/usr/share/hol88-2.02.19940316/contrib/Z/TelephoneBook.ml is in hol88-contrib-source 2.02.19940316-19.
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% HOL theory to represent the "Telephone Book" from the paper %
% "A Simple Demonstration of Balzac" by Rodger Collinson %
%============================================================================%
%----------------------------------------------------------------------------%
% First load the ML code to support Z. %
%----------------------------------------------------------------------------%
loadf `SCHEMA`;;
%----------------------------------------------------------------------------%
% Declare a new theory called `TelephoneBook`. %
%----------------------------------------------------------------------------%
force_new_theory `TelephoneBook`;;
%----------------------------------------------------------------------------%
% The telphone book %
%----------------------------------------------------------------------------%
sets `NUMBER SUBSCRIBER`;;
%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!%
% The following two axioms are NOT assumed in Balzac, since it assumes %
% ALL basic types are non-empty. This is not standard! %
%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!%
declare_axiom "NUMBER =/= {}";;
declare_axiom "SUBSCRIBER =/= {}";;
declare
`TelephoneBook`
"SCHEMA
[book :: SUBSCRIBER -+> NUMBER;
known :: (P SUBSCRIBER);
free :: (P NUMBER)]
%---------------------------%
[known = dom book;
free = NUMBER DIFF (ran book)]";;
%----------------------------------------------------------------------------%
% A general purpose tactic for proving goals of the form: %
% %
% |-? EXISTS [<schema>] true %
%----------------------------------------------------------------------------%
let EXISTS_CONSISTENT_TAC =
REWRITE_TAC[::;-+>;<->;><;P;SUBSET_DEF;NOT_IN_EMPTY;true_DEF]
THEN SET_SPEC_TAC
THEN REWRITE_TAC
[NOT_IN_EMPTY;dom_EMPTY;ran_EMPTY;DIFF_EMPTY;IN_SING;UNION_EMPTY;
dom_SING;ran_SING;NOT_IN;IN_DIFF;|->;PAIR_EQ]
THEN SET_SPEC_TAC
THEN REPEAT STRIP_TAC
THEN ((EXISTS_TAC "CHOICE SUBSCRIBER" THEN EXISTS_TAC "CHOICE NUMBER")
ORELSE ALL_TAC)
THEN ASM_REWRITE_TAC
[GSYM CHOICE;Axiom_1;Axiom_2;GSYM |->;RangeAntiResSING;Ap_SING];;
prove_theorem
(`TelephoneBook_consistent`,
"EXISTS [TelephoneBook] true",
EXISTS_TAC "{}:(SUBSCRIBER # NUMBER)set"
THEN EXISTS_TAC "{}:(SUBSCRIBER)set"
THEN EXISTS_TAC "NUMBER"
THEN EXISTS_CONSISTENT_TAC);;
declare
`Connect`
"SCHEMA
[DELTA TelephoneBook;
subscriber? :: SUBSCRIBER;
number! :: NUMBER]
%-------------------------------------------%
[free =/= {};
subscriber? NOT_IN known;
number! IN free;
book' = book UNION{subscriber? |-> number!}]";;
prove_theorem
(`Connect_consistent`,
"EXISTS [Connect] true",
EXISTS_TAC "{}:(SUBSCRIBER # NUMBER)set"
THEN EXISTS_TAC "{}:(SUBSCRIBER)set"
THEN EXISTS_TAC "NUMBER"
THEN EXISTS_TAC "{CHOICE SUBSCRIBER |-> CHOICE NUMBER}"
THEN EXISTS_TAC "{CHOICE SUBSCRIBER}"
THEN EXISTS_TAC "NUMBER DIFF {CHOICE NUMBER}"
THEN EXISTS_TAC "CHOICE SUBSCRIBER"
THEN EXISTS_TAC "CHOICE NUMBER"
THEN EXISTS_CONSISTENT_TAC);;
prove_theorem
(`Connect_proof_1`,
"FORALL [Connect] (known' = known UNION {subscriber?})",
REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[dom_UNION;dom_SING]);;
prove_theorem
(`Connect_proof_2`,
"FORALL [Connect] (free' = free DIFF {number!})",
REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[ran_UNION;ran_SING;EXTENSION;DIFF_UNION]);;
declare
`Disconnect`
"SCHEMA
[DELTA TelephoneBook;
number? :: NUMBER]
%-------------------------%
[number? IN ran book;
book' = book +> {number?}]";;
prove_theorem
(`Disconnect_consistent`,
"EXISTS [Disconnect] true",
EXISTS_TAC "{CHOICE SUBSCRIBER |-> CHOICE NUMBER}"
THEN EXISTS_TAC "{CHOICE SUBSCRIBER}"
