--- a/src/FOL/FOL.ML Mon Dec 12 10:26:05 1994 +0100
+++ b/src/FOL/FOL.ML Tue Dec 13 11:51:12 1994 +0100
@@ -8,25 +8,10 @@
open FOL;
-signature FOL_LEMMAS =
- sig
- val disjCI : thm
- val excluded_middle : thm
- val excluded_middle_tac: string -> int -> tactic
- val exCI : thm
- val ex_classical : thm
- val iffCE : thm
- val impCE : thm
- val notnotD : thm
- end;
-
-
-structure FOL_Lemmas : FOL_LEMMAS =
-struct
(*** Classical introduction rules for | and EX ***)
-val disjCI = prove_goal FOL.thy
+qed_goal "disjCI" FOL.thy
"(~Q ==> P) ==> P|Q"
(fn prems=>
[ (resolve_tac [classical] 1),
@@ -34,14 +19,14 @@
(REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
(*introduction rule involving only EX*)
-val ex_classical = prove_goal FOL.thy
+qed_goal "ex_classical" FOL.thy
"( ~(EX x. P(x)) ==> P(a)) ==> EX x.P(x)"
(fn prems=>
[ (resolve_tac [classical] 1),
(eresolve_tac (prems RL [exI]) 1) ]);
(*version of above, simplifying ~EX to ALL~ *)
-val exCI = prove_goal FOL.thy
+qed_goal "exCI" FOL.thy
"(ALL x. ~P(x) ==> P(a)) ==> EX x.P(x)"
(fn [prem]=>
[ (resolve_tac [ex_classical] 1),
@@ -49,7 +34,7 @@
(eresolve_tac [notE] 1),
(eresolve_tac [exI] 1) ]);
-val excluded_middle = prove_goal FOL.thy "~P | P"
+qed_goal "excluded_middle" FOL.thy "~P | P"
(fn _=> [ rtac disjCI 1, assume_tac 1 ]);
(*For disjunctive case analysis*)
@@ -60,14 +45,14 @@
(*Classical implies (-->) elimination. *)
-val impCE = prove_goal FOL.thy
+qed_goal "impCE" FOL.thy
"[| P-->Q; ~P ==> R; Q ==> R |] ==> R"
(fn major::prems=>
[ (resolve_tac [excluded_middle RS disjE] 1),
(DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);
(*Double negation law*)
-val notnotD = prove_goal FOL.thy "~~P ==> P"
+qed_goal "notnotD" FOL.thy "~~P ==> P"
(fn [major]=>
[ (resolve_tac [classical] 1), (eresolve_tac [major RS notE] 1) ]);
@@ -76,14 +61,9 @@
(*Classical <-> elimination. Proof substitutes P=Q in
~P ==> ~Q and P ==> Q *)
-val iffCE = prove_goalw FOL.thy [iff_def]
+qed_goalw "iffCE" FOL.thy [iff_def]
"[| P<->Q; [| P; Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R"
(fn prems =>
[ (resolve_tac [conjE] 1),
(REPEAT (DEPTH_SOLVE_1
(etac impCE 1 ORELSE mp_tac 1 ORELSE ares_tac prems 1))) ]);
-
-
-end;
-
-open FOL_Lemmas;
--- a/src/FOL/IFOL.ML Mon Dec 12 10:26:05 1994 +0100
+++ b/src/FOL/IFOL.ML Tue Dec 13 11:51:12 1994 +0100
@@ -8,88 +8,29 @@
open IFOL;
-signature IFOL_LEMMAS =
- sig
- val allE: thm
- val all_cong: thm
- val all_dupE: thm
- val all_impE: thm
- val box_equals: thm
- val conjE: thm
- val conj_cong: thm
- val conj_impE: thm
- val contrapos: thm
- val disj_cong: thm
- val disj_impE: thm
- val eq_cong: thm
- val eq_mp_tac: int -> tactic
- val ex1I: thm
- val ex_ex1I: thm
- val ex1E: thm
- val ex1_equalsE: thm
- val ex1_cong: thm
- val ex_cong: thm
- val ex_impE: thm
- val iffD1: thm
- val iffD2: thm
- val iffE: thm
- val iffI: thm
- val iff_cong: thm
- val iff_impE: thm
- val iff_refl: thm
- val iff_sym: thm
- val iff_trans: thm
- val impE: thm
- val imp_cong: thm
- val imp_impE: thm
- val mp_tac: int -> tactic
- val notE: thm
- val notI: thm
- val not_cong: thm
- val not_impE: thm
- val not_sym: thm
