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+(* Title: FOL/ex/Intuitionistic
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1991 University of Cambridge
+*)
+
+header{*Intuitionistic First-Order Logic*}
+
+theory Intuitionistic = IFOL:
+
+(*
+Single-step ML 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);
+*)
+
+
+text{*Metatheorem (for \emph{propositional} formulae):
+ $P$ is classically provable iff $\neg\neg P$ is intuitionistically provable.
+ Therefore $\neg P$ is classically provable iff it is intuitionistically
+ provable.
+
+Proof: Let $Q$ be the conjuction of the propositions $A\vee\neg A$, one for
+each atom $A$ in $P$. Now $\neg\neg Q$ is intuitionistically provable because
+$\neg\neg(A\vee\neg A)$ is and because double-negation distributes over
+conjunction. If $P$ is provable classically, then clearly $Q\rightarrow P$ is
+provable intuitionistically, so $\neg\neg(Q\rightarrow P)$ is also provable
+intuitionistically. The latter is intuitionistically equivalent to $\neg\neg
+Q\rightarrow\neg\neg P$, hence to $\neg\neg P$, since $\neg\neg Q$ is
+intuitionistically provable. Finally, if $P$ is a negation then $\neg\neg P$
+is intuitionstically equivalent to $P$. [Andy Pitts] *}
+
+lemma "~~(P&Q) <-> ~~P & ~~Q"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "~~ ((~P --> Q) --> (~P --> ~Q) --> P)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*Double-negation does NOT distribute over disjunction*}
+
+lemma "~~(P-->Q) <-> (~~P --> ~~Q)"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "~~~P <-> ~P"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "~~((P --> Q | R) --> (P-->Q) | (P-->R))"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "(P<->Q) <-> (Q<->P)"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "((P --> (Q | (Q-->R))) --> R) --> R"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "(((G-->A) --> J) --> D --> E) --> (((H-->B)-->I)-->C-->J)
+ --> (A-->H) --> F --> G --> (((C-->B)-->I)-->D)-->(A-->C)
+ --> (((F-->A)-->B) --> I) --> E"
+by (tactic{*IntPr.fast_tac 1*})
+
+
+text{*Lemmas for the propositional double-negation translation*}
+
+lemma "P --> ~~P"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "~~(~~P --> P)"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "~~P & ~~(P --> Q) --> ~~Q"
+by (tactic{*IntPr.fast_tac 1*})
+
+
+text{*The following are classically but not constructively valid.
+ The attempt to prove them terminates quickly!*}
+lemma "((P-->Q) --> P) --> P"
+apply (tactic{*IntPr.fast_tac 1*} | -)
+apply (rule asm_rl) --{*Checks that subgoals remain: proof failed.*}
+oops
+
+lemma "(P&Q-->R) --> (P-->R) | (Q-->R)"
+apply (tactic{*IntPr.fast_tac 1*} | -)
+apply (rule asm_rl) --{*Checks that subgoals remain: proof failed.*}
+oops
+
+
+subsection{*de Bruijn formulae*}
+
+text{*de Bruijn formula with three predicates*}
+lemma "((P<->Q) --> P&Q&R) &
+ ((Q<->R) --> P&Q&R) &
+ ((R<->P) --> P&Q&R) --> P&Q&R"
+by (tactic{*IntPr.fast_tac 1*})
+
+
+text{*de Bruijn formula with five predicates*}
+lemma "((P<->Q) --> P&Q&R&S&T) &
+ ((Q<->R) --> P&Q&R&S&T) &
+ ((R<->S) --> P&Q&R&S&T) &
+ ((S<->T) --> P&Q&R&S&T) &
+ ((T<->P) --> P&Q&R&S&T) --> P&Q&R&S&T"
+by (tactic{*IntPr.fast_tac 1*})
+
+
+(*** Problems from of Sahlin, Franzen and Haridi,
+ An Intuitionistic Predicate Logic Theorem Prover.
+ J. Logic and Comp. 2 (5), October 1992, 619-656.
