi,t())
    ver#(Or(phi,psi),t()) -> or#(ver(phi,t()),ver(psi,t()))
    ver#(Not(phi),t()) -> ver#(phi,t())
    ver#(Not(phi),t()) -> not#(ver(phi,t()))
    ver#(Exists(n,phi),t()) -> ver#(phi,t())
    ver#(Exists(n,phi),t()) -> ver#(phi,cons(n,t()))
    ver#(Exists(n,phi),t()) -> or#(ver(phi,cons(n,t())),ver(phi,t()))
   TRS:
    in(x,nil()) -> ff()
    in(x,cons(a,l)) -> or(eq(x,a),in(x,l))
    or(tt(),x) -> tt()
    or(ff(),x) -> x
    eq(0(),0()) -> tt()
    eq(s(x),0()) -> ff()
    eq(0(),s(y)) -> ff()
    eq(s(x),s(y)) -> eq(x,y)
    ver(Var(x),t()) -> in(x,t())
    ver(Or(phi,psi),t()) -> or(ver(phi,t()),ver(psi,t()))
    ver(Not(phi),t()) -> not(ver(phi,t()))
    ver(Exists(n,phi),t()) -> or(ver(phi,cons(n,t())),ver(phi,t()))
    not(tt()) -> ff()
    not(ff()) -> tt()
   Matrix Interpretation Processor: dim=1
    
    usable rules:
     in(x,nil()) -> ff()
     in(x,cons(a,l)) -> or(eq(x,a),in(x,l))
     or(tt(),x) -> tt()
     or(ff(),x) -> x
     eq(0(),0()) -> tt()
     eq(s(x),0()) -> ff()
     eq(0(),s(y)) -> ff()
     eq(s(x),s(y)) -> eq(x,y)
     ver(Var(x),t()) -> in(x,t())
     ver(Or(phi,psi),t()) -> or(ver(phi,t()),ver(psi,t()))
     ver(Not(phi),t()) -> not(ver(phi,t()))
     ver(Exists(n,phi),t()) -> or(ver(phi,cons(n,t())),ver(phi,t()))
     not(tt()) -> ff()
     not(ff()) -> tt()
    interpretation:
     [in#](x0, x1) = 5x1,
     
     [ver#](x0, x1) = 4x0 + 2x1,
     
     [main#](x0) = 4x0 + 1,
     
     [eq#](x0, x1) = 3x1 + 6,
     
     [or#](x0, x1) = x0 + 3,
     
     [not#](x0) = 4x0,
     
     [Exists](x0, x1) = 2x0 + 2x1 + 6,
     
     [Not](x0) = 6x0 + 2,
     
     [Or](x0, x1) = x0 + x1 + 4,
     
     [t] = 0,
     
     [Var](x0) = 4,
     
     [cons](x0, x1) = 2x0 + 4x1 + 4,
     
     [in](x0, x1) = 4x1 + 3,
     
     [ver](x0, x1) = 4x0 + 4x1 + 1,
     
     [nil] = 0,
     
     [s](x0) = x0 + 3,
     
     [eq](x0, x1) = 6x1 + 6,
     
     [0] = 1,
     
     [or](x0, x1) = x0 + x1 + 3,
     
     [ff] = 0,
     
     [not](x0) = 7,
     
     [tt] = 7
    orientation:
     eq#(s(x),s(y)) = 3y + 15 >= 3y + 6 = eq#(x,y)
     
     main#(phi) = 4phi + 1 >= 4phi = ver#(phi,nil())
     
     in#(x,cons(a,l)) = 10a + 20l + 20 >= 5l = in#(x,l)
     
     in#(x,cons(a,l)) = 10a + 20l + 20 >= 3a + 6 = eq#(x,a)
     
     in#(x,cons(a,l)) = 10a + 20l + 20 >= 6a + 9 = or#(eq(x,a),in(x,l))
     
     ver#(Var(x),t()) = 16 >= 0 = in#(x,t())
     
     ver#(Or(phi,psi),t()) = 4phi + 4psi + 16 >= 4psi = ver#(psi,t())
     
     ver#(Or(phi,psi),t()) = 4phi + 4psi + 16 >= 4phi = ver#(phi,t())
     
     ver#(Or(phi,psi),t()) = 4phi + 4psi + 16 >= 4phi + 4 = or#(ver(phi,t()),ver(psi,t()))
     
     ver#(Not(phi),t()) = 24phi + 8 >= 4phi = ver#(phi,t())
     
     ver#(Not(phi),t()) = 24phi + 8 >= 16phi + 4 = not#(ver(phi,t()))
     
     ver#(Exists(n,phi),t()) = 8n + 8phi + 24 >= 4phi = ver#(phi,t())
     
     ver#(Exists(n,phi),t()) = 8n + 8phi + 24 >= 4n + 4phi + 8 = ver#(phi,cons(n,t()))
     
     ver#(Exists(n,phi),t()) = 8n + 8phi + 24 >= 8n + 4phi + 20 = or#(ver(phi,cons(n,t())),ver(phi,t()))
     
     in(x,nil()) = 3 >= 0 = ff()
     
     in(x,cons(a,l)) = 8a + 16l + 19 >= 6a + 4l + 12 = or(eq(x,a),in(x,l))
     
     or(tt(),x) = x + 10 >= 7 = tt()
     
     or(ff(),x) = x + 3 >= x = x
     
     eq(0(),0()) = 12 >= 7 = tt()
     
     eq(s(x),0()) = 12 >= 0 = ff()
     
     eq(0(),s(y)) = 6y + 24 >= 0 = ff()
     
     eq(s(x),s(y)) = 6y + 24 >= 6y + 6 = eq(x,y)
     
     ver(Var(x),t()) = 17 >= 3 = in(x,t())
     
     ver(Or(phi,psi),t()) = 4phi + 4psi + 17 >= 4phi + 4psi + 5 = or(ver(phi,t()),ver(psi,t()))
     
     ver(Not(phi),t()) = 24phi + 9 >= 7 = not(ver(phi,t()))
     
     ver(Exists(n,phi),t()) = 8n + 8phi + 25 >= 8n + 8phi + 21 = or(ver(phi,cons(n,t())),ver(phi,t()))
     
     not(tt()) = 7 >= 0 = ff()
     
     not(ff()) = 7 >= 7 = tt()
    problem:
     DPs:
      
     TRS:
      in(x,nil()) -> ff()
      in(x,cons(a,l)) -> or(eq(x,a),in(x,l))
      or(tt(),x) -> tt()
      or(ff(),x) -> x
      eq(0(),0()) -> tt()
      eq(s(x),0()) -> ff()
      eq(0(),s(y)) -> ff()
      eq(s(x),s(y)) -> eq(x,y)
      ver(Var(x),t()) -> in(x,t())
      ver(Or(phi,psi),t()) -> or(ver(phi,t()),ver(psi,t()))
      ver(Not(phi),t()) -> not(ver(phi,t()))
      ver(Exists(n,phi),t()) -> or(ver(phi,cons(n,t())),ver(phi,t()))
      not(tt()) -> ff()
      not(ff()) -> tt()
    Qed