- val not_to_imp: thm
- val pred1_cong: thm
- val pred2_cong: thm
- val pred3_cong: thm
- val pred_congs: thm list
- val rev_mp: thm
- val simp_equals: thm
- val ssubst: thm
- val subst_context: thm
- val subst_context2: thm
- val subst_context3: thm
- val sym: thm
- val trans: thm
- val TrueI: thm
- end;
-
-structure IFOL_Lemmas : IFOL_LEMMAS =
-struct
-
-val TrueI = prove_goalw IFOL.thy [True_def] "True"
+qed_goalw "TrueI" IFOL.thy [True_def] "True"
(fn _ => [ (REPEAT (ares_tac [impI] 1)) ]);
(*** Sequent-style elimination rules for & --> and ALL ***)
-val conjE = prove_goal IFOL.thy
+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))) ]);
-val impE = prove_goal IFOL.thy
+qed_goal "impE" IFOL.thy
"[| P-->Q; P; Q ==> R |] ==> R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
-val allE = prove_goal IFOL.thy
+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*)
-val all_dupE = prove_goal IFOL.thy
+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)) ]);
@@ -97,16 +38,16 @@
(*** Negation rules, which translate between ~P and P-->False ***)
-val notI = prove_goalw IFOL.thy [not_def] "(P ==> False) ==> ~P"
+qed_goalw "notI" IFOL.thy [not_def] "(P ==> False) ==> ~P"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
-val notE = prove_goalw IFOL.thy [not_def] "[| ~P; P |] ==> R"
+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. *)
-val not_to_imp = prove_goal IFOL.thy
+qed_goal "not_to_imp" IFOL.thy
"[| ~P; (P-->False) ==> Q |] ==> Q"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);
@@ -115,12 +56,12 @@
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 *)
-val rev_mp = prove_goal IFOL.thy "[| P; P --> Q |] ==> Q"
+qed_goal "rev_mp" IFOL.thy "[| P; P --> Q |] ==> Q"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
(*Contrapositive of an inference rule*)
-val contrapos = prove_goal IFOL.thy "[| ~Q; P==>Q |] ==> ~P"
+qed_goal "contrapos" IFOL.thy "[| ~Q; P==>Q |] ==> ~P"
(fn [major,minor]=>
[ (rtac (major RS notE RS notI) 1),
(etac minor 1) ]);
@@ -137,34 +78,34 @@
(*** If-and-only-if ***)
-val iffI = prove_goalw IFOL.thy [iff_def]
+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) *)
-val iffE = prove_goalw IFOL.thy [iff_def]
+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 *)
-val iffD1 = prove_goalw IFOL.thy [iff_def] "[| P <-> Q; P |] ==> Q"
+qed_goalw "iffD1" IFOL.thy [iff_def] "[| P <-> Q; P |] ==> Q"
(fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
-val iffD2 = prove_goalw IFOL.thy [iff_def] "[| P <-> Q; Q |] ==> P"
+qed_goalw "iffD2" IFOL.thy [iff_def] "[| P <-> Q; Q |] ==> P"
(fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
-val iff_refl = prove_goal IFOL.thy "P <-> P"
+qed_goal "iff_refl" IFOL.thy "P <-> P"
(fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
-val iff_sym = prove_goal IFOL.thy "Q <-> P ==> P <-> Q"
+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)) ]);
-val iff_trans = prove_goal IFOL.thy
+qed_goal "iff_trans" IFOL.thy
"!!P Q R. [| P <-> Q; Q<-> R |] ==> P <-> R"
(fn _ =>
[ (rtac iffI 1),
@@ -177,17 +118,17 @@
do NOT mean the same thing. The parser treats EX!x y.P(x,y) as sequential.
