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|
(*---------------------------------------------------------------------------*)
(*
File: mk_ensures.sml
Description: This file defines ENSURES and the theorems and corrollaries
described in [CM88].
Author: (c) Copyright 1989-2008 by Flemming Andersen
Date: June 29, 1989
Last Update: December 30, 2007
*)
(*---------------------------------------------------------------------------*)
(*---------------------------------------------------------------------------*)
(* The definition of ENSURES is based on the definition:
p ensures q in Pr = <p unless q in Pr /\ (?s in Pr: {p /\ ~q} s {q})>
where p and q are state dependent first order logic predicates and s
in the program Pr are conditionally enabled statements transforming
a state into a new state. ENSURES then requires safety and the
existance of at least one state transition statement s which makes q
valid.
*)
let EXIST_TRANSITION_term =
`(!p q. EXIST_TRANSITION (p:'a->bool) q [] <=> F) /\
(!p q. EXIST_TRANSITION p q (CONS (st:'a->'a) Pr) <=>
((!s. (p s /\ ~q s) ==> q (st s)) \/ (EXIST_TRANSITION p q Pr)))`;;
let EXIST_TRANSITION = new_recursive_definition
list_RECURSION EXIST_TRANSITION_term;;
parse_as_infix ( "EXIST_TRANSITION", (TL_FIX, "right") );;
let ENSURES = new_infix_definition
("ENSURES", "<=>",
`!(p:'a->bool) q (Pr:('a->'a)list).
ENSURES p q Pr = (((p UNLESS q) Pr) /\ ((p EXIST_TRANSITION q) Pr))`,
TL_FIX);;
let ENSURES_STMT = new_infix_definition
("ENSURES_STMT", "<=>",
`!(p:'a->bool) q (st:'a->'a).
ENSURES_STMT p q st = (\s. p s /\ ~(q s) ==> q (st s))`,
TL_FIX);;
(*-------------------------------------------------------------------------*)
(*
Lemmas
*)
(*-------------------------------------------------------------------------*)
let ENSURES_lemma0 = TAC_PROOF
(([],
(`!(p:'a->bool) q r st.
((!s. p s /\ ~q s ==> q (st s)) /\ (!s. q s ==> r s)) ==>
(!s. p s /\ ~r s ==> r (st s))`)),
REPEAT STRIP_TAC THEN
ASSUME_TAC (CONTRAPOS (SPEC_ALL (ASSUME (`!s:'a. q s ==> r s`)))) THEN
ASSUME_TAC (SPEC (`(st:'a->'a) s`) (ASSUME (`!s:'a. q s ==> r s`))) THEN
RES_TAC THEN
RES_TAC);;
set_goal([],
(`!(p:'a->bool) p' q q' h.
(!s. (p UNLESS_STMT q) h s) ==>
(!s. (p' UNLESS_STMT q') h s) ==>
(!s. p' s /\ ~q' s ==> q' (h s)) ==>
(!s. (p /\* p') s /\ ~((p /\* q' \/* p' /\* q) \/* q /\* q') s) ==>
(((p /\* q' \/* p' /\* q) \/* q /\* q') (h s))`)
);;
let ENSURES_lemma1 = TAC_PROOF
(([],
`!(p:'a->bool) p' q q' h.
(!s. (p UNLESS_STMT q) h s) ==>
(!s. (p' UNLESS_STMT q') h s) ==>
(!s. p' s /\ ~q' s ==> q' (h s)) ==>
(!s. (p /\* p') s /\ ~((p /\* q' \/* p' /\* q) \/* q /\* q') s
==> ((p /\* q' \/* p' /\* q) \/* q /\* q') (h s))`),
REWRITE_TAC [UNLESS_STMT; AND_def; OR_def] THEN
CONV_TAC (DEPTH_CONV BETA_CONV) THEN
MESON_TAC []);;
let ENSURES_lemma2 = TAC_PROOF
(([],
(`!(p:'a->bool) q r st.