+***)
+
+text{*Problem 1.1*}
+lemma "(ALL x. EX y. ALL z. p(x) & q(y) & r(z)) <->
+ (ALL z. EX y. ALL x. p(x) & q(y) & r(z))"
+by (tactic{*IntPr.best_dup_tac 1*}) --{*SLOW*}
+
+text{*Problem 3.1*}
+lemma "~ (EX x. ALL y. mem(y,x) <-> ~ mem(x,x))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*Problem 4.1: hopeless!*}
+lemma "(ALL x. p(x) --> p(h(x)) | p(g(x))) & (EX x. p(x)) & (ALL x. ~p(h(x)))
+ --> (EX x. p(g(g(g(g(g(x)))))))"
+oops
+
+
+subsection{*Intuitionistic FOL: propositional problems based on Pelletier.*}
+
+text{*~~1*}
+lemma "~~((P-->Q) <-> (~Q --> ~P))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~2*}
+lemma "~~(~~P <-> P)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*3*}
+lemma "~(P-->Q) --> (Q-->P)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~4*}
+lemma "~~((~P-->Q) <-> (~Q --> P))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~5*}
+lemma "~~((P|Q-->P|R) --> P|(Q-->R))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~6*}
+lemma "~~(P | ~P)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~7*}
+lemma "~~(P | ~~~P)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~8. Peirce's law*}
+lemma "~~(((P-->Q) --> P) --> P)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*9*}
+lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*10*}
+lemma "(Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)"
+by (tactic{*IntPr.fast_tac 1*})
+
+subsection{*11. Proved in each direction (incorrectly, says Pelletier!!) *}
+lemma "P<->P"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~12. Dijkstra's law *}
+lemma "~~(((P <-> Q) <-> R) <-> (P <-> (Q <-> R)))"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "((P <-> Q) <-> R) --> ~~(P <-> (Q <-> R))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*13. Distributive law*}
+lemma "P | (Q & R) <-> (P | Q) & (P | R)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~14*}
+lemma "~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~15*}
+lemma "~~((P --> Q) <-> (~P | Q))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~16*}
+lemma "~~((P-->Q) | (Q-->P))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~17*}
+lemma "~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*Dijkstra's "Golden Rule"*}
+lemma "(P&Q) <-> P <-> Q <-> (P|Q)"
+by (tactic{*IntPr.fast_tac 1*})
+
+
+subsection{*****Examples with quantifiers*****}
+
+
+subsection{*The converse is classical in the following implications...*}
+
+lemma "(EX x. P(x)-->Q) --> (ALL x. P(x)) --> Q"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "((ALL x. ~P(x))-->Q) --> ~ (ALL x. ~ (P(x)|Q))"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "(ALL x. P(x)) | Q --> (ALL x. P(x) | Q)"
+by (tactic{*IntPr.fast_tac 1*})
+
+lemma "(EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"
+by (tactic{*IntPr.fast_tac 1*})
+
+
+
+
+subsection{*The following are not constructively valid!*}
+text{*The attempt to prove them terminates quickly!*}
+
+lemma "((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)"
+apply (tactic{*IntPr.fast_tac 1*} | -)
+apply (rule asm_rl) --{*Checks that subgoals remain: proof failed.*}
+oops
+
+lemma "(P --> (EX x. Q(x))) --> (EX x. P-->Q(x))"
+apply (tactic{*IntPr.fast_tac 1*} | -)
+apply (rule asm_rl) --{*Checks that subgoals remain: proof failed.*}
+oops
+
+lemma "(ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)"
+apply (tactic{*IntPr.fast_tac 1*} | -)
+apply (rule asm_rl) --{*Checks that subgoals remain: proof failed.*}
+oops
+
+lemma "(ALL x. ~~P(x)) --> ~~(ALL x. P(x))"
+apply (tactic{*IntPr.fast_tac 1*} | -)
+apply (rule asm_rl) --{*Checks that subgoals remain: proof failed.*}
+oops
+
+text{*Classically but not intuitionistically valid. Proved by a bug in 1986!*}
+lemma "EX x. Q(x) --> (ALL x. Q(x))"
