(* Title: FOLP/IFOLP.ML
ID: $Id$
Author: Martin D Coen, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
(*** Sequent-style elimination rules for & --> and ALL ***)
val prems= Goal
"[| p:P&Q; !!x y.[| x:P; y:Q |] ==> f(x,y):R |] ==> ?a:R";
by (REPEAT (resolve_tac prems 1
ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN resolve_tac prems 1))) ;
qed "conjE";
val prems= Goal
"[| p:P-->Q; q:P; !!x. x:Q ==> r(x):R |] ==> ?p:R";
by (REPEAT (resolve_tac (prems@[mp]) 1)) ;
qed "impE";
val prems= Goal
"[| p:ALL x. P(x); !!y. y:P(x) ==> q(y):R |] ==> ?p:R";
by (REPEAT (resolve_tac (prems@[spec]) 1)) ;
qed "allE";
(*Duplicates the quantifier; for use with eresolve_tac*)
val prems= Goal
"[| p:ALL x. P(x); !!y z.[| y:P(x); z:ALL x. P(x) |] ==> q(y,z):R \
\ |] ==> ?p:R";
by (REPEAT (resolve_tac (prems@[spec]) 1)) ;
qed "all_dupE";
(*** Negation rules, which translate between ~P and P-->False ***)
val notI = prove_goalw (the_context ()) [not_def] "(!!x. x:P ==> q(x):False) ==> ?p:~P"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
val notE = prove_goalw (the_context ()) [not_def] "[| p:~P; q:P |] ==> ?p:R"
(fn prems=>
[ (resolve_tac [mp RS FalseE] 1),
(REPEAT (resolve_tac prems 1)) ]);
(*This is useful with the special implication rules for each kind of P. *)
val prems= Goal
"[| p:~P; !!x. x:(P-->False) ==> q(x):Q |] ==> ?p:Q";
by (REPEAT (ares_tac (prems@[impI,notE]) 1)) ;
qed "not_to_imp";
(* For substitution int an assumption P, reduce Q to P-->Q, substitute into
this implication, then apply impI to move P back into the assumptions.
To specify P use something like
eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1 *)
Goal "[| p:P; q:P --> Q |] ==> ?p:Q";
by (REPEAT (ares_tac [mp] 1)) ;
qed "rev_mp";
(*Contrapositive of an inference rule*)
val [major,minor]= Goal "[| p:~Q; !!y. y:P==>q(y):Q |] ==> ?a:~P";
by (rtac (major RS notE RS notI) 1);
by (etac minor 1) ;
qed "contrapos";
(** Unique assumption tactic.
Ignores proof objects.
Fails unless one assumption is equal and exactly one is unifiable
**)
local
fun discard_proof (Const("Proof",_) $ P $ _) = P;
in
val uniq_assume_tac =
SUBGOAL
(fn (prem,i) =>
let val hyps = map discard_proof (Logic.strip_assums_hyp prem)
and concl = discard_proof (Logic.strip_assums_concl prem)
in
if exists (fn hyp => hyp aconv concl) hyps
then case distinct (op =) (filter (fn hyp => could_unify (hyp, concl)) hyps) of
[_] => assume_tac i
| _ => no_tac
else no_tac
end);
end;
(*** Modus Ponens Tactics ***)
(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
fun mp_tac i = eresolve_tac [notE,make_elim mp] i THEN assume_tac i;
(*Like mp_tac but instantiates no variables*)
fun int_uniq_mp_tac i = eresolve_tac [notE,impE] i THEN uniq_assume_tac i;
(*** If-and-only-if ***)
val iffI = prove_goalw (the_context ()) [iff_def]
"[| !!x. x:P ==> q(x):Q; !!x. x:Q ==> r(x):P |] ==> ?p:P<->Q"
(fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);
(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
val iffE = prove_goalw (the_context ()) [iff_def]
"[| p:P <-> Q; !!x y.[| x:P-->Q; y:Q-->P |] ==> q(x,y):R |] ==> ?p:R"
(fn prems => [ (rtac conjE 1), (REPEAT (ares_tac prems 1)) ]);
(* Destruct rules for <-> similar to Modus Ponens *)
val iffD1 = prove_goalw (the_context ()) [iff_def] "[| p:P <-> Q; q:P |] ==> ?p:Q"
(fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
val iffD2 = prove_goalw (the_context ()) [iff_def] "[| p:P <-> Q; q:Q |] ==> ?p:P"
(fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
Goal "?p:P <-> P";
by (REPEAT (ares_tac [iffI] 1)) ;
qed "iff_refl";
Goal "p:Q <-> P ==> ?p:P <-> Q";
by (etac iffE 1);
by (rtac iffI 1);
by (REPEAT (eresolve_tac [asm_rl,mp] 1)) ;
qed "iff_sym";
Goal "[| p:P <-> Q; q:Q<-> R |] ==> ?p:P <-> R";
by (rtac iffI 1);
by (REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ;
qed "iff_trans";
(*** Unique existence. NOTE THAT the following 2 quantifications
EX!x such that [EX!y such that P(x,y)] (sequential)
EX!x,y such that P(x,y) (simultaneous)
do NOT mean the same thing. The parser treats EX!x y.P(x,y) as sequential.
