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|
#! /bin/sh
# This is a shell archive. Remove anything before this line, then unpack
# it by saving it into a file and typing "sh file". To overwrite existing
# files, type "sh file -c". You can also feed this as standard input via
# unshar, or by typing "sh <file", e.g.. If this archive is complete, you
# will see the following message at the end:
# "End of shell archive."
# Contents: READ-ME Makefile mk_set_defs.ml mk_set.ml set.ml
# Wrapped by kalker@tomcat on Mon Sep 25 11:35:44 1989
PATH=/bin:/usr/bin:/usr/ucb ; export PATH
if test -f 'READ-ME' -a "${1}" != "-c" ; then
echo shar: Will not clobber existing file \"'READ-ME'\"
else
echo shar: Extracting \"'READ-ME'\" \(1819 characters\)
sed "s/^X//" >'READ-ME' <<'END_OF_FILE'
X+ ===================================================================== +
X| |
X| LIBRARY : set |
X| |
X| DESCRIPTION : Definition of set theory. |
X| |
X| AUTHOR : T. kalker |
X| DATE : 8 JUNE 1989 |
X| |
X+ ===================================================================== +
X
X
X+ --------------------------------------------------------------------- +
X| |
X| FILES: |
X| |
X+ --------------------------------------------------------------------- +
X
X
X mk_set_defs.ml elementary definitions of set theory.
X
X mk_set.ml elementary theorems of set theory.
X
X set.ml loads the library into hol.
X
X+ --------------------------------------------------------------------- +
X| |
X| DOCUMENTATION: |
X| |
X+ --------------------------------------------------------------------- +
X
XSets are modeled as predicates on types.
XThe empty set is borrowed from "fixpoints" where
Xit is called BOT.
XThe constructed is mainly concerned with theorems on
Xfunctions on sets. The boolean algebra of sets is hardly
Xconsidered, and is left to the taste of the user.
X
X+ --------------------------------------------------------------------- +
X| |
X| TO REBUILD THE LIBRARY: |
X| |
X+ --------------------------------------------------------------------- +
X
X 0) necessary libraries: fixpoints, auxiliary
X
X 1) edit the pathnames in the Makefile (if necessary)
X
X 2) type "make clean"
X
X 3) type "make all"
X
X+ --------------------------------------------------------------------- +
X| |
X| TO USE THE LIBRARY: |
X| |
X+ --------------------------------------------------------------------- +
X
X load_library `set`;;
X
X
END_OF_FILE
if test 1819 -ne `wc -c <'READ-ME'`; then
echo shar: \"'READ-ME'\" unpacked with wrong size!
fi
chmod +x 'READ-ME'
# end of 'READ-ME'
fi
if test -f 'Makefile' -a "${1}" != "-c" ; then
echo shar: Will not clobber existing file \"'Makefile'\"
else
echo shar: Extracting \"'Makefile'\" \(977 characters\)
sed "s/^X//" >'Makefile' <<'END_OF_FILE'
X# =====================================================================
X#
X# MAKEFILE FOR THE HOL LIBRARY: SET
X#
X# =====================================================================
X
X# =====================================================================
X#
X# MAIN ENTRIES:
X#
X# make all : create theories and compile code
X#
X# make clean : remove theories and compiled code
X#
X# =====================================================================
X
X# =====================================================================
X# MACROS:
X#
X# Hol : the pathname of the version of hol used
X# =====================================================================
X
XHol=~/com/hol
X
Xclean:
X rm -f *_ml.o *_ml.l *.th
X @echo "===> library set: all object code and theory files deleted"
X
Xset.th: mk_set_defs.ml mk_set.ml
X rm -f set.th
X echo 'set_flag(`abort_when_fail`,true);;'\
X 'loadt `mk_set_defs`;;'\
X 'loadt `mk_set`;;'\
X 'quit();;' | ${Hol}
X
Xall: set.th
END_OF_FILE
if test 977 -ne `wc -c <'Makefile'`; then
echo shar: \"'Makefile'\" unpacked with wrong size!
fi
chmod +x 'Makefile'
# end of 'Makefile'
fi
if test -f 'mk_set_defs.ml' -a "${1}" != "-c" ; then
echo shar: Will not clobber existing file \"'mk_set_defs.ml'\"
else
echo shar: Extracting \"'mk_set_defs.ml'\" \(3410 characters\)
sed "s/^X//" >'mk_set_defs.ml' <<'END_OF_FILE'
X%<
XFILE: mk_set_defs.ml
X
XAUTHOR: Ton Kalker
X
XDATE: 10 july 1989
X
XCOMMENT: a preliminary version
X of set theory!