+apply (tactic{*IntPr.fast_tac 1*} | -)
+apply (rule asm_rl) --{*Checks that subgoals remain: proof failed.*}
+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{*~~18*}
+lemma "~~(EX y. ALL x. P(y)-->P(x))"
+oops --{*NOT PROVED*}
+
+text{*~~19*}
+lemma "~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))"
+oops --{*NOT PROVED*}
+
+text{*20*}
+lemma "(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{*21*}
+lemma "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))"
+oops --{*NOT PROVED; needs quantifier duplication*}
+
+text{*22*}
+lemma "(ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~23*}
+lemma "~~ ((ALL x. P | Q(x)) <-> (P | (ALL x. Q(x))))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*24*}
+lemma "~(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))"
+txt{*Not clear why @{text fast_tac}, @{text best_tac}, @{text ASTAR} and
+ @{text ITER_DEEPEN} all take forever*}
+apply (tactic{* IntPr.safe_tac*})
+apply (erule impE)
+apply (tactic{*IntPr.fast_tac 1*})
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*25*}
+lemma "(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.fast_tac 1*})
+
+text{*~~26*}
+lemma "(~~(EX x. p(x)) <-> ~~(EX x. q(x))) &
+ (ALL x. ALL y. p(x) & q(y) --> (r(x) <-> s(y)))
+ --> ((ALL x. p(x)-->r(x)) <-> (ALL x. q(x)-->s(x)))"
+oops --{*NOT PROVED*}
+
+text{*27*}
+lemma "(EX x. P(x) & ~Q(x)) &
+ (ALL x. P(x) --> R(x)) &
+ (ALL x. M(x) & L(x) --> P(x)) &
+ ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x)))
+ --> (ALL x. M(x) --> ~L(x))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*~~28. AMENDED*}
+lemma "(ALL x. P(x) --> (ALL x. Q(x))) &
+ (~~(ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) &
+ (~~(EX x. S(x)) --> (ALL x. L(x) --> M(x)))
+ --> (ALL x. P(x) & L(x) --> M(x))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*29. Essentially the same as Principia Mathematica *11.71*}
+lemma "(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{*~~30*}
+lemma "(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{*31*}
+lemma "~(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{*32*}
+lemma "(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.fast_tac 1*})
+
+text{*~~33*}
+lemma "(ALL x. ~~(P(a) & (P(x)-->P(b))-->P(c))) <->
+ (ALL x. ~~((~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c))))"
+apply (tactic{*IntPr.best_tac 1*})
+done
+
+
+text{*36*}
+lemma "(ALL x. EX y. J(x,y)) &
+ (ALL x. EX y. G(x,y)) &
+ (ALL x y. J(x,y) | G(x,y) --> (ALL z. J(y,z) | G(y,z) --> H(x,z)))
+ --> (ALL x. EX y. H(x,y))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*37*}
+lemma "(ALL z. EX w. ALL x. EX y.
+ ~~(P(x,z)-->P(y,w)) & P(y,z) & (P(y,w) --> (EX u. Q(u,w)))) &
+ (ALL x z. ~P(x,z) --> (EX y. Q(y,z))) &
+ (~~(EX x y. Q(x,y)) --> (ALL x. R(x,x)))
+ --> ~~(ALL x. EX y. R(x,y))"
+oops --{*NOT PROVED*}
+
+text{*39*}
+lemma "~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*40. AMENDED*}
+lemma "(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.fast_tac 1*})
+
+text{*44*}
+lemma "(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.fast_tac 1*})
+
+text{*48*}
+lemma "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*51*}
+lemma "(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.fast_tac 1*})
+
+text{*52*}
+text{*Almost the same as 51. *}
+lemma "(EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) -->
+ (EX w. ALL y. EX z. (ALL x. P(x,y) <-> x=z) <-> y=w)"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*56*}
+lemma "(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
+by (tactic{*IntPr.fast_tac 1*})
+
+text{*57*}
+lemma "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.fast_tac 1*})
+
+text{*60*}
+lemma "ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
+by (tactic{*IntPr.fast_tac 1*})
+
+end
+
+