***)
val prems = goalw (the_context ()) [ex1_def]
"[| p:P(a); !!x u. u:P(x) ==> f(u) : x=a |] ==> ?p:EX! x. P(x)";
by (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ;
qed "ex1I";
val prems = goalw (the_context ()) [ex1_def]
"[| p:EX! x. P(x); \
\ !!x u v. [| u:P(x); v:ALL y. P(y) --> y=x |] ==> f(x,u,v):R |] ==>\
\ ?a : R";
by (cut_facts_tac prems 1);
by (REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ;
qed "ex1E";
(*** <-> congruence rules for simplification ***)
(*Use iffE on a premise. For conj_cong, imp_cong, all_cong, ex_cong*)
fun iff_tac prems i =
resolve_tac (prems RL [iffE]) i THEN
REPEAT1 (eresolve_tac [asm_rl,mp] i);
val conj_cong = prove_goal (the_context ())
"[| p:P <-> P'; !!x. x:P' ==> q(x):Q <-> Q' |] ==> ?p:(P&Q) <-> (P'&Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (ares_tac [iffI,conjI] 1
ORELSE eresolve_tac [iffE,conjE,mp] 1
ORELSE iff_tac prems 1)) ]);
val disj_cong = prove_goal (the_context ())
"[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P|Q) <-> (P'|Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
ORELSE ares_tac [iffI] 1
ORELSE mp_tac 1)) ]);
val imp_cong = prove_goal (the_context ())
"[| p:P <-> P'; !!x. x:P' ==> q(x):Q <-> Q' |] ==> ?p:(P-->Q) <-> (P'-->Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (ares_tac [iffI,impI] 1
ORELSE etac iffE 1
ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ]);
val iff_cong = prove_goal (the_context ())
"[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P<->Q) <-> (P'<->Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (etac iffE 1
ORELSE ares_tac [iffI] 1
ORELSE mp_tac 1)) ]);
val not_cong = prove_goal (the_context ())
"p:P <-> P' ==> ?p:~P <-> ~P'"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (ares_tac [iffI,notI] 1
ORELSE mp_tac 1
ORELSE eresolve_tac [iffE,notE] 1)) ]);
val all_cong = prove_goal (the_context ())
"(!!x. f(x):P(x) <-> Q(x)) ==> ?p:(ALL x. P(x)) <-> (ALL x. Q(x))"
(fn prems =>
[ (REPEAT (ares_tac [iffI,allI] 1
ORELSE mp_tac 1
ORELSE etac allE 1 ORELSE iff_tac prems 1)) ]);
val ex_cong = prove_goal (the_context ())
"(!!x. f(x):P(x) <-> Q(x)) ==> ?p:(EX x. P(x)) <-> (EX x. Q(x))"
(fn prems =>
[ (REPEAT (etac exE 1 ORELSE ares_tac [iffI,exI] 1
ORELSE mp_tac 1
ORELSE iff_tac prems 1)) ]);
(*NOT PROVED
val ex1_cong = prove_goal (the_context ())
"(!!x.f(x):P(x) <-> Q(x)) ==> ?p:(EX! x.P(x)) <-> (EX! x.Q(x))"
(fn prems =>
[ (REPEAT (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1
ORELSE mp_tac 1
ORELSE iff_tac prems 1)) ]);
*)
(*** Equality rules ***)
val refl = ieqI;
val subst = prove_goal (the_context ()) "[| p:a=b; q:P(a) |] ==> ?p : P(b)"
(fn [prem1,prem2] => [ rtac (prem2 RS rev_mp) 1, (rtac (prem1 RS ieqE) 1),
rtac impI 1, atac 1 ]);
Goal "q:a=b ==> ?c:b=a";
by (etac subst 1);
by (rtac refl 1) ;
qed "sym";
Goal "[| p:a=b; q:b=c |] ==> ?d:a=c";
by (etac subst 1 THEN assume_tac 1);
qed "trans";
(** ~ b=a ==> ~ a=b **)
Goal "p:~ b=a ==> ?q:~ a=b";
by (etac contrapos 1);
by (etac sym 1) ;
qed "not_sym";
(*calling "standard" reduces maxidx to 0*)
val ssubst = standard (sym RS subst);
(*A special case of ex1E that would otherwise need quantifier expansion*)
Goal "[| p:EX! x. P(x); q:P(a); r:P(b) |] ==> ?d:a=b";
by (etac ex1E 1);
by (rtac trans 1);
by (rtac sym 2);
by (REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ;
qed "ex1_equalsE";
(** Polymorphic congruence rules **)
Goal "[| p:a=b |] ==> ?d:t(a)=t(b)";
by (etac ssubst 1);
by (rtac refl 1) ;
qed "subst_context";
Goal "[| p:a=b; q:c=d |] ==> ?p:t(a,c)=t(b,d)";
by (REPEAT (etac ssubst 1));
by (rtac refl 1) ;
qed "subst_context2";
Goal "[| p:a=b; q:c=d; r:e=f |] ==> ?p:t(a,c,e)=t(b,d,f)";
by (REPEAT (etac ssubst 1));
by (rtac refl 1) ;
qed "subst_context3";
(*Useful with eresolve_tac for proving equalties from known equalities.