(!s. p s /\ ~q s ==> q (st s)) ==>
(!s. (p s \/ r s) /\ ~(q s \/ r s) ==> q (st s) \/ r (st s))`)),
REWRITE_TAC [(GEN_ALL (SYM (SPEC_ALL CONJ_ASSOC)));
(SYM (SPEC_ALL DISJ_ASSOC));NOT_CLAUSES;DE_MORGAN_THM] THEN
REPEAT STRIP_TAC THEN RES_TAC THEN
ASM_REWRITE_TAC []);;
let ENSURES_lemma3 = TAC_PROOF
(([],
(`!(p:'a->bool) q r Pr. (p ENSURES (q \/* r)) Pr ==>
(((p /\* (Not q)) \/* (p /\* q)) ENSURES (q \/* r)) Pr`)),
REWRITE_TAC [AND_COMPL_OR_lemma]);;
let ENSURES_lemma4 = TAC_PROOF
(([],
`!(p:'a->bool) q r (st:'a->'a).
(!s. p s /\ ~q s ==> q (st s)) ==>
(!s. (p \/* r) s /\ ~(q \/* r) s ==> (q \/* r) (st s))`),
REPEAT GEN_TAC THEN
REWRITE_TAC [OR_def] THEN
MESON_TAC []);;
(*---------------------------------------------------------------------------*)
(*
Theorems about EXIST_TRANSITION
*)
(*---------------------------------------------------------------------------*)
(*
EXIST_TRANSITION Consequence Weakening Theorem:
p EXIST_TRANSITION q in Pr; q ==> r
-------------------------------------
p EXIST_TRANSITION r in Pr
*)
let EXIST_TRANSITION_thm1 = prove_thm
("EXIST_TRANSITION_thm1",
(`!(p:'a->bool) q r Pr.
((p EXIST_TRANSITION q) Pr /\ (!s. (q s) ==> (r s))) ==>
((p EXIST_TRANSITION r) Pr)`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [EXIST_TRANSITION] THEN
STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [] THEN
REWRITE_TAC [REWRITE_RULE
[ASSUME `!s:'a. p s /\ ~q s ==> q (h s)`; ASSUME `!s:'a. q s ==> r s`]
(SPECL [`p:'a->bool`;`q:'a->bool`;`r:'a->bool`;`h:'a->'a`] ENSURES_lemma0)]);;
(*
Impossibility EXIST_TRANSITION Theorem:
p EXIST_TRANSITION false in Pr
--------------------------------
~p
*)
let EXIST_TRANSITION_thm2 = prove_thm
("EXIST_TRANSITION_thm2",
(`!(p:'a->bool) Pr.
((p EXIST_TRANSITION False) Pr) ==> !s. (Not p) s`),
GEN_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [EXIST_TRANSITION; NOT_def1] THEN
STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [] THENL
[
UNDISCH_TAC (`!s:'a. ((p:'a->bool) s) /\ ~(False s)
==> (False ((h:'a->'a) s))`) THEN
REWRITE_TAC [FALSE_def] THEN
CONV_TAC (DEPTH_CONV BETA_CONV)
;
UNDISCH_TAC (`!s:'a. (Not (p:'a->bool)) s`) THEN
REWRITE_TAC [NOT_def1] THEN
CONV_TAC (DEPTH_CONV BETA_CONV)
]);;
(*
Always EXIST_TRANSITION Theorem:
false EXIST_TRANSITION p in Pr
*)
let EXIST_TRANSITION_thm3 = prove_thm
("EXIST_TRANSITION_thm3",
(`!(p:'a->bool) st Pr. (False EXIST_TRANSITION p) (CONS st Pr)`),
REPEAT GEN_TAC THEN
REWRITE_TAC [EXIST_TRANSITION; FALSE_def]);;
let EXIST_TRANSITION_thm4 = prove_thm
("EXIST_TRANSITION_thm4",
(`!(p:'a->bool) q r Pr.