THEN EXISTS_TAC "NUMBER DIFF {CHOICE NUMBER}"
THEN EXISTS_TAC "{}:(SUBSCRIBER # NUMBER)set"
THEN EXISTS_TAC "{}:(SUBSCRIBER)set"
THEN EXISTS_TAC "NUMBER"
THEN EXISTS_TAC "CHOICE NUMBER"
THEN EXISTS_CONSISTENT_TAC);;
prove_theorem
(`Disconnect_proof_1`,
"FORALL
[Disconnect]
(known' = known DIFF {s | s IN SUBSCRIBER /\ (book s = number?)})",
REWRITE_TAC[::]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[DIFF_DEF;EXTENSION;+>;|->;dom;PAIR_EQ]
THEN SET_SPEC_TAC
THEN REWRITE_TAC[PAIR_EQ;IN_SING]
THEN GEN_TAC
THEN EQ_TAC
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[]
THEN SMART_ELIMINATE_TAC
THENL
[EXISTS_TAC "y':NUMBER"
THEN ASM_REWRITE_TAC[];
LITE_IMP_RES_TAC ApFunCor
THEN POP_ASSUM(ASSUME_TAC o SYM)
THEN RES_TAC;
EXISTS_TAC "y:NUMBER"
THEN EXISTS_TAC "x:SUBSCRIBER"
THEN EXISTS_TAC "y:NUMBER"
THEN ASM_REWRITE_TAC[]
THEN LITE_IMP_RES_TAC ApFunCor
THEN LITE_IMP_RES_TAC IN_dom_ran
THEN RW_ASM_THEN ACCEPT_TAC [el 2;el 3] (el 4)]);;
prove_theorem
(`Disconnect_proof_2`,
"FORALL
[Disconnect]
(free' = free UNION {number?})",
REWRITE_TAC[::]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[DIFF_DEF;EXTENSION;+>;|->;ran;PAIR_EQ]
THEN SET_SPEC_TAC
THEN REWRITE_TAC[PAIR_EQ;IN_UNION;IN_SING]
THEN GEN_TAC
THEN SET_SPEC_TAC
THEN CONV_TAC(TOP_DEPTH_CONV NOT_EXISTS_CONV)
THEN EQ_TAC
THEN REPEAT STRIP_TAC
THEN SMART_ELIMINATE_TAC
THEN ASM_REWRITE_TAC[]
THEN REPEAT STRIP_TAC
THEN RES_TAC
THEN POP_ASSUM
(STRIP_ASSUME_TAC o
CONV_RULE(DEPTH_CONV FORALL_OR_CONV) o
GEN_ALL o
REWRITE_RULE[DE_MORGAN_THM] o
SPECL["x:SUBSCRIBER";"x:SUBSCRIBER";"x:NUMBER"])
THEN ASM_REWRITE_TAC[]);;
declare
`FindNumber`
"SCHEMA
[XI TelephoneBook;
subscriber? :: SUBSCRIBER;
number! :: NUMBER]
%--------------------------%
[subscriber? IN known;
number! = book subscriber?]";;
prove_theorem
(`FindNumber_consistent`,
"EXISTS [FindNumber] true",
REWRITE_TAC[PAIR_EQ]
THEN EXISTS_TAC "{CHOICE SUBSCRIBER |-> CHOICE NUMBER}"
THEN EXISTS_TAC "{CHOICE SUBSCRIBER}"
THEN EXISTS_TAC "NUMBER DIFF {CHOICE NUMBER}"
THEN EXISTS_TAC "{CHOICE SUBSCRIBER |-> CHOICE NUMBER}"
THEN EXISTS_TAC "{CHOICE SUBSCRIBER}"
THEN EXISTS_TAC "NUMBER DIFF {CHOICE NUMBER}"
THEN EXISTS_TAC "CHOICE SUBSCRIBER"
THEN EXISTS_TAC "CHOICE NUMBER"
THEN EXISTS_CONSISTENT_TAC);;
declare
`initTelephoneBook`
"SCHEMA
[TelephoneBook]
%-------------%
[known = {}]";;
prove_theorem
(`initTelephoneBook_consistent`,
"EXISTS [initTelephoneBook] true",
EXISTS_TAC "{}:(SUBSCRIBER # NUMBER)set"
THEN EXISTS_TAC "{}:(SUBSCRIBER)set"
THEN EXISTS_TAC "NUMBER"
THEN EXISTS_CONSISTENT_TAC);;
free_set `REPORT = ok
| full_book
| already_known
| unknown_number
| unknown_subscriber`;;
declare
`Success`
"SCHEMA
[result! :: REPORT]
%-----------------%
[result! = ok]";;
declare
`FullBook`
"SCHEMA
[XI TelephoneBook;
result! :: REPORT]
%-----------------%
[free = {};
result! = full_book]";;
declare
`AlreadyKnown`
"SCHEMA
[XI TelephoneBook;
subscriber? :: SUBSCRIBER;
result! :: REPORT]
%-----------------%
[subscriber? IN known;
result! = already_known]";;
declare
`RConnect`
"(Connect AND Success) OR FullBook OR AlreadyKnown";;
prove_theorem
(`RConnect_total`,
"FORALL [RConnect] (pre RConnect)",
REWRITE_TAC[::;SCHEMA;CONJL;NOT_IN]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[DIFF_DEF;EXTENSION;+>;|->;dom;PAIR_EQ]
THEN SET_SPEC_TAC
THEN REWRITE_TAC[PAIR_EQ;IN_SING]
THEN EXISTS_TAC
"((free =/= {}) /\ (subscriber? NOT_IN known))
=> (book UNION {subscriber? |-> number!})
| book"
THEN EXISTS_TAC
"((free =/= {}) /\ (subscriber? NOT_IN known))
=> dom(book UNION {subscriber? |-> number!}) |
dom book"
THEN EXISTS_TAC
"((free =/= {}) /\ (subscriber? NOT_IN known))
=> NUMBER DIFF ran(book UNION {subscriber? |-> number!}) |
NUMBER DIFF ran book"
THEN EXISTS_TAC "number!"