a = b
| |
c = d *)
Goal "[| p:a=b; q:a=c; r:b=d |] ==> ?p:c=d";
by (rtac trans 1);
by (rtac trans 1);
by (rtac sym 1);
by (REPEAT (assume_tac 1)) ;
qed "box_equals";
(*Dual of box_equals: for proving equalities backwards*)
Goal "[| p:a=c; q:b=d; r:c=d |] ==> ?p:a=b";
by (rtac trans 1);
by (rtac trans 1);
by (REPEAT (eresolve_tac [asm_rl, sym] 1)) ;
qed "simp_equals";
(** Congruence rules for predicate letters **)
Goal "p:a=a' ==> ?p:P(a) <-> P(a')";
by (rtac iffI 1);
by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
qed "pred1_cong";
Goal "[| p:a=a'; q:b=b' |] ==> ?p:P(a,b) <-> P(a',b')";
by (rtac iffI 1);
by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
qed "pred2_cong";
Goal "[| p:a=a'; q:b=b'; r:c=c' |] ==> ?p:P(a,b,c) <-> P(a',b',c')";
by (rtac iffI 1);
by (DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ;
qed "pred3_cong";
(*special cases for free variables P, Q, R, S -- up to 3 arguments*)
val pred_congs =
List.concat (map (fn c =>
map (fn th => read_instantiate [("P",c)] th)
[pred1_cong,pred2_cong,pred3_cong])
(explode"PQRS"));
(*special case for the equality predicate!*)
val eq_cong = read_instantiate [("P","op =")] pred2_cong;
(*** Simplifications of assumed implications.
Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
used with mp_tac (restricted to atomic formulae) is COMPLETE for
intuitionistic propositional logic. See
R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
(preprint, University of St Andrews, 1991) ***)
val major::prems= Goal
"[| p:(P&Q)-->S; !!x. x:P-->(Q-->S) ==> q(x):R |] ==> ?p:R";
by (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ;
qed "conj_impE";
val major::prems= Goal
"[| p:(P|Q)-->S; !!x y.[| x:P-->S; y:Q-->S |] ==> q(x,y):R |] ==> ?p:R";
by (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ;
qed "disj_impE";
(*Simplifies the implication. Classical version is stronger.
Still UNSAFE since Q must be provable -- backtracking needed. *)
val major::prems= Goal
"[| p:(P-->Q)-->S; !!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q; !!x. x:S ==> r(x):R |] ==> \
\ ?p:R";
by (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ;
qed "imp_impE";
(*Simplifies the implication. Classical version is stronger.
Still UNSAFE since ~P must be provable -- backtracking needed. *)
val major::prems= Goal
"[| p:~P --> S; !!y. y:P ==> q(y):False; !!y. y:S ==> r(y):R |] ==> ?p:R";
by (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ;
qed "not_impE";
(*Simplifies the implication. UNSAFE. *)
val major::prems= Goal
"[| p:(P<->Q)-->S; !!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q; \
\ !!x y.[| x:Q; y:P-->S |] ==> r(x,y):P; !!x. x:S ==> s(x):R |] ==> ?p:R";
by (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ;
qed "iff_impE";
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
val major::prems= Goal
"[| p:(ALL x. P(x))-->S; !!x. q:P(x); !!y. y:S ==> r(y):R |] ==> ?p:R";
by (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ;
qed "all_impE";
(*Unsafe: (EX x.P(x))-->S is equivalent to ALL x.P(x)-->S. *)
val major::prems= Goal
"[| p:(EX x. P(x))-->S; !!y. y:P(a)-->S ==> q(y):R |] ==> ?p:R";
by (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ;
qed "ex_impE";