(*  Title:      FOLP/ex/int.ML
    ID:         $Id: int.ML,v 1.9 2005/09/18 12:25:50 wenzelm Exp $
    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]
*)

goal (theory "IFOLP") "?p : ~~(P&Q) <-> ~~P & ~~Q";
by (IntPr.fast_tac 1);
result();

goal (theory "IFOLP") "?p : ~~~P <-> ~P";
by (IntPr.fast_tac 1);
result();

goal (theory "IFOLP") "?p : ~~((P --> Q | R)  -->  (P-->Q) | (P-->R))";
by (IntPr.fast_tac 1);
result();

goal (theory "IFOLP") "?p : (P<->Q) <-> (Q<->P)";
by (IntPr.fast_tac 1);
result();


writeln"Lemmas for the propositional double-negation translation";

goal (theory "IFOLP") "?p : P --> ~~P";
by (IntPr.fast_tac 1);
result();

goal (theory "IFOLP") "?p : ~~(~~P --> P)";
by (IntPr.fast_tac 1);
result();

goal (theory "IFOLP") "?p : ~~P & ~~(P --> Q) --> ~~Q";
by (IntPr.fast_tac 1);
result();


writeln"The following are classically but not constructively valid.";

(*The attempt to prove them terminates quickly!*)
goal (theory "IFOLP") "?p : ((P-->Q) --> P)  -->  P";
by (IntPr.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
(*Check that subgoals remain: proof failed.*)
getgoal 1;  

goal (theory "IFOLP") "?p : (P&Q-->R)  -->  (P-->R) | (Q-->R)";
by (IntPr.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
getgoal 1;  


writeln"Intuitionistic FOL: propositional problems based on Pelletier.";

writeln"Problem ~~1";
goal (theory "IFOLP") "?p : ~~((P-->Q)  <->  (~Q --> ~P))";
by (IntPr.fast_tac 1);
result();
(*5 secs*)


writeln"Problem ~~2";
goal (theory "IFOLP") "?p : ~~(~~P  <->  P)";
by (IntPr.fast_tac 1);
result();
(*1 secs*)


writeln"Problem 3";
goal (theory "IFOLP") "?p : ~(P-->Q) --> (Q-->P)";
by (IntPr.fast_tac 1);
result();

writeln"Problem ~~4";
goal (theory "IFOLP") "?p : ~~((~P-->Q)  <->  (~Q --> P))";
by (IntPr.fast_tac 1);
result();
(*9 secs*)

writeln"Problem ~~5";
goal (theory "IFOLP") "?p : ~~((P|Q-->P|R) --> P|(Q-->R))";
by (IntPr.fast_tac 1);
result();
(*10 secs*)


writeln"Problem ~~6";
goal (theory "IFOLP") "?p : ~~(P | ~P)";
by (IntPr.fast_tac 1);
result();

writeln"Problem ~~7";
goal (theory "IFOLP") "?p : ~~(P | ~~~P)";
by (IntPr.fast_tac 1);
result();

writeln"Problem ~~8.  Peirce's law";
goal (theory "IFOLP") "?p : ~~(((P-->Q) --> P)  -->  P)";
by (IntPr.fast_tac 1);
result();

writeln"Problem 9";
goal (theory "IFOLP") "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)";
by (IntPr.fast_tac 1);
result();
(*9 secs*)


writeln"Problem 10";
goal (theory "IFOLP") "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)";
by (IntPr.fast_tac 1);
result();

writeln"11.  Proved in each direction (incorrectly, says Pelletier!!) ";
goal (theory "IFOLP") "?p : P<->P";
by (IntPr.fast_tac 1);

writeln"Problem ~~12.  Dijkstra's law  ";
goal (theory "IFOLP") "?p : ~~(((P <-> Q) <-> R)  <->  (P <-> (Q <-> R)))";
by (IntPr.fast_tac 1);
result();

goal (theory "IFOLP") "?p : ((P <-> Q) <-> R)  -->  ~~(P <-> (Q <-> R))";
by (IntPr.fast_tac 1);
result();

writeln"Problem 13.  Distributive law";
goal (theory "IFOLP") "?p : P | (Q & R)  <-> (P | Q) & (P | R)";
by (IntPr.fast_tac 1);
result();

writeln"Problem ~~14";
goal (theory "IFOLP") "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))";
by (IntPr.fast_tac 1);
result();

writeln"Problem ~~15";
goal (theory "IFOLP") "?p : ~~((P --> Q) <-> (~P | Q))";
by (IntPr.fast_tac 1);
result();

writeln"Problem ~~16";
goal (theory "IFOLP") "?p : ~~((P-->Q) | (Q-->P))";
by (IntPr.fast_tac 1);
result();