(p EXIST_TRANSITION q) Pr ==>
((p \/* r) EXIST_TRANSITION (q \/* r)) Pr`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [EXIST_TRANSITION] THEN
STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [] THEN
REWRITE_TAC [REWRITE_RULE
[ASSUME `!s:'a. (p:'a->bool) s /\ ~q s ==> q (h s)`]
(SPECL [`p:'a->bool`;`q:'a->bool`;`r:'a->bool`;`h:'a->'a`]
ENSURES_lemma4)]);;
let APPEND_lemma01 = TAC_PROOF
(([],
`!(l:('a)list). (APPEND l []) = l`),
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC [APPEND]);;
let EXIST_TRANSITION_thm5 = prove_thm
("EXIST_TRANSITION_thm5",
(`!(p:'a->bool) q st Pr.
(!s. p s /\ ~q s ==> q (st s))
==> (p EXIST_TRANSITION q) (CONS st Pr)`),
REPEAT GEN_TAC THEN
REWRITE_TAC [EXIST_TRANSITION] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC []);;
let APPEND_lemma02 = TAC_PROOF
(([],
`!st (l:('a)list). (APPEND [st] l) = (CONS st l)`),
GEN_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [APPEND]);;
let APPEND_lemma03 = TAC_PROOF
(([],
`!st (l1:('a)list) l2.
(APPEND (APPEND l1 [st]) l2) = (APPEND l1 (CONS st l2))`),
GEN_TAC THEN
LIST_INDUCT_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [APPEND] THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [APPEND]);;
let APPEND_lemma04 = TAC_PROOF
(([],
`!st (l1:('a)list) l2.
(APPEND (CONS st l1) l2) = (CONS st (APPEND l1 l2))`),
GEN_TAC THEN
LIST_INDUCT_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [APPEND] THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [APPEND]);;
let EXIST_TRANSITION_thm6 = prove_thm
("EXIST_TRANSITION_thm6",
(`!(p:'a->bool) q st Pr1 Pr2.
(!s. p s /\ ~q s ==> q (st s))
==> (p EXIST_TRANSITION q) (APPEND Pr1 (CONS st Pr2))`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [EXIST_TRANSITION;APPEND] THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC []);;
let EXIST_TRANSITION_thm7 = prove_thm
("EXIST_TRANSITION_thm7",
(`!(p:'a->bool) q FPr GPr.
(p EXIST_TRANSITION q) FPr
==> (p EXIST_TRANSITION q) (APPEND FPr GPr)`),
GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [EXIST_TRANSITION;APPEND] THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;EXIST_TRANSITION;APPEND]);;
let EXIST_TRANSITION_thm8 = prove_thm
("EXIST_TRANSITION_thm8",
(`!(p:'a->bool) q FPr GPr.
(p EXIST_TRANSITION q) FPr
==> (p EXIST_TRANSITION q) (APPEND GPr FPr)`),
GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [APPEND_lemma01;EXIST_TRANSITION;APPEND] THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;EXIST_TRANSITION;APPEND] THENL
[
REWRITE_TAC [UNDISCH_ALL (SPECL
[`p:'a->bool`;`q:'a->bool`;`h:'a->'a`;`t':('a->'a)list`;`t:('a->'a)list`]
EXIST_TRANSITION_thm6)]
;
REWRITE_TAC [REWRITE_RULE [APPEND_lemma03] (SPECL
[`(APPEND (t':('a->'a)list) [h])`]
(ASSUME `!GPr:('a->'a)list. (p EXIST_TRANSITION q) (APPEND GPr t)`))]
]);;
let EXIST_TRANSITION_thm9 = prove_thm
("EXIST_TRANSITION_thm9",
(`!(p:'a->bool) q st FPr GPr.
