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()))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [in#](x0, x1) = x0 + 1/2x1 + 2,
    
    [ver#](x0, x1) = 2x0,
    
    [main#](x0) = 2x0 + 1/2,
    
    [eq#](x0, x1) = x0 + x1,
    
    [or#](x0, x1) = 1/2x0 + 1/2,
    
    [not#](x0) = 0,
    
    [Exists](x0, x1) = 7/2x0 + 2x1 + 1/2,
    
    [Not](x0) = x0 + 1,
    
    [Or](x0, x1) = 2x0 + 3/2x1 + 5/2,
    
    [t] = 0,
    
    [Var](x0) = 2x0 + 7/2,
    
    [cons](x0, x1) = 3x0 + 2x1 + 1,
    
    [in](x0, x1) = 0,
    
    [ver](x0, x1) = 0,
    
    [nil] = 0,
    
    [main](x0) = 2,
    
    [s](x0) = 5/2x0 + 5/2,
    
    [eq](x0, x1) = 2x0 + 2,
    
    [0] = 1,
    
    [or](x0, x1) = 2x1,
    
    [ff] = 0,
    
    [not](x0) = 0,
    
    [tt] = 0
   orientation:
    eq#(s(x),s(y)) = 5/2x + 5/2y + 5 >= x + y = eq#(x,y)
    
    main#(phi) = 2phi + 1/2 >= 2phi = ver#(phi,nil())
    
    in#(x,cons(a,l)) = 3/2a + l + x + 5/2 >= 1/2l + x + 2 = in#(x,l)
    
    in#(x,cons(a,l)) = 3/2a + l + x + 5/2 >= a + x = eq#(x,a)
    
    in#(x,cons(a,l)) = 3/2a + l + x + 5/2 >= x + 3/2 = or#(eq(x,a),in(x,l))
    
    ver#(Var(x),t()) = 4x + 7 >= x + 2 = in#(x,t())
    
    ver#(Or(phi,psi),t()) = 4phi + 3psi + 5 >= 2psi = ver#(psi,t())
    
    ver#(Or(phi,psi),t()) = 4phi + 3psi + 5 >= 2phi = ver#(phi,t())
    
    ver#(Or(phi,psi),t()) = 4phi + 3psi + 5 >= 1/2 = or#(ver(phi,t()),ver(psi,t()))
    
    ver#(Not(phi),t()) = 2phi + 2 >= 2phi = ver#(phi,t())
    
    ver#(Not(phi),t()) = 2phi + 2 >= 0 = not#(ver(phi,t()))
    
    ver#(Exists(n,phi),t()) = 7n + 4phi + 1 >= 2phi = ver#(phi,t())
    
    ver#(Exists(n,phi),t()) = 7n + 4phi + 1 >= 2phi = ver#(phi,cons(n,t()))
    
    ver#(Exists(n,phi),t()) = 7n + 4phi + 1 >= 1/2 = or#(ver(phi,cons(n,t())),ver(phi,t()))
    
    not(tt()) = 0 >= 0 = ff()
    
    not(ff()) = 0 >= 0 = tt()
    
    or(tt(),x) = 2x >= 0 = tt()
    
    or(ff(),x) = 2x >= x = x
    
    eq(0(),0()) = 4 >= 0 = tt()
    
    eq(s(x),0()) = 5x + 7 >= 0 = ff()
    
    eq(0(),s(y)) = 4 >= 0 = ff()
    
    eq(s(x),s(y)) = 5x + 7 >= 2x + 2 = eq(x,y)
    
    main(phi) = 2 >= 0 = ver(phi,nil())
    
    in(x,nil()) = 0 >= 0 = ff()
    
    in(x,cons(a,l)) = 0 >= 0 = or(eq(x,a),in(x,l))
    
    ver(Var(x),t()) = 0 >= 0 = in(x,t())
    
    ver(Or(phi,psi),t()) = 0 >= 0 = or(ver(phi,t()),ver(psi,t()))
    
    ver(Not(phi),t()) = 0 >= 0 = not(ver(phi,t()))
    
    ver(Exists(n,phi),t()) = 0 >= 0 = or(ver(phi,cons(n,t())),ver(phi,t()))
   problem:
    DPs:
     
    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()))
   Qed