X Should in the end be fitted
X in with a standarized HOL
X set theory.
X The constant "empty set" is
X borrowed from "fixpoints" where
X is is called BOT.
X>%
X
Xnew_theory `set`;;
X
Xnew_parent `fixpoints`;;
X
Xlet TOP =
X new_definition
X (`TOP`,
X "TOP = \x:*.T");;
X
Xlet MEMBER =
X new_infix_definition
X (`MEMBER`,
X "MEMBER a (A:*->bool) = A a");;
X
Xlet SING =
X new_definition
X (`SING`,
X "SING (a:*) = (\x.x = a)");;
X
Xlet UNION =
X new_infix_definition
X (`UNION`,
X "UNION IN (OUT:*->bool) = \a.((IN a) \/ (OUT a))");;
X
Xlet INTERSECT =
X new_infix_definition
X (`INTERSECT`,
X "INTERSECT A (B:*->bool) = \a.((A a) /\ (B a))");;
X
Xlet DELETE =
X new_infix_definition
X (`DELETE`,
X "DELETE A (B:*->bool) = \a.((A a) /\ ~(B a))");;
X
Xlet DIFF =
X new_infix_definition
X (`DIFF`,
X "DIFF A (B:*->bool) = \a.(((A a) /\ ~(B a)) \/ (~(A a) /\ (B a)))");;
X
Xlet SUBSET =
X new_infix_definition
X (`SUBSET`,
X "SUBSET A (B:*->bool) = (!a. (A a) ==> (B a))");;
X
Xlet SUPSET =
X new_infix_definition
X (`SUPSET`,
X "SUPSET A (B:*->bool) = (!a. (B a) ==> (A a))");;
X
Xlet DISJOINT =
X new_definition
X (`DISJOINT`,
X "DISJOINT A B = (!a:*.~((A a) /\ (B a)))");;
X
Xlet PSUBSET =
X new_infix_definition(`PSUBSET`,
X "PSUBSET A (B:*->bool) = ~(A = B) /\ (A SUBSET B)");;
X
Xlet PSUPSET =
X new_infix_definition(`PSUPSET`,
X "PSUPSET A (B:*->bool) = ~(A = B) /\ (A SUPSET B)");;
X
Xlet GINTERSECT = new_definition(`GINTERSECT`,
X "GINTERSECT (SS:(*->bool)->bool) x = !X.(SS X) ==> (X x)");;
X
Xlet GUNION = new_definition(`GUNION`,
X "GUNION (SS:(*->bool)->bool) x = ?X.(SS X) /\ (X x)");;
X
Xlet FUN_TY_DEF =
X new_infix_definition
X (`FUN_TY_DEF`,
X "--> (A:*->bool) (B:*->bool) f = !x.((A x) ==> (B (f x)))");;
X
Xnew_special_symbol `-->>`;;
X
Xnew_special_symbol `>-->`;;
X
Xnew_special_symbol `<-->`;;
X
Xlet FUN_ONTO =
X new_infix_definition
X (`FUN_ONTO`,
X " -->> (A:*->bool) (B:*->bool) f = ((!x.((A x) ==> (B (f x)))) /\
X (!x.(B x) ==> (?y.(A y) /\ (x = (f y)))))");;
X
Xlet FUN_ONE_ONE =
X new_infix_definition
X (`FUN_ONE_ONE`,
X ">--> (A:*->bool) (B:*->bool) f = ((!x.((A x) ==> (B (f x)))) /\
X (! x y.((A x) /\ (A y) /\ ((f x) = (f y)))
X ==> (x = y)))");;
X
Xlet FUN_ISO =
X new_infix_definition
X (`FUN_ISO`,
X "<--> A (B:*->bool) f = (((A >--> B) f) /\ ((A -->> B) f) )");;
X
Xlet FUN_INV =
X new_definition
X (`FUN_INV`,
X "FINV A B f (y:*):* = ((B y) /\ (?x.((A x) /\ (y = (f x))))) =>
X (@x.((A x) /\ (y = (f x)))) | @x.A x");;
X
Xlet PINVERSE =
X new_definition
X (`PINVERSE`,
X "PINVERSE (A,B) f (g:*->*) = ((A --> B) f) /\
X ((B --> A) g) /\
X (!x:*.((A x) ==> (((g o f) x) = x)))");;
X
Xlet INVERSE =
X new_definition
X (`INVERSE`,
X "INVERSE (A,B) f (g:*->*) = (PINVERSE (A,B) f g) /\ (PINVERSE (B,A) g f)");;
X
Xlet IMAGE =
X new_definition
X (`IMAGE`,
X "IMAGE (f:*->*) (A:*->bool) a = (?b.((A b) /\ (a = (f b))))");;
X
X
END_OF_FILE
if test 3410 -ne `wc -c <'mk_set_defs.ml'`; then
echo shar: \"'mk_set_defs.ml'\" unpacked with wrong size!