(p EXIST_TRANSITION q) (APPEND FPr GPr)
==> (p EXIST_TRANSITION q) (APPEND FPr (CONS st GPr))`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [APPEND_lemma01;EXIST_TRANSITION;APPEND] THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;EXIST_TRANSITION;APPEND]);;
let EXIST_TRANSITION_thm10 = prove_thm
("EXIST_TRANSITION_thm10",
(`!(p:'a->bool) q st Pr.
(p EXIST_TRANSITION q) Pr ==> (p EXIST_TRANSITION q) (CONS st Pr)`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;APPEND;EXIST_TRANSITION] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC []);;
let EXIST_TRANSITION_thm11 = prove_thm
("EXIST_TRANSITION_thm11",
(`!(p:'a->bool) q st Pr.
(p EXIST_TRANSITION q) (APPEND [st] Pr) =
(p EXIST_TRANSITION q) (APPEND Pr [st])`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;APPEND;EXIST_TRANSITION] THEN
EQ_TAC THEN
STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;APPEND;EXIST_TRANSITION] THENL
[
REWRITE_TAC [REWRITE_RULE [APPEND_lemma02] (SYM (ASSUME
`(((p:'a->bool) EXIST_TRANSITION q) (APPEND [st] t)) <=>
((p EXIST_TRANSITION q) (APPEND t [st]))`))] THEN
ASM_REWRITE_TAC [EXIST_TRANSITION]
;
REWRITE_TAC [REWRITE_RULE [APPEND_lemma02] (SYM (ASSUME
`(((p:'a->bool) EXIST_TRANSITION q) (APPEND [st] t)) <=>
((p EXIST_TRANSITION q) (APPEND t [st]))`))] THEN
ASM_REWRITE_TAC [UNDISCH_ALL (SPECL
[`p:'a->bool`;`q:'a->bool`;`st:'a->'a`;`t:('a->'a)list`]
EXIST_TRANSITION_thm10)]
;
STRIP_ASSUME_TAC (REWRITE_RULE [APPEND_lemma02;EXIST_TRANSITION]
(ASSUME `((p:'a->bool) EXIST_TRANSITION q) (APPEND [st] t)`)) THEN
ASM_REWRITE_TAC []
]);;
let EXIST_TRANSITION_thm12a = prove_thm
("EXIST_TRANSITION_thm12a",
(`!(p:'a->bool) q FPr GPr.
(p EXIST_TRANSITION q) (APPEND FPr GPr) ==>
(p EXIST_TRANSITION q) (APPEND GPr FPr)`),
GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;APPEND;EXIST_TRANSITION] THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;EXIST_TRANSITION;APPEND] THENL
[
REWRITE_TAC [UNDISCH_ALL (SPECL [`p:'a->bool`;`q:'a->bool`;`h:'a->'a`;
`GPr:('a->'a)list`;`t:('a->'a)list`] EXIST_TRANSITION_thm6)]
;
REWRITE_TAC [UNDISCH_ALL (SPECL [`p:'a->bool`;`q:'a->bool`;`h:'a->'a`;
`GPr:('a->'a)list`;`t:('a->'a)list`] EXIST_TRANSITION_thm9)]
]);;
let EXIST_TRANSITION_thm12b = prove_thm
("EXIST_TRANSITION_thm12b",
(`!(p:'a->bool) q FPr GPr.
(p EXIST_TRANSITION q) (APPEND GPr FPr) ==>
(p EXIST_TRANSITION q) (APPEND FPr GPr)`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;APPEND;EXIST_TRANSITION] THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
ASM_REWRITE_TAC [APPEND_lemma01;EXIST_TRANSITION;APPEND] THENL
[
REWRITE_TAC [UNDISCH_ALL (SPECL [`p:'a->bool`;`q:'a->bool`;`h:'a->'a`;
`FPr:('a->'a)list`;`t:('a->'a)list`] EXIST_TRANSITION_thm6)]
;
REWRITE_TAC [UNDISCH_ALL (SPECL [`p:'a->bool`;`q:'a->bool`;`h:'a->'a`;
`FPr:('a->'a)list`;`t:('a->'a)list`] EXIST_TRANSITION_thm9)]
]);;
let EXIST_TRANSITION_thm12 = prove_thm
("EXIST_TRANSITION_thm12",
(`!(p:'a->bool) q FPr GPr.
