Theory Hard_Quantifiers

(*  Title:      Sequents/LK/Hard_Quantifiers.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

Hard examples with quantifiers.  Can be read to test the LK system.
From  F. J. Pelletier,
  Seventy-Five Problems for Testing Automatic Theorem Provers,
  J. Automated Reasoning 2 (1986), 191-216.
  Errata, JAR 4 (1988), 236-236.

Uses pc_tac rather than fast_tac when the former is significantly faster.

theory Hard_Quantifiers
imports "../LK"

lemma " (x. P(x)  Q(x))  (x. P(x))  (x. Q(x))"
  by fast

lemma " (x. P  Q(x))  (P  (x. Q(x)))"
  by fast

lemma " (x. P(x)  Q)  (x. P(x))  Q"
  by fast

lemma " (x. P(x))  Q  (x. P(x)  Q)"
  by fast

text "Problems requiring quantifier duplication"

(*Not provable by fast: needs multiple instantiation of ∀*)
lemma " (x. P(x)  P(f(x)))  P(d)  P(f(f(f(d))))"
  by best_dup

(*Needs double instantiation of the quantifier*)
lemma " x. P(x)  P(a)  P(b)"
  by fast_dup

lemma " z. P(z)  (x. P(x))"
  by best_dup

text "Hard examples with quantifiers"

text "Problem 18"
lemma " y. x. P(y)P(x)"
  by best_dup

text "Problem 19"
lemma " x. y z. (P(y)Q(z))  (P(x)Q(x))"
  by best_dup

text "Problem 20"
lemma " (x y. z. w. (P(x)  Q(y)R(z)  S(w)))
     (x y. P(x)  Q(y))  (z. R(z))"
  by fast

text "Problem 21"
lemma " (x. P  Q(x))  (x. Q(x)  P)  (x. P  Q(x))"
  by best_dup

text "Problem 22"
lemma " (x. P  Q(x))  (P  (x. Q(x)))"
  by fast

text "Problem 23"
lemma " (x. P  Q(x))  (P  (x. Q(x)))"
  by best

text "Problem 24"
lemma " ¬ (x. S(x)  Q(x))  (x. P(x)  Q(x)  R(x)) 
     ¬ (x. P(x))  (x. Q(x))  (x. Q(x)  R(x)  S(x))
     (x. P(x)  R(x))"
  by pc

text "Problem 25"
lemma " (x. P(x)) 
        (x. L(x)  ¬ (M(x)  R(x))) 
        (x. P(x)  (M(x)  L(x))) 
        ((x. P(x)Q(x))  (x. P(x)  R(x)))
     (x. Q(x)  P(x))"
  by best

text "Problem 26"
lemma " ((x. p(x))  (x. q(x))) 
      (x. y. p(x)  q(y)  (r(x)  s(y)))
   ((x. p(x)r(x))  (x. q(x)s(x)))"
  by pc

text "Problem 27"
lemma " (x. P(x)  ¬ Q(x)) 
              (x. P(x)  R(x)) 
              (x. M(x)  L(x)  P(x)) 
              ((x. R(x)  ¬ Q(x))  (x. L(x)  ¬ R(x)))
           (x. M(x)  ¬ L(x))"
  by pc

text "Problem 28.  AMENDED"
lemma " (x. P(x)  (x. Q(x))) 
        ((x. Q(x)  R(x))  (x. Q(x)  S(x))) 
        ((x. S(x))  (x. L(x)  M(x)))
     (x. P(x)  L(x)  M(x))"
  by pc

text "Problem 29.  Essentially the same as Principia Mathematica *11.71"
lemma " (x. P(x))  (y. Q(y))
     ((x. P(x)  R(x))  (y. Q(y)  S(y)) 
         (x y. P(x)  Q(y)  R(x)  S(y)))"
  by pc

text "Problem 30"
lemma " (x. P(x)  Q(x)  ¬ R(x)) 
        (x. (Q(x)  ¬ S(x))  P(x)  R(x))
     (x. S(x))"
  by fast

text "Problem 31"
lemma " ¬ (x. P(x)  (Q(x)  R(x))) 
        (x. L(x)  P(x)) 
        (x. ¬ R(x)  M(x))
     (x. L(x)  M(x))"
  by fast

text "Problem 32"
lemma " (x. P(x)  (Q(x)  R(x))  S(x)) 
        (x. S(x)  R(x)  L(x)) 
        (x. M(x)  R(x))
     (x. P(x)  M(x)  L(x))"
  by best

text "Problem 33"
lemma " (x. P(a)  (P(x)  P(b))  P(c)) 
     (x. (¬ P(a)  P(x)  P(c))  (¬ P(a)  ¬ P(b)  P(c)))"
  by fast

text "Problem 34  AMENDED (TWICE!!)"
(*Andrews's challenge*)
lemma " ((x. y. p(x)  p(y))  
               ((x. q(x))  (y. p(y))))     
              ((x. y. q(x)  q(y))  
               ((x. p(x))  (y. q(y))))"
  by best_dup