writeln"Problem ~~17";
goal (theory "IFOLP")
  "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))";
by (IntPr.fast_tac 1);
(*over 5 minutes?? -- printing the proof term takes 40 secs!!*)
result();


writeln"** Examples with quantifiers **";

writeln"The converse is classical in the following implications...";

goal (theory "IFOLP") "?p : (EX x. P(x)-->Q)  -->  (ALL x. P(x)) --> Q";
by (IntPr.fast_tac 1); 
result();  

goal (theory "IFOLP") "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)";
by (IntPr.fast_tac 1); 
result();  

goal (theory "IFOLP") "?p : ((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))";
by (IntPr.fast_tac 1); 
result();  

goal (theory "IFOLP") "?p : (ALL x. P(x)) | Q  -->  (ALL x. P(x) | Q)";
by (IntPr.fast_tac 1); 
result();  

goal (theory "IFOLP") "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))";
by (IntPr.fast_tac 1);
result();  




writeln"The following are not constructively valid!";
(*The attempt to prove them terminates quickly!*)

goal (theory "IFOLP") "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)";
by (IntPr.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
getgoal 1; 

goal (theory "IFOLP") "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))";
by (IntPr.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
getgoal 1; 

goal (theory "IFOLP") "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)";
by (IntPr.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
getgoal 1; 

goal (theory "IFOLP") "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))";
by (IntPr.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
getgoal 1; 

(*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
goal (theory "IFOLP") "?p : EX x. Q(x) --> (ALL x. Q(x))";
by (IntPr.fast_tac 1) handle ERROR => writeln"Failed, as expected";  
getgoal 1; 


writeln"Hard examples with quantifiers";

(*The ones that have not been proved are not known to be valid!
  Some will require quantifier duplication -- not currently available*)

writeln"Problem ~~18";
goal (theory "IFOLP") "?p : ~~(EX y. ALL x. P(y)-->P(x))";
(*NOT PROVED*)

writeln"Problem ~~19";
goal (theory "IFOLP") "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))";
(*NOT PROVED*)

writeln"Problem 20";
goal (theory "IFOLP") "?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 (IntPr.fast_tac 1); 
result();

writeln"Problem 21";
goal (theory "IFOLP")
    "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))";
(*NOT PROVED*)

writeln"Problem 22";
goal (theory "IFOLP") "?p : (ALL x. P <-> Q(x))  -->  (P <-> (ALL x. Q(x)))";
by (IntPr.fast_tac 1); 
result();

writeln"Problem ~~23";
goal (theory "IFOLP") "?p : ~~ ((ALL x. P | Q(x))  <->  (P | (ALL x. Q(x))))";
by (IntPr.best_tac 1);  
result();

writeln"Problem 24";
goal (theory "IFOLP") "?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*)
by IntPr.safe_tac; 
by (etac impE 1);
by (IntPr.fast_tac 1); 
by (IntPr.fast_tac 1); 
result();

writeln"Problem 25";
goal (theory "IFOLP") "?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 (IntPr.best_tac 1);
result();

writeln"Problem 29.  Essentially the same as Principia Mathematica *11.71";
goal (theory "IFOLP") "?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 (IntPr.fast_tac 1); 
result();

writeln"Problem ~~30";
goal (theory "IFOLP") "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) & \
\       (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))  \
\   --> (ALL x. ~~S(x))";
by (IntPr.fast_tac 1);  
result();

writeln"Problem 31";
goal (theory "IFOLP") "?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 (IntPr.fast_tac 1);
result();

writeln"Problem 32";
goal (theory "IFOLP") "?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 (IntPr.best_tac 1);  (*SLOW*)
result();

writeln"Problem 39";
goal (theory "IFOLP") "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))";
by (IntPr.fast_tac 1);
result();

writeln"Problem 40.  AMENDED";
goal (theory "IFOLP") "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->  \
\             ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))";
by (IntPr.fast_tac 1);
result();

writeln"Problem 44";
goal (theory "IFOLP") "?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 (IntPr.fast_tac 1);
result();

writeln"Problem 48";
goal (theory "IFOLP") "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c";
by (IntPr.fast_tac 1);
result();

writeln"Problem 51";
goal (theory "IFOLP")
    "?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 (IntPr.best_tac 1);  (*60 seconds*)
result();

writeln"Problem 56";
goal (theory "IFOLP")
    "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))";
by (IntPr.fast_tac 1);
result();

writeln"Problem 57";
goal (theory "IFOLP")
    "?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 (IntPr.fast_tac 1);
result();

writeln"Problem 60";
goal (theory "IFOLP")
    "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))";
by (IntPr.fast_tac 1);
result();