***)
-val ex1I = prove_goalw IFOL.thy [ex1_def]
+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*)
-val ex_ex1I = prove_goal IFOL.thy
+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)) ]);
-val ex1E = prove_goalw IFOL.thy [ex1_def]
+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),
@@ -201,7 +142,7 @@
resolve_tac (prems RL [iffE]) i THEN
REPEAT1 (eresolve_tac [asm_rl,mp] i);
-val conj_cong = prove_goal IFOL.thy
+qed_goal "conj_cong" IFOL.thy
"[| P <-> P'; P' ==> Q <-> Q' |] ==> (P&Q) <-> (P'&Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
@@ -209,7 +150,7 @@
ORELSE eresolve_tac [iffE,conjE,mp] 1
ORELSE iff_tac prems 1)) ]);
-val disj_cong = prove_goal IFOL.thy
+qed_goal "disj_cong" IFOL.thy
"[| P <-> P'; Q <-> Q' |] ==> (P|Q) <-> (P'|Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
@@ -217,7 +158,7 @@
ORELSE ares_tac [iffI] 1
ORELSE mp_tac 1)) ]);
-val imp_cong = prove_goal IFOL.thy
+qed_goal "imp_cong" IFOL.thy
"[| P <-> P'; P' ==> Q <-> Q' |] ==> (P-->Q) <-> (P'-->Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
@@ -225,7 +166,7 @@
ORELSE eresolve_tac [iffE] 1
ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ]);
-val iff_cong = prove_goal IFOL.thy
+qed_goal "iff_cong" IFOL.thy
"[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
@@ -233,7 +174,7 @@
ORELSE ares_tac [iffI] 1
ORELSE mp_tac 1)) ]);
-val not_cong = prove_goal IFOL.thy
+qed_goal "not_cong" IFOL.thy
"P <-> P' ==> ~P <-> ~P'"
(fn prems =>
[ (cut_facts_tac prems 1),
@@ -241,21 +182,21 @@
ORELSE mp_tac 1
ORELSE eresolve_tac [iffE,notE] 1)) ]);
-val all_cong = prove_goal IFOL.thy
+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)) ]);
-val ex_cong = prove_goal IFOL.thy
+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)) ]);
-val ex1_cong = prove_goal IFOL.thy
+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
@@ -264,20 +205,20 @@
(*** Equality rules ***)
-val sym = prove_goal IFOL.thy "a=b ==> b=a"
+qed_goal "sym" IFOL.thy "a=b ==> b=a"
(fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
-val trans = prove_goal IFOL.thy "[| a=b; b=c |] ==> a=c"
+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*)
-val ssubst = standard (sym RS subst);
+bind_thm ("ssubst", (sym RS subst));
(*A special case of ex1E that would otherwise need quantifier expansion*)
-val ex1_equalsE = prove_goal IFOL.thy
+qed_goal "ex1_equalsE" IFOL.thy
"[| EX! x.P(x); P(a); P(b) |] ==> a=b"
(fn prems =>
[ (cut_facts_tac prems 1),
@@ -288,18 +229,18 @@
(** Polymorphic congruence rules **)
-val subst_context = prove_goal IFOL.thy
+qed_goal "subst_context" IFOL.thy
"[| a=b |] ==> t(a)=t(b)"
(fn prems=>
[ (resolve_tac (prems RL [ssubst]) 1),
(resolve_tac [refl] 1) ]);
-val subst_context2 = prove_goal IFOL.thy
+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]))) ]);
-val subst_context3 = prove_goal IFOL.thy
+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]))) ]);
@@ -308,7 +249,7 @@
a = b
| |
c = d *)
-val box_equals = prove_goal IFOL.thy
+qed_goal "box_equals" IFOL.thy
"[| a=b; a=c; b=d |] ==> c=d"
(fn prems=>
[ (resolve_tac [trans] 1),
@@ -317,7 +258,7 @@
(REPEAT (resolve_tac prems 1)) ]);
(*Dual of box_equals: for proving equalities backwards*)
-val simp_equals = prove_goal IFOL.thy
+qed_goal "simp_equals" IFOL.thy
"[| a=c; b=d; c=d |] ==> a=b"
(fn prems=>
[ (resolve_tac [trans] 1),
@@ -326,21 +267,21 @@
(** Congruence rules for predicate letters **)
-val pred1_cong = prove_goal IFOL.thy
+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)) ]);
-val pred2_cong = prove_goal IFOL.thy
+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)) ]);
-val pred3_cong = prove_goal IFOL.thy
+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),
@@ -366,50 +307,45 @@
R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
(preprint, University of St Andrews, 1991) ***)
-val conj_impE = prove_goal IFOL.thy
+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)) ]);
-val disj_impE = prove_goal IFOL.thy
+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. *)
-val imp_impE = prove_goal IFOL.thy
+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. *)
-val not_impE = prove_goal IFOL.thy
+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. *)
-val iff_impE = prove_goal IFOL.thy
+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*)
-val all_impE = prove_goal IFOL.thy
+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. *)
-val ex_impE = prove_goal IFOL.thy
+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)) ]);
-
-end;
-
-open IFOL_Lemmas;
-