src/FOLP/ex/Intuitionistic.thy
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+(*  Title:      FOLP/ex/Intuitionistic.thy
+    ID:         \$Id\$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1991  University of Cambridge
+
+Intuitionistic First-Order Logic.
+
+Single-step commands:
+by (IntPr.step_tac 1)
+by (biresolve_tac safe_brls 1);
+by (biresolve_tac haz_brls 1);
+by (assume_tac 1);
+by (IntPr.safe_tac 1);
+by (IntPr.mp_tac 1);
+by (IntPr.fast_tac 1);
+*)
+
+(*Note: for PROPOSITIONAL formulae...
+  ~A is classically provable iff it is intuitionistically provable.
+  Therefore A is classically provable iff ~~A is intuitionistically provable.
+
+Let Q be the conjuction of the propositions A|~A, one for each atom A in
+P.  If P is provable classically, then clearly P&Q is provable
+intuitionistically, so ~~(P&Q) is also provable intuitionistically.
+The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P,
+since ~~Q is intuitionistically provable.  Finally, if P is a negation then
+~~P is intuitionstically equivalent to P.  [Andy Pitts]
+*)
+
+theory Intuitionistic
+imports IFOLP
+begin
+
+lemma "?p : ~~(P&Q) <-> ~~P & ~~Q"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : ~~~P <-> ~P"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : ~~((P --> Q | R)  -->  (P-->Q) | (P-->R))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : (P<->Q) <-> (Q<->P)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+
+subsection {* Lemmas for the propositional double-negation translation *}
+
+lemma "?p : P --> ~~P"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : ~~(~~P --> P)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : ~~P & ~~(P --> Q) --> ~~Q"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+
+subsection {* The following are classically but not constructively valid *}
+
+(*The attempt to prove them terminates quickly!*)
+lemma "?p : ((P-->Q) --> P)  -->  P"
+  apply (tactic {* IntPr.fast_tac 1 *})?
+  oops
+
+lemma "?p : (P&Q-->R)  -->  (P-->R) | (Q-->R)"
+  apply (tactic {* IntPr.fast_tac 1 *})?
+  oops
+
+
+subsection {* Intuitionistic FOL: propositional problems based on Pelletier *}
+
+text "Problem ~~1"
+lemma "?p : ~~((P-->Q)  <->  (~Q --> ~P))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~2"
+lemma "?p : ~~(~~P  <->  P)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem 3"
+lemma "?p : ~(P-->Q) --> (Q-->P)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~4"
+lemma "?p : ~~((~P-->Q)  <->  (~Q --> P))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~5"
+lemma "?p : ~~((P|Q-->P|R) --> P|(Q-->R))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~6"
+lemma "?p : ~~(P | ~P)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~7"
+lemma "?p : ~~(P | ~~~P)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~8.  Peirce's law"
+lemma "?p : ~~(((P-->Q) --> P)  -->  P)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem 9"
+lemma "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem 10"
+lemma "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "11.  Proved in each direction (incorrectly, says Pelletier!!) "
+lemma "?p : P<->P"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~12.  Dijkstra's law  "
+lemma "?p : ~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : ((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem 13.  Distributive law"
+lemma "?p : P | (Q & R)  <-> (P | Q) & (P | R)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~14"
+lemma "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~15"
+lemma "?p : ~~((P --> Q) <-> (~P | Q))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~16"
+lemma "?p : ~~((P-->Q) | (Q-->P))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~17"
+lemma "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"
+  by (tactic {* IntPr.fast_tac 1 *})  -- slow
+
+
+subsection {* Examples with quantifiers *}
+
+text "The converse is classical in the following implications..."
+
+lemma "?p : (EX x. P(x)-->Q)  -->  (ALL x. P(x)) --> Q"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : ((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : (ALL x. P(x)) | Q  -->  (ALL x. P(x) | Q)"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+lemma "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+
+text "The following are not constructively valid!"
+text "The attempt to prove them terminates quickly!"
+
+lemma "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)"
+  apply (tactic {* IntPr.fast_tac 1 *})?
+  oops
+
+lemma "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))"
+  apply (tactic {* IntPr.fast_tac 1 *})?