(p EXIST_TRANSITION q) (APPEND GPr FPr) =
(p EXIST_TRANSITION q) (APPEND FPr GPr)`),
REPEAT GEN_TAC THEN
EQ_TAC THEN
REPEAT STRIP_TAC THEN
RES_TAC THEN
REWRITE_TAC [UNDISCH_ALL (SPEC_ALL EXIST_TRANSITION_thm12a);
UNDISCH_ALL (SPEC_ALL EXIST_TRANSITION_thm12b)]);;
(*---------------------------------------------------------------------------*)
(*
Theorems about ENSURES
*)
(*---------------------------------------------------------------------------*)
(*
Reflexivity Theorem:
p ensures p in Pr
The theorem is only valid for non-empty programs
*)
let ENSURES_thm0 = prove_thm
("ENSURES_thm0",
(`!(p:'a->bool) q. (p ENSURES q) [] = F`),
REWRITE_TAC [ENSURES] THEN
STRIP_TAC THEN
REWRITE_TAC [UNLESS;EXIST_TRANSITION]);;
let ENSURES_thm1 = prove_thm
("ENSURES_thm1",
(`!(p:'a->bool) st Pr. (p ENSURES p) (CONS st Pr)`),
REWRITE_TAC [ENSURES] THEN
STRIP_TAC THEN
REWRITE_TAC [UNLESS;EXIST_TRANSITION] THEN
REWRITE_TAC [UNLESS_thm1;UNLESS_STMT] THEN
REWRITE_TAC [BETA_CONV (`(\s:'a. (p s /\ ~p s) ==> p (st s))s`)] THEN
REWRITE_TAC[NOT_AND;IMP_CLAUSES]);;
(*
Consequence Weakening Theorem:
p ensures q in Pr; q ==> r
----------------------------
p ensures r in Pr
*)
let ENSURES_thm2 = prove_thm
("ENSURES_thm2",
(`!(p:'a->bool) q r Pr.
((p ENSURES q) Pr /\ (!s:'a. (q s) ==> (r s)))
==>
((p ENSURES r) Pr)`),
REWRITE_TAC [ENSURES] THEN
REPEAT STRIP_TAC THENL
[
ASSUME_TAC (UNDISCH_ALL (SPEC (`!s:'a. q s ==> r s`)
(SPEC (`((p:'a->bool) UNLESS q) Pr`) AND_INTRO_THM))) THEN
STRIP_ASSUME_TAC (UNDISCH_ALL (SPEC_ALL UNLESS_thm3))
;
ASSUME_TAC (UNDISCH_ALL (SPEC (`!s:'a. q s ==> r s`)
(SPEC (`((p:'a->bool) EXIST_TRANSITION q) Pr`) AND_INTRO_THM))) THEN
STRIP_ASSUME_TAC (UNDISCH_ALL (SPEC_ALL EXIST_TRANSITION_thm1))
]);;
(*
Impossibility Theorem:
p ensures false in Pr
----------------------
~p
*)
let ENSURES_thm3 = prove_thm
("ENSURES_thm3",
(`!(p:'a->bool) Pr. ((p ENSURES False) Pr) ==> !s. (Not p)s`),
GEN_TAC THEN
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC [ENSURES; UNLESS; EXIST_TRANSITION] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC [] THENL
[
UNDISCH_TAC `!s:'a. (p:'a->bool) s /\ ~(False s) ==> False ((h:'a->'a) s)` THEN
REWRITE_TAC [FALSE_def; NOT_def1] THEN
CONV_TAC (DEPTH_CONV BETA_CONV)
;
IMP_RES_TAC EXIST_TRANSITION_thm2
]);;
(*
Conjunction Theorem:
p unless q in Pr; p' ensures q' in Pr
-----------------------------------------------
p/\p' ensures (p/\q')\/(p'/\q)\/(q/\q') in Pr
*)
let ENSURES_thm4 = prove_thm
("ENSURES_thm4",
(`!(p:'a->bool) q p' q' Pr.