text "Problem 35"
lemma " x y. P(x,y)  (u v. P(u,v))"
  by best_dup

text "Problem 36"
lemma " (x. y. J(x,y)) 
         (x. y. G(x,y)) 
         (x y. J(x,y)  G(x,y) 
         (z. J(y,z)  G(y,z)  H(x,z)))
          (x. y. H(x,y))"
  by fast

text "Problem 37"
lemma " (z. w. x. y.
           (P(x,z)P(y,w))  P(y,z)  (P(y,w)  (u. Q(u,w)))) 
        (x z. ¬ P(x,z)  (y. Q(y,z))) 
        ((x y. Q(x,y))  (x. R(x,x)))
     (x. y. R(x,y))"
  by pc

text "Problem 38"
lemma " (x. p(a)  (p(x)  (y. p(y)  r(x,y))) 
                 (z. w. p(z)  r(x,w)  r(w,z)))  
         (x. (¬ p(a)  p(x)  (z. w. p(z)  r(x,w)  r(w,z))) 
                 (¬ p(a)  ¬ (y. p(y)  r(x,y)) 
                 (z. w. p(z)  r(x,w)  r(w,z))))"
  by pc

text "Problem 39"
lemma " ¬ (x. y. F(y,x)  ¬ F(y,y))"
  by fast

text "Problem 40.  AMENDED"
lemma " (y. x. F(x,y)  F(x,x)) 
         ¬ (x. y. z. F(z,y)  ¬ F(z,x))"
  by fast

text "Problem 41"
lemma " (z. y. x. f(x,y)  f(x,z)  ¬ f(x,x))
          ¬ (z. x. f(x,z))"
  by fast

text "Problem 42"
lemma " ¬ (y. x. p(x,y)  ¬ (z. p(x,z)  p(z,x)))"

text "Problem 43"
lemma " (x. y. q(x,y)  (z. p(z,x)  p(z,y)))
           (x. (y. q(x,y)  q(y,x)))"

text "Problem 44"
lemma " (x. f(x) 
                 (y. g(y)  h(x,y)  (y. g(y)  ¬ h(x,y)))) 
         (x. j(x)  (y. g(y)  h(x,y)))
          (x. j(x)  ¬ f(x))"
  by fast

text "Problem 45"
lemma " (x. f(x)  (y. g(y)  h(x,y)  j(x,y))
                       (y. g(y)  h(x,y)  k(y))) 
      ¬ (y. l(y)  k(y)) 
      (x. f(x)  (y. h(x,y)  l(y))
                    (y. g(y)  h(x,y)  j(x,y)))
       (x. f(x)  ¬ (y. g(y)  h(x,y)))"
  by best

text "Problems (mainly) involving equality or functions"

text "Problem 48"
lemma " (a = b  c = d)  (a = c  b = d)  a = d  b = c"
  by (fast add!: subst)

text "Problem 50"
lemma " (x. P(a,x)  (y. P(x,y)))  (x. y. P(x,y))"
  by best_dup

text "Problem 51"
lemma " (z w. x y. P(x,y)  (x = z  y = w)) 
         (z. x. w. (y. P(x,y)  y = w)  x = z)"
  by (fast add!: subst)

text "Problem 52"  (*Almost the same as 51. *)
lemma " (z w. x y. P(x,y)  (x = z  y = w)) 
         (w. y. z. (x. P(x,y)  x = z)  y = w)"
  by (fast add!: subst)

text "Problem 56"
lemma " (x.(y. P(y)  x = f(y))  P(x))  (x. P(x)  P(f(x)))"
  by (best add: symL subst)
  (*requires tricker to orient the equality properly*)

text "Problem 57"
lemma " P(f(a,b), f(b,c))  P(f(b,c), f(a,c)) 
         (x y z. P(x,y)  P(y,z)  P(x,z))  P(f(a,b), f(a,c))"
  by fast

text "Problem 58!"
lemma " (x y. f(x) = g(y))  (x y. f(f(x)) = f(g(y)))"
  by (fast add!: subst)

text "Problem 59"
(*Unification works poorly here -- the abstraction %sobj prevents efficient
  operation of the occurs check*)
lemma " (x. P(x)  ¬ P(f(x)))  (x. P(x)  ¬ P(f(x)))"
  using [[unify_trace_bound = 50]]
  by best_dup

text "Problem 60"
lemma " x. P(x,f(x))  (y. (z. P(z,y)  P(z,f(x)))  P(x,y))"
  by fast

text "Problem 62 as corrected in JAR 18 (1997), page 135"
lemma " (x. p(a)  (p(x)  p(f(x)))  p(f(f(x)))) 
      (x. (¬ p(a)  p(x)  p(f(f(x)))) 
              (¬ p(a)  ¬ p(f(x))  p(f(f(x)))))"
  by fast

(*18 June 92: loaded in 372 secs*)
(*19 June 92: loaded in 166 secs except #34, using repeat_goal_tac*)
(*29 June 92: loaded in 370 secs*)
(*18 September 2005: loaded in 1.809 secs*)