fi
chmod +x 'mk_set_defs.ml'
# end of 'mk_set_defs.ml'
fi
if test -f 'mk_set.ml' -a "${1}" != "-c" ; then
echo shar: Will not clobber existing file \"'mk_set.ml'\"
else
echo shar: Extracting \"'mk_set.ml'\" \(10958 characters\)
sed "s/^X//" >'mk_set.ml' <<'END_OF_FILE'
X%<
XAUTHOR: Ton Kalker
X
XDATE: 13 september 1989
X
XCOMMENT: proves some basic theorems
X on functions on sets.
X>%
X
X%<
Xload_theory `set`;;
X
Xautoload_defs_and_thms `set`;;
X>%
X
Xload_library `auxiliary`;;
X
Xautoload_defs_and_thms `fixpoints`;;
X
X%<
Xnew_special_symbol `-->>`;;
X
Xnew_special_symbol `>-->`;;
X
Xnew_special_symbol `<-->`;;
X>%
X
Xlet w = "!A B (f:*->*).(((A >-->B) f) \/ ((A -->> B) f)) ==> ((A --> B) f)";;
X
Xlet FUN_TY = prove_thm(
X `FUN_TY`,
X w,
X REPEAT STRIP_TAC THEN
X RULE_ASSUM_TAC (REWRITE_RULE[FUN_ONTO;FUN_ONE_ONE]) THEN
X ASM_REWRITE_TAC[FUN_TY_DEF]);;
X
Xlet w = "!A (B:*->bool).!f.((A --> B) f) = ((IMAGE f A) SUBSET B)";;
X
Xlet FUN_TY_IMAGE =
X prove_thm(
X `FUN_TY_IMAGE`,
X w,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[IMAGE;SUBSET;FUN_TY_DEF] THEN
X EQ_TAC THEN
X REPEAT STRIP_TAC THENL
X [RES_TAC THEN
X ASM_REWRITE_TAC[];
X ASSUM_LIST ((MAP_EVERY MATCH_MP_TAC) o (sublist [2])) THEN
X EXISTS_TAC "x:*" THEN
X ASM_REWRITE_TAC[]]);;
X
Xlet w = "!A (B:*->bool).!f:*->*.((A -->> B) f) = ((IMAGE f A) = B)";;
X
Xlet ONTO_IMAGE =
X prove_thm(
X `ONTO_IMAGE`,
X w,
X REPEAT GEN_TAC THEN
X CONV_TAC (RAND_CONV FUN_EQ_CONV) THEN
X REWRITE_TAC [IMAGE;FUN_ONTO] THEN
X EQ_TAC THEN
X REPEAT STRIP_TAC THENL
X [EQ_TAC THEN
X REPEAT STRIP_TAC THEN
X RES_TAC THEN
X ASM_REWRITE_TAC[];
X FILTER_RULE_ASSUM_TAC [2] (SPEC "f (x:*):*") THEN
X FILTER_RULE_ASSUM_TAC [2] SYM THEN
X ASM_REWRITE_TAC[] THEN
X EXISTS_TAC "x:*" THEN
X ASM_REWRITE_TAC[];
X FILTER_RULE_ASSUM_TAC [2] (SPEC "x:*") THEN
X REWRITE_RULE_ASSUM_TAC [2] [1] THEN
X ASM_REWRITE_TAC[]]);;
X
Xlet w= "!A:*->bool.!f g. (IMAGE g (IMAGE f A)) = IMAGE (g o f) A";;
X
Xlet IMAGE_COMP =
X prove_thm(
X `IMAGE_COMP`,
X w,
X CONV_TAC (ONCE_DEPTH_CONV FUN_EQ_CONV) THEN
X REPEAT GEN_TAC THEN
X REWRITE_TAC [IMAGE;o_DEF] THEN
X BETA_TAC THEN
X EQ_TAC THEN
X REPEAT STRIP_TAC THENL
X [EXISTS_TAC "b':*" THEN
X ASM_REWRITE_TAC[];
X EXISTS_TAC "f (b:*):*" THEN