(p UNLESS q) Pr /\ (p' ENSURES q') Pr ==>
((p /\* p') ENSURES (((p /\* q') \/* (p' /\* q)) \/* (q /\* q')))
Pr`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [ENSURES;UNLESS;EXIST_TRANSITION] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC [] THENL
[
REWRITE_TAC
[REWRITE_RULE [ASSUME `!s:'a. ((p:'a->bool) UNLESS_STMT q) (h:'a->'a) s`;
ASSUME `!s:'a. ((p':'a->bool) UNLESS_STMT q') (h:'a->'a) s`]
(SPECL [`p:'a->bool`;`q:'a->bool`;`p':'a->bool`;`q':'a->bool`;`h:'a->'a`]
UNLESS_STMT_thm3)]
;
REWRITE_TAC
[REWRITE_RULE [ASSUME `((p:'a->bool) UNLESS q) (t:('a->'a)list)`;
ASSUME `((p':'a->bool) UNLESS q') (t:('a->'a)list)`]
(SPECL [`p:'a->bool`;`q:'a->bool`;`p':'a->bool`;`q':'a->bool`;`t:('a->'a)list`]
UNLESS_thm4)]
;
REWRITE_TAC [REWRITE_RULE
[ASSUME `!s:'a. ((p:'a->bool) UNLESS_STMT q) (h:'a->'a) s`;
ASSUME `!s:'a. ((p':'a->bool) UNLESS_STMT q') (h:'a->'a) s`;
ASSUME `!s:'a. (p':'a->bool) s /\ ~(q' s) ==> q' ((h:'a->'a) s)`]
(SPEC_ALL ENSURES_lemma1)]
;
REWRITE_TAC
[REWRITE_RULE [ASSUME `!s:'a. ((p:'a->bool) UNLESS_STMT q) (h:'a->'a) s`;
ASSUME `!s:'a. ((p':'a->bool) UNLESS_STMT q') (h:'a->'a) s`]
(SPECL [`p:'a->bool`;`q:'a->bool`;`p':'a->bool`;`q':'a->bool`;`h:'a->'a`]
UNLESS_STMT_thm3)]
;
UNDISCH_TAC `((p:'a->bool) UNLESS q) t /\ (p' ENSURES q') (t:('a->'a)list)
==> (p /\* p' ENSURES (p /\* q' \/* p' /\* q) \/* q /\* q') t` THEN
ASM_REWRITE_TAC [ENSURES] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC []
;
UNDISCH_TAC `((p:'a->bool) UNLESS q) t /\ (p' ENSURES q') (t:('a->'a)list)
==> (p /\* p' ENSURES (p /\* q' \/* p' /\* q) \/* q /\* q') t` THEN
ASM_REWRITE_TAC [ENSURES] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC []
]);;
(*
Conjunction Theorem:
p ensures q in Pr
-------------------------
p\/r ensures q\/r in Pr
*)
let ENSURES_thm5 = prove_thm
("ENSURES_thm5",
(`!(p:'a->bool) q r Pr.