+  oops
+
+lemma "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)"
+  apply (tactic {* IntPr.fast_tac 1 *})?
+  oops
+
+lemma "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))"
+  apply (tactic {* IntPr.fast_tac 1 *})?
+  oops
+
+(*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
+lemma "?p : EX x. Q(x) --> (ALL x. Q(x))"
+  apply (tactic {* IntPr.fast_tac 1 *})?
+  oops
+
+
+subsection "Hard examples with quantifiers"
+
+text {*
+  The ones that have not been proved are not known to be valid!
+  Some will require quantifier duplication -- not currently available.
+*}
+
+text "Problem ~~18"
+lemma "?p : ~~(EX y. ALL x. P(y)-->P(x))" oops
+(*NOT PROVED*)
+
+text "Problem ~~19"
+lemma "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))" oops
+(*NOT PROVED*)
+
+text "Problem 20"
+lemma "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))
+    --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem 21"
+lemma "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))" oops
+(*NOT PROVED*)
+
+text "Problem 22"
+lemma "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem ~~23"
+lemma "?p : ~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))"
+  by (tactic {* IntPr.fast_tac 1 *})
+
+text "Problem 24"
+lemma "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &
+     (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x))
+    --> ~~(EX x. P(x)&R(x))"
+(*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*)
+  apply (tactic "IntPr.safe_tac")
+  apply (erule impE)
+   apply (tactic "IntPr.fast_tac 1")
+  apply (tactic "IntPr.fast_tac 1")
+  done
+
+text "Problem 25"
+lemma "?p : (EX x. P(x)) &
+        (ALL x. L(x) --> ~ (M(x) & R(x))) &
+        (ALL x. P(x) --> (M(x) & L(x))) &
+        ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))
+    --> (EX x. Q(x)&P(x))"
+  by (tactic "IntPr.best_tac 1")
+
+text "Problem 29.  Essentially the same as Principia Mathematica *11.71"
+lemma "?p : (EX x. P(x)) & (EX y. Q(y))
+    --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->
+         (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"
+  by (tactic "IntPr.fast_tac 1")
+
+text "Problem ~~30"
+lemma "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) &
+        (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))
+    --> (ALL x. ~~S(x))"
+  by (tactic "IntPr.fast_tac 1")
+
+text "Problem 31"
+lemma "?p : ~(EX x. P(x) & (Q(x) | R(x))) &
+        (EX x. L(x) & P(x)) &
+        (ALL x. ~ R(x) --> M(x))
+    --> (EX x. L(x) & M(x))"
+  by (tactic "IntPr.fast_tac 1")
+
+text "Problem 32"
+lemma "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) &
+        (ALL x. S(x) & R(x) --> L(x)) &
+        (ALL x. M(x) --> R(x))
+    --> (ALL x. P(x) & M(x) --> L(x))"
+  by (tactic "IntPr.best_tac 1") -- slow
+
+text "Problem 39"
+lemma "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
+  by (tactic "IntPr.best_tac 1")
+
+text "Problem 40.  AMENDED"
+lemma "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->
+              ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"
+  by (tactic "IntPr.best_tac 1") -- slow
+
+text "Problem 44"
+lemma "?p : (ALL x. f(x) -->
+              (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &
+              (EX x. j(x) & (ALL y. g(y) --> h(x,y)))
+              --> (EX x. j(x) & ~f(x))"
+  by (tactic "IntPr.best_tac 1")
+
+text "Problem 48"
+lemma "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"
+  by (tactic "IntPr.best_tac 1")
+
+text "Problem 51"
+lemma
+    "?p : (EX z w. ALL x y. P(x,y) <->  (x=z & y=w)) -->
+     (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"
+  by (tactic "IntPr.best_tac 1") -- {*60 seconds*}
+
+text "Problem 56"
+lemma "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
+  by (tactic "IntPr.best_tac 1")
+
+text "Problem 57"
+lemma
+    "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) &
+     (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))"
+  by (tactic "IntPr.best_tac 1")
+
+text "Problem 60"
+lemma "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
+  by (tactic "IntPr.best_tac 1")
+
+end```