X ASM_REWRITE_TAC[] THEN
X EXISTS_TAC "b:*" THEN
X ASM_REWRITE_TAC[]]);;
X
Xlet w1 = "!A:*->bool.(A --> A) I";;
X
Xlet thm1 =
X prove
X (w1,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_TY_DEF] THEN
X BETA_TAC THEN
X REWRITE_TAC[I_THM]);;
X
Xlet w2 = "!A:*->bool.(A >--> A) I";;
X
Xlet thm2 =
X prove
X (w2,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_ONE_ONE;thm1] THEN
X BETA_TAC THEN
X REWRITE_TAC[I_THM] THEN
X REPEAT STRIP_TAC THEN
X ASM_REWRITE_TAC[]);;
X
Xlet w3 = "!A:*->bool.(A -->> A) I";;
X
Xlet thm3 =
X prove
X (w3,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_ONTO;thm1] THEN
X BETA_TAC THEN
X REWRITE_TAC[I_THM] THEN
X REPEAT STRIP_TAC THEN
X EXISTS_TAC "x:*" THEN
X ASM_REWRITE_TAC[]);;
X
Xlet w4 = "!A:*->bool.(A <--> A) I";;
X
Xlet thm4 =
X prove
X (w4,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_ISO] THEN
X BETA_TAC THEN
X REWRITE_TAC[thm2;thm3]);;
X
Xlet FUN_IDEN =
X save_thm
X (`FUN_IDEN`,
X GEN_ALL (LIST_CONJ (map SPEC_ALL [thm1;thm2;thm3;thm4])));;
X
Xlet w1 = "!B.!f:*->*.((BOT --> B) f)";;
X
Xlet thm1 =
X prove(
X w1,
X REWRITE_TAC[FUN_TY_DEF;BOT]);;
X
Xlet w2 = "!B.!f:*->*.((BOT >--> B) f)";;
X
Xlet thm2 =
X prove(
X w2,
X REWRITE_TAC[BOT;FUN_ONE_ONE]);;
X
X
Xlet w3 = "!B.!f:*->*.((BOT -->> B)f) = (B = BOT)";;
X
Xlet thm3 =
X prove(
X w3,
X REWRITE_TAC[BOT;FUN_ONTO] THEN
X GEN_TAC THEN
X CONV_TAC SYM_CONV THEN
X CONV_TAC (LHS_CONV (FUN_EQ_CONV THENC (REDEPTH_CONV BETA_CONV))) THEN
X REWRITE_TAC[]);;
X
Xlet w4 = "!B.!f:*->*.((BOT <--> B)f) = (B = BOT)";;
X
Xlet thm4 =
X prove(
X w4,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_ISO;thm2;thm3]);;
X
Xlet BOTTOM_LEFT =
X save_thm(
X `BOTTOM_LEFT`,
X GEN_ALL (LIST_CONJ (map SPEC_ALL [thm1;thm2;thm3;thm4])));;
X
Xlet w1 = "!A.!f:*->*.((A --> BOT) f) = (A = BOT)";;
X
Xlet thm1 =
X prove(
X w1,
X REWRITE_TAC[BOT;FUN_TY_DEF] THEN
X GEN_TAC THEN
X CONV_TAC SYM_CONV THEN
X CONV_TAC (LHS_CONV (FUN_EQ_CONV THENC (REDEPTH_CONV BETA_CONV))) THEN
X REWRITE_TAC[]);;
X
Xlet w2 = "!A.!f:*->*.((A >--> BOT) f) = (A = BOT)";;
X
Xlet thm2 =
X prove(
X w2,
X REWRITE_TAC[BOT;FUN_ONE_ONE] THEN
X BETA_TAC THEN
X REPEAT GEN_TAC THEN
X CONV_TAC SYM_CONV THEN
X CONV_TAC (LHS_CONV (FUN_EQ_CONV THENC (REDEPTH_CONV BETA_CONV))) THEN