((p ENSURES q) Pr) ==> (((p \/* r) ENSURES (q \/* r)) Pr)`),
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC [ENSURES;UNLESS;EXIST_TRANSITION] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC [] THENL
[
IMP_RES_TAC UNLESS_STMT_thm6 THEN
ASM_REWRITE_TAC []
;
IMP_RES_TAC UNLESS_cor23 THEN
ASM_REWRITE_TAC []
;
REWRITE_TAC [REWRITE_RULE
[ASSUME `!s:'a. (p:'a->bool) s /\ ~q s ==> q (h s)`]
(SPECL [`p:'a->bool`;`q:'a->bool`;`r:'a->bool`;`h:'a->'a`]
ENSURES_lemma4)]
;
IMP_RES_TAC UNLESS_STMT_thm6 THEN
ASM_REWRITE_TAC []
;
IMP_RES_TAC UNLESS_cor23 THEN
ASM_REWRITE_TAC []
;
IMP_RES_TAC EXIST_TRANSITION_thm4 THEN
ASM_REWRITE_TAC []
]);;
(*
-----------------------------------------------------------------------------
Corollaries about ENSURES
-----------------------------------------------------------------------------
*)
(*
Implies Corollary:
p => q
-------------------
p ensures q in Pr
This corollary is only valid for non-empty programs.
*)
let ENSURES_cor1 = prove_thm
("ENSURES_cor1",
(`!(p:'a->bool) q st Pr.
(!s. p s ==> q s) ==> (p ENSURES q) (CONS st Pr)`),
REPEAT GEN_TAC THEN
DISCH_TAC THEN
ASSUME_TAC (SPEC_ALL ENSURES_thm1) THEN
ASSUME_TAC (UNDISCH_ALL (SPECL
[(`((p:'a->bool) ENSURES p)(CONS st Pr)`);(`!s:'a. p s ==> q s`)]
AND_INTRO_THM)) THEN
STRIP_ASSUME_TAC (UNDISCH_ALL (SPECL
[(`p:'a->bool`);(`p:'a->bool`);(`q:'a->bool`);
(`CONS (st:'a->'a) Pr`)]
ENSURES_thm2)));;
let ENSURES_cor2 = prove_thm
("ENSURES_cor2",
(`!(p:'a->bool) q Pr. (p ENSURES q) Pr ==> (p UNLESS q) Pr`),
REWRITE_TAC [ENSURES] THEN
REPEAT STRIP_TAC);;
let ENSURES_cor3 = prove_thm
("ENSURES_cor3",
(`!(p:'a->bool) q r Pr.
((p \/* q) ENSURES r)Pr ==> (p ENSURES (q \/* r))Pr`),
REPEAT GEN_TAC THEN
DISCH_TAC THEN
ASSUME_TAC (UNDISCH_ALL (SPECL
[(`((p:'a->bool) \/* q)`);(`r:'a->bool`);
(`Pr:('a->'a)list`)] ENSURES_cor2)) THEN
ASSUME_TAC (UNDISCH_ALL (SPECL
[(`p:'a->bool`);(`q:'a->bool`);(`r:'a->bool`);
(`Pr:('a->'a)list`)] UNLESS_cor4)) THEN
ASSUME_TAC (UNDISCH_ALL (SPECL
[(`((p:'a->bool) UNLESS (q \/* r))Pr`);
(`(((p:'a->bool) \/* q) ENSURES r)Pr`)]
AND_INTRO_THM)) THEN
ASSUME_TAC (UNDISCH_ALL (SPECL
[(`p:'a->bool`);(`((q:'a->bool) \/* r)`);
(`((p:'a->bool) \/* q)`);(`r:'a->bool`);
(`Pr:('a->'a)list`)] ENSURES_thm4)) THEN
UNDISCH_TAC (`(((p:'a->bool) /\* (p \/* q)) ENSURES
(((p /\* r) \/* ((p \/* q) /\* (q \/* r))) \/*