THEN EXISTS_TAC "result!"
THEN ASM_CASES_TAC "(free =/= {}) /\ (subscriber? NOT_IN known)"
THEN LITE_IMP_RES_TAC UNION_SING_IN_P
THEN LITE_IMP_RES_TAC domPfunIN
THEN LITE_IMP_RES_TAC ranPfunIN
THEN SMART_ELIMINATE_TAC
THEN LITE_IMP_RES_TAC UNION_SING_Pfun
THEN ASM_REWRITE_TAC[DIFF_IN_P]
THEN RW_ASM_THEN ASSUME_TAC [K =/=;K NOT_IN;K DE_MORGAN_THM;el 10] (el 7)
THEN RES_TAC
THEN ASM_F_TAC);;
declare
`UnknownNumber`
"SCHEMA
[XI TelephoneBook;
number? :: NUMBER;
result! :: REPORT]
%-----------------%
[number? IN free;
result! = unknown_number]";;
declare
`RDisconnect`
"(Disconnect AND Success) OR UnknownNumber";;
prove_theorem
(`RDisconnect_total`,
"FORALL [RDisconnect] (pre RDisconnect)",
REWRITE_TAC[::;SCHEMA;CONJL;NOT_IN]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[DIFF_DEF;EXTENSION;+>;|->;dom;PAIR_EQ]
THEN SET_SPEC_TAC
THEN REWRITE_TAC[PAIR_EQ;IN_SING]
THEN EXISTS_TAC
"(number? IN ran book /\ (result! = ok))
=> book +> {number?}
| book"
THEN EXISTS_TAC
"(number? IN ran book /\ (result! = ok))
=> dom(book +> {number?}) |
dom book"
THEN EXISTS_TAC
"(number? IN ran book /\ (result! = ok))
=> NUMBER DIFF ran(book +> {number?}) |
NUMBER DIFF ran book"
THEN EXISTS_TAC "result!"
THEN ASM_CASES_TAC "number? IN ran book /\ (result! = ok)"
THEN LITE_IMP_RES_TAC UNION_SING_IN_P
THEN LITE_IMP_RES_TAC domPfunIN
THEN LITE_IMP_RES_TAC ranPfunIN
THEN SMART_ELIMINATE_TAC
THEN LITE_IMP_RES_TAC UNION_SING_Pfun
THEN LITE_IMP_RES_TAC RangeAntiResPfun
THEN LITE_IMP_RES_TAC domRangeAntiResPfun
THEN REWRITE_ASMS_TAC[]
THEN ASM_REWRITE_TAC[DIFF_IN_P]
THEN RES_TAC);;
declare
`UnknownSubscriber`
"SCHEMA
[XI TelephoneBook;
subscriber? :: SUBSCRIBER;
result! :: REPORT]
%-----------------%
[subscriber? NOT_IN known;
result! = unknown_subscriber]";;
declare
`RFindNumber`
"(FindNumber AND Success) OR UnknownSubscriber";;
prove_theorem
(`RFindNumber_total`,
"FORALL [RFindNumber] (pre RFindNumber)",
REWRITE_TAC[::;SCHEMA;CONJL;NOT_IN]
THEN REPEAT STRIP_TAC
THEN ASM_REWRITE_TAC[DIFF_DEF;EXTENSION;+>;|->;dom;PAIR_EQ]
THEN SET_SPEC_TAC
THEN REWRITE_TAC[PAIR_EQ;IN_SING]
THEN EXISTS_TAC "book"
THEN EXISTS_TAC "known"
THEN EXISTS_TAC "free"
THEN EXISTS_TAC
"(subscriber? IN dom book) => book^^subscriber? | number!"
THEN EXISTS_TAC "(subscriber? IN dom book) => ok | unknown_subscriber"
THEN ASSUM_LIST(STRIP_ASSUME_TAC o REWRITE_RULE[PAIR_EQ] o el 4)
THEN SMART_ELIMINATE_TAC
THEN (SMART_ELIMINATE_TAC ORELSE ALL_TAC)
THEN ASM_REWRITE_TAC[REPORT;IN_UNIV]);;
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