X REWRITE_TAC[] THEN
X EQ_TAC THEN
X STRIP_TAC THEN
X REPEAT CONJ_TAC THEN
X ASM_REWRITE_TAC[]);;
X
X
X
Xlet w3 = "!A.!f:*->*.((A -->> BOT) f) = (A = BOT)";;
X
Xlet thm3 =
X prove(
X w3,
X REWRITE_TAC[BOT;FUN_ONTO] THEN
X BETA_TAC THEN
X REPEAT GEN_TAC THEN
X CONV_TAC SYM_CONV THEN
X CONV_TAC (LHS_CONV (FUN_EQ_CONV THENC (REDEPTH_CONV BETA_CONV))) THEN
X REWRITE_TAC[]);;
X
Xlet w4 = "!A.!f:*->*.((A <--> BOT) f) = (A = BOT)";;
X
Xlet thm4 =
X prove(
X w4,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_ISO;thm2;thm3]);;
X
X
Xlet BOTTOM_RIGHT =
X save_thm(
X `BOTTOM_RIGHT`,
X GEN_ALL (LIST_CONJ (map SPEC_ALL [thm1;thm2;thm3;thm4])));;
X
Xlet w1 = "!A B (D:*->bool).!f g.((A --> B)f) /\ ((B --> D)g) ==>
X ((A --> D)(g o f))";;
X
Xlet thm1 =
X prove
X (w1,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_TY_DEF;o_DEF] THEN
X (2 TIMES BETA_TAC) THEN
X REPEAT STRIP_TAC THEN
X (2 TIMES RES_TAC));;
X
Xlet w2 = "!A B (D:*->bool).!f g.((A >--> B)f) /\ ((B >--> D)g) ==>
X ((A >--> D)(g o f))";;
X
Xlet thm2 =
X prove
X (w2,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_ONE_ONE;o_DEF] THEN
X (2 TIMES BETA_TAC) THEN
X REPEAT STRIP_TAC THEN
X (3 TIMES RES_TAC));;
X
Xlet w3 = "!A B (D:*->bool).!f g.((A -->> B)f) /\ ((B -->> D)g) ==>
X ((A -->> D)(g o f))";;
X
Xlet thm3 =
X prove
X (w3,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[ONTO_IMAGE;SYM (SPEC_ALL IMAGE_COMP)] THEN
X REPEAT STRIP_TAC THEN
X ASM_REWRITE_TAC[]);;
X
X
X
Xlet w4 = "!A B (D:*->bool).!f g.((A <--> B)f) /\ ((B <--> D)g) ==>
X ((A <--> D)(g o f))";;
X
Xlet thm4 =
X prove
X (w4,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_ISO] THEN
X BETA_TAC THEN
X REPEAT STRIP_TAC THEN
X ((MAP_EVERY IMP_RES_TAC) [thm2;thm3]));;
X
Xlet FUN_COMP =
X save_thm
X (`FUN_COMP`,
X GEN_ALL (LIST_CONJ (map SPEC_ALL [thm1;thm2;thm3;thm4])));;
X
Xlet w = "!A.~(A = BOT) = (?x:*.A x)";;
X
Xlet NOT_BOT =
X prove_thm(
X `NOT_BOT`,
X w,
X GEN_TAC THEN
X REWRITE_TAC[BOT] THEN
X BETA_TAC THEN
X CONV_TAC ((ONCE_DEPTH_CONV FUN_EQ_CONV) THENC
X (ONCE_DEPTH_CONV NOT_FORALL_CONV)) THEN
X REWRITE_TAC[]);;
X
Xlet w =" !A (B:*->bool).!f g g'.((PINVERSE (A,B) f g) /\ (PINVERSE (B,A) g' f))
X ==>
X ((PINVERSE (A,B) f g') /\ (PINVERSE (B,A) g f))";;
X
Xlet LR_INV =
