src/FOL/IFOL.ML
 author lcp Fri, 16 Dec 1994 13:30:34 +0100 changeset 793 0b5c5f568d74 parent 779 4ab9176b45b7 child 821 650ee089809b permissions -rw-r--r--
conj_cong2: new congruence rule
```
(*  Title: 	FOL/ifol.ML
ID:         \$Id\$
Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

Tactics and lemmas for ifol.thy (intuitionistic first-order logic)
*)

open IFOL;

qed_goalw "TrueI" IFOL.thy [True_def] "True"
(fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);

(*** Sequent-style elimination rules for & --> and ALL ***)

qed_goal "conjE" IFOL.thy
"[| P&Q; [| P; Q |] ==> R |] ==> R"
(fn prems=>
[ (REPEAT (resolve_tac prems 1
ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN
resolve_tac prems 1))) ]);

qed_goal "impE" IFOL.thy
"[| P-->Q;  P;  Q ==> R |] ==> R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);

qed_goal "allE" IFOL.thy
"[| ALL x.P(x); P(x) ==> R |] ==> R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);

(*Duplicates the quantifier; for use with eresolve_tac*)
qed_goal "all_dupE" IFOL.thy
"[| ALL x.P(x);  [| P(x); ALL x.P(x) |] ==> R \
\    |] ==> R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);

(*** Negation rules, which translate between ~P and P-->False ***)

qed_goalw "notI" IFOL.thy [not_def] "(P ==> False) ==> ~P"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);

qed_goalw "notE" IFOL.thy [not_def] "[| ~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. *)
qed_goal "not_to_imp" IFOL.thy
"[| ~P;  (P-->False) ==> Q |] ==> Q"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);

(* 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   *)
qed_goal "rev_mp" IFOL.thy "[| P;  P --> Q |] ==> Q"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);

(*Contrapositive of an inference rule*)
qed_goal "contrapos" IFOL.thy "[| ~Q;  P==>Q |] ==> ~P"
(fn [major,minor]=>
[ (rtac (major RS notE RS notI) 1),
(etac minor 1) ]);

(*** Modus Ponens Tactics ***)

(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
fun mp_tac i = eresolve_tac [notE,impE] i  THEN  assume_tac i;

(*Like mp_tac but instantiates no variables*)
fun eq_mp_tac i = eresolve_tac [notE,impE] i  THEN  eq_assume_tac i;

(*** If-and-only-if ***)

qed_goalw "iffI" IFOL.thy [iff_def]
"[| P ==> Q;  Q ==> P |] ==> P<->Q"
(fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);

(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
qed_goalw "iffE" IFOL.thy [iff_def]
"[| P <-> Q;  [| P-->Q; Q-->P |] ==> R |] ==> R"
(fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]);

(* Destruct rules for <-> similar to Modus Ponens *)

qed_goalw "iffD1" IFOL.thy [iff_def] "[| P <-> Q;  P |] ==> Q"
(fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);

qed_goalw "iffD2" IFOL.thy [iff_def] "[| P <-> Q;  Q |] ==> P"
(fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);

qed_goal "iff_refl" IFOL.thy "P <-> P"
(fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);

qed_goal "iff_sym" IFOL.thy "Q <-> P ==> P <-> Q"
(fn [major] =>
[ (rtac (major RS iffE) 1),
(rtac iffI 1),
(REPEAT (eresolve_tac [asm_rl,mp] 1)) ]);

qed_goal "iff_trans" IFOL.thy
"!!P Q R. [| P <-> Q;  Q<-> R |] ==> P <-> R"
(fn _ =>
[ (rtac iffI 1),
(REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ]);

(*** 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.
***)

qed_goalw "ex1I" IFOL.thy [ex1_def]
"[| P(a);  !!x. P(x) ==> x=a |] ==> EX! x. P(x)"
(fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]);

(*Sometimes easier to use: the premises have no shared variables*)
qed_goal "ex_ex1I" IFOL.thy
"[| EX x.P(x);  !!x y. [| P(x); P(y) |] ==> x=y |] ==> EX! x. P(x)"
(fn [ex,eq] => [ (rtac (ex RS exE) 1),
(REPEAT (ares_tac [ex1I,eq] 1)) ]);

qed_goalw "ex1E" IFOL.thy [ex1_def]
"[| EX! x.P(x);  !!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R |] ==> R"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ]);

(*** <-> 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);

qed_goal "conj_cong" IFOL.thy
"[| P <-> P';  P' ==> Q <-> Q' |] ==> (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)) ]);

(*Reversed congruence rule!   Used in ZF/Order*)
qed_goal "conj_cong2" IFOL.thy
"[| P <-> P';  P' ==> Q <-> Q' |] ==> (Q&P) <-> (Q'&P')"
(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)) ]);

qed_goal "disj_cong" IFOL.thy
"[| P <-> P';  Q <-> Q' |] ==> (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)) ]);

qed_goal "imp_cong" IFOL.thy
"[| P <-> P';  P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT   (ares_tac [iffI,impI] 1
ORELSE  eresolve_tac [iffE] 1
ORELSE  mp_tac 1 ORELSE iff_tac prems 1)) ]);

qed_goal "iff_cong" IFOL.thy
"[| P <-> P';  Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT   (eresolve_tac [iffE] 1