X prove_thm(
X `LR_INV`,
X w,
X REWRITE_TAC[PINVERSE;FUN_TY_DEF;o_DEF] THEN
X BETA_TAC THEN
X REPEAT STRIP_TAC THEN
X ASM_REWRITE_TAC[] THEN
X RES_TAC THENL
X [RES_TAC THEN
X SUB_ASSUM_TAC [5;6;8;12] THEN
X RES_TAC THEN
X REWRITE_RULE_ASSUM_TAC [1] [3;4] THEN
X FILTER_RULE_ASSUM_TAC [1] SYM THEN
X ASM_REWRITE_TAC[];
X REWRITE_RULE_ASSUM_TAC [4] [6] THEN
X ASM_REWRITE_TAC[]]);;
X
Xlet w = "! A B (f:*->*).(~(A = BOT)) ==> (B --> A) (FINV A B f)";;
X
Xlet FUN_INV_TY =
X prove_thm(
X `FUN_INV_TY`,
X w,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[NOT_BOT;FUN_TY_DEF;FUN_INV] THEN
X DISCH_THEN ASSUME_TAC THEN
X REPEAT STRIP_TAC THEN
X FILTER_RULE_ASSUM_TAC [2] SELECT_RULE THEN
X ASM_CASES_TAC "(?x':*. A x' /\ ((x:*) = f x'))" THEN
X ASM_REWRITE_TAC[] THEN
X FILTER_RULE_ASSUM_TAC [1] SELECT_RULE THEN
X ASM_REWRITE_TAC[]);;
X
X
X
Xlet w = "!A B (f:*->*).(( ~(A = BOT)) /\ ((A >--> B) f)) ==>
X (PINVERSE (A,B) f (FINV A B f))";;
X
Xlet INJ_FINV =
X prove_thm(
X `INJ_FINV`,
X w,
X REPEAT GEN_TAC THEN
X REWRITE_TAC [FUN_ONE_ONE;PINVERSE;FUN_TY_DEF;o_DEF] THEN
X BETA_TAC THEN
X REPEAT STRIP_TAC THENL
X [RES_TAC;
X IMP_RES_TAC (REWRITE_RULE[FUN_TY_DEF] FUN_INV_TY);
X REWRITE_TAC[FUN_INV] THEN
X SUBGOAL_THEN "?x':*. A x' /\ (f x = (f x':*))" ASSUME_TAC THENL
X [EXISTS_TAC "x:*" THEN ASM_REWRITE_TAC[];ALL_TAC] THEN
X ASM_REWRITE_TAC[] THEN
X FILTER_RULE_ASSUM_TAC [1] SELECT_RULE THEN
X FILTER_STRIP_ASSUM_TAC [1] THEN
X FILTER_RULE_ASSUM_TAC [1] SYM THEN
X RES_TAC THEN
X ASM_REWRITE_TAC[]]);;
X
Xlet w = "!A B (f:*->*).(( ~(A = BOT)) /\ ((A -->> B) f)) ==>
X (PINVERSE (B,A) (FINV A B f) f)";;
X
Xlet SURJ_FINV =
X prove_thm(
X `SURJ_FINV`,
X w,
X REPEAT GEN_TAC THEN
X REWRITE_TAC [FUN_ONTO;PINVERSE;FUN_TY_DEF;o_DEF] THEN
X BETA_TAC THEN
X REPEAT STRIP_TAC THENL
X [IMP_RES_TAC (REWRITE_RULE[FUN_TY_DEF] FUN_INV_TY);
X RES_TAC;
X REWRITE_TAC[FUN_INV] THEN
X RES_TAC THEN
X ASM_REWRITE_TAC[] THEN
X FILTER_RULE_ASSUM_TAC [1] (SYM o CONJUNCT2 o SELECT_RULE) THEN
X ASM_REWRITE_TAC[]]);;
X
Xlet w = "!A B (f:*->*) g.(((A --> B) f) /\ ((B --> A) g) /\ (PINVERSE (A,B) f g)) ==>
X (((A >--> B) f) /\ ((B -->> A) g))";;
X
Xlet INJ_SURJ_PINVERSE = prove_thm(
X `INJ_SURJ_PINVERSE`,
X w,
X REPEAT GEN_TAC THEN
X REWRITE_TAC[FUN_TY_DEF;FUN_ONE_ONE;FUN_ONTO;PINVERSE;o_DEF] THEN
X BETA_TAC THEN
X REPEAT STRIP_TAC THEN
X RES_TAC THENL
X [REWRITE_RULE_ASSUM_TAC [3] [5;4] THEN
X ASM_REWRITE_TAC[];
X EXISTS_TAC "f (x:*):*" THEN
X ASM_REWRITE_TAC[]]);;
X
Xlet w = "!A (B:*->bool) f g.((A --> B) f) /\ ((B --> A) g) /\ (INVERSE (A,B) f g) ==>
X (((A <--> B) f) /\ ((B <--> A) g))";;
X
Xlet ISO_INVERSE =
X prove_thm(
X `ISO_INVERSE`,
X w,
X REWRITE_TAC[INVERSE;FUN_ISO] THEN
X REPEAT STRIP_TAC THEN
X NEW_IMP_RES_TAC INJ_SURJ_PINVERSE);;
X
X
Xlet w = "!A (B:*->bool).!f.((A <--> B)f) ==> ((B <--> A) (FINV A B f))";;
X
Xlet ISO_FINV =
X prove_thm(
X `ISO_FINV`,
X w,
X REPEAT GEN_TAC THEN
X REWRITE_TAC [FUN_ISO] THEN
X ASM_CASES_TAC "(A:*->bool) = BOT" THEN
X ASM_REWRITE_TAC[BOTTOM_LEFT;BOTTOM_RIGHT] THEN
X ASM_CASES_TAC "(B:*->bool) = BOT" THEN
X ASM_REWRITE_TAC[BOTTOM_LEFT;BOTTOM_RIGHT] THEN
X REPEAT STRIP_TAC THEN
X MAP_EVERY IMP_RES_TAC [FUN_TY;FUN_INV_TY;SURJ_FINV;INJ_FINV;INJ_SURJ_PINVERSE]);;
X
X
X
X
X
X
END_OF_FILE
if test 10958 -ne `wc -c <'mk_set.ml'`; then
echo shar: \"'mk_set.ml'\" unpacked with wrong size!
fi
chmod +x 'mk_set.ml'
# end of 'mk_set.ml'
fi
if test -f 'set.ml' -a "${1}" != "-c" ; then
echo shar: Will not clobber existing file \"'set.ml'\"
else
echo shar: Extracting \"'set.ml'\" \(826 characters\)
sed "s/^X//" >'set.ml' <<'END_OF_FILE'
X% FILE : set.ml %
X% DESCRIPTION : loads the library set into hol. %
X% %
X% AUTHOR : T. Kalker %
X% DATE : 18 September 1989 %
X
X
Xif draft_mode()
X then (if mem `set` (parents (current_theory()))
X then
X (print_newline();
X print_string`set is parent`;
X print_newline())
X else
X (print_newline();
X print_string`set declared a new parent`;
X print_newline();
X new_parent`set`))
X else (load_theory`set` ? print_newline());;
X
Xnew_special_symbol `-->>`;;
X
Xnew_special_symbol `>-->`;;
X
Xnew_special_symbol `<-->`;;
X
Xload_from_lib false `auxiliary` `functions`;;
X
Xautoload_theory(`definition`,`fixpoints`,`BOT`);;
X
Xautoload_defs_and_thms `set`;;
X
Xlet goal = g
Xand apply = expandf;;
END_OF_FILE
if test 826 -ne `wc -c <'set.ml'`; then
echo shar: \"'set.ml'\" unpacked with wrong size!
fi
chmod +x 'set.ml'
# end of 'set.ml'
fi
echo shar: End of shell archive.
exit 0
|