ORELSE  ares_tac [iffI] 1
ORELSE  mp_tac 1)) ]);

qed_goal "not_cong" IFOL.thy
"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)) ]);

qed_goal "all_cong" IFOL.thy
"(!!x.P(x) <-> Q(x)) ==> (ALL x.P(x)) <-> (ALL x.Q(x))"
(fn prems =>
[ (REPEAT   (ares_tac [iffI,allI] 1
ORELSE   mp_tac 1
ORELSE   eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]);

qed_goal "ex_cong" IFOL.thy
"(!!x.P(x) <-> Q(x)) ==> (EX x.P(x)) <-> (EX x.Q(x))"
(fn prems =>
[ (REPEAT   (eresolve_tac [exE] 1 ORELSE ares_tac [iffI,exI] 1
ORELSE   mp_tac 1
ORELSE   iff_tac prems 1)) ]);

qed_goal "ex1_cong" IFOL.thy
"(!!x.P(x) <-> Q(x)) ==> (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 ***)

qed_goal "sym" IFOL.thy "a=b ==> b=a"
(fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);

qed_goal "trans" IFOL.thy "[| a=b;  b=c |] ==> a=c"
(fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);

(** ~ b=a ==> ~ a=b **)
val [not_sym] = compose(sym,2,contrapos);

(*calling "standard" reduces maxidx to 0*)
bind_thm ("ssubst", (sym RS subst));

(*A special case of ex1E that would otherwise need quantifier expansion*)
qed_goal "ex1_equalsE" IFOL.thy
"[| EX! x.P(x);  P(a);  P(b) |] ==> a=b"
(fn prems =>
[ (cut_facts_tac prems 1),
(etac ex1E 1),
(rtac trans 1),
(rtac sym 2),
(REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ]);

(** Polymorphic congruence rules **)

qed_goal "subst_context" IFOL.thy
"[| a=b |]  ==>  t(a)=t(b)"
(fn prems=>
[ (resolve_tac (prems RL [ssubst]) 1),
(resolve_tac [refl] 1) ]);

qed_goal "subst_context2" IFOL.thy
"[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
(fn prems=>
[ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);

qed_goal "subst_context3" IFOL.thy
"[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
(fn prems=>
[ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);

(*Useful with eresolve_tac for proving equalties from known equalities.
a = b
|   |
c = d	*)
qed_goal "box_equals" IFOL.thy
"[| a=b;  a=c;  b=d |] ==> c=d"
(fn prems=>
[ (resolve_tac [trans] 1),
(resolve_tac [trans] 1),
(resolve_tac [sym] 1),
(REPEAT (resolve_tac prems 1)) ]);

(*Dual of box_equals: for proving equalities backwards*)
qed_goal "simp_equals" IFOL.thy
"[| a=c;  b=d;  c=d |] ==> a=b"
(fn prems=>
[ (resolve_tac [trans] 1),
(resolve_tac [trans] 1),
(REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]);

(** Congruence rules for predicate letters **)

qed_goal "pred1_cong" IFOL.thy
"a=a' ==> P(a) <-> P(a')"
(fn prems =>
[ (cut_facts_tac prems 1),
(rtac iffI 1),
(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);

qed_goal "pred2_cong" IFOL.thy
"[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
(fn prems =>
[ (cut_facts_tac prems 1),
(rtac iffI 1),
(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);

qed_goal "pred3_cong" IFOL.thy
"[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
(fn prems =>
[ (cut_facts_tac prems 1),
(rtac iffI 1),
(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);

(*special cases for free variables P, Q, R, S -- up to 3 arguments*)

val pred_congs =
flat (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)  ***)

qed_goal "conj_impE" IFOL.thy
"[| (P&Q)-->S;  P-->(Q-->S) ==> R |] ==> R"
(fn major::prems=>
[ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);

qed_goal "disj_impE" IFOL.thy
"[| (P|Q)-->S;  [| P-->S; Q-->S |] ==> R |] ==> R"
(fn major::prems=>
[ (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ]);

(*Simplifies the implication.  Classical version is stronger.
Still UNSAFE since Q must be provable -- backtracking needed.  *)
qed_goal "imp_impE" IFOL.thy
"[| (P-->Q)-->S;  [| P; Q-->S |] ==> Q;  S ==> R |] ==> R"
(fn major::prems=>
[ (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ]);

(*Simplifies the implication.  Classical version is stronger.
Still UNSAFE since ~P must be provable -- backtracking needed.  *)
qed_goal "not_impE" IFOL.thy
"[| ~P --> S;  P ==> False;  S ==> R |] ==> R"
(fn major::prems=>
[ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]);

(*Simplifies the implication.   UNSAFE.  *)
qed_goal "iff_impE" IFOL.thy
"[| (P<->Q)-->S;  [| P; Q-->S |] ==> Q;  [| Q; P-->S |] ==> P;  \
\       S ==> R |] ==> R"
(fn major::prems=>
[ (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ]);

(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
qed_goal "all_impE" IFOL.thy
"[| (ALL x.P(x))-->S;  !!x.P(x);  S ==> R |] ==> R"
(fn major::prems=>
[ (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ]);

(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
qed_goal "ex_impE" IFOL.thy
"[| (EX x.P(x))-->S;  P(x)-->S ==> R |] ==> R"
(fn major::prems=>
[ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]);
```