((q \/* r) /\* r))) Pr`) THEN
REWRITE_TAC [AND_OR_EQ_lemma] THEN
REWRITE_TAC [OR_ASSOC_lemma;AND_ASSOC_lemma] THEN
PURE_ONCE_REWRITE_TAC [SPECL
[(`((q:'a->bool) \/* r)`);
(`r:'a->bool`)] AND_COMM_lemma] THEN
ONCE_REWRITE_TAC [AND_OR_EQ_AND_COMM_OR_lemma] THEN
REWRITE_TAC [AND_OR_EQ_lemma] THEN
DISCH_TAC THEN
ASSUME_TAC (SPECL [(`p:'a->bool`);(`q:'a->bool`);(`r:'a->bool`)]
IMPLY_WEAK_lemma5) THEN
ASSUME_TAC (UNDISCH_ALL (SPECL
[(`((p:'a->bool) ENSURES
((p /\* r) \/* (((p \/* q) /\* (q \/* r)) \/* r)))Pr`);
(`!s:'a. ((p /\* r) \/* (((p \/* q) /\* (q \/* r)) \/* r))s ==>
(q \/* r)s`)]
AND_INTRO_THM)) THEN
STRIP_ASSUME_TAC (UNDISCH_ALL (SPECL
[(`p:'a->bool`);
(`(((p:'a->bool) /\* r) \/* (((p \/* q) /\* (q \/* r)) \/* r))`);
(`((q:'a->bool) \/* r)`);(`Pr:('a->'a)list`)]
ENSURES_thm2)));;
let ENSURES_cor4 = prove_thm
("ENSURES_cor4",
(`!(p:'a->bool) q r Pr. (p ENSURES (q \/* r)) Pr ==>
((p /\* (Not q)) ENSURES (q \/* r)) Pr`),
REPEAT STRIP_TAC THEN
ASSUME_TAC (UNDISCH_ALL (SPEC_ALL ENSURES_lemma3)) THEN
ASSUME_TAC (UNDISCH_ALL (SPECL
[(`((p:'a->bool) /\* (Not q))`);(`((p:'a->bool) /\* q)`);
(`((q:'a->bool) \/* r)`);(`Pr:('a->'a)list`)] ENSURES_cor3)) THEN
UNDISCH_TAC
(`(((p:'a->bool) /\* (Not q)) ENSURES
((p /\* q) \/* (q \/* r)))Pr`) THEN
REWRITE_TAC [GEN_ALL (SYM (SPEC_ALL OR_ASSOC_lemma))] THEN
REWRITE_TAC [P_AND_Q_OR_Q_lemma]);;
(*
Consequence Weakening Corollary:
p ensures q in Pr
-------------------------
p ensures (q \/ r) in Pr
*)
let ENSURES_cor5 = prove_thm
("ENSURES_cor5",
(`!(p:'a->bool) q r Pr.
(p ENSURES q) Pr ==> (p ENSURES (q \/* r)) Pr`),
REPEAT STRIP_TAC THEN
ASSUME_TAC (SPECL [(`q:'a->bool`);(`r:'a->bool`)]
IMPLY_WEAK_lemma_b) THEN
ASSUME_TAC (SPECL
[(`p:'a->bool`);(`q:'a->bool`);(`(q:'a->bool) \/* r`)]
ENSURES_thm2) THEN
RES_TAC);;
(*
Always Corollary:
false ensures p in Pr
*)
let ENSURES_cor6 = prove_thm
("ENSURES_cor6",
(`!(p:'a->bool) st Pr. (False ENSURES p) (CONS st Pr)`),
REWRITE_TAC [ENSURES;UNLESS_cor7;EXIST_TRANSITION_thm3]);;
let ENSURES_cor7 = prove_thm
("ENSURES_cor7",
(`!(p:'a->bool) q r Pr.
(p ENSURES q) Pr /\ (r STABLE Pr)
==>
((p /\* r) ENSURES (q /\*
r))Pr`),
REPEAT GEN_TAC THEN
REWRITE_TAC [STABLE] THEN
REPEAT STRIP_TAC THEN
IMP_RES_TAC (ONCE_REWRITE_RULE [AND_COMM_lemma]
(REWRITE_RULE [AND_False_lemma;OR_False_lemma]
(ONCE_REWRITE_RULE [OR_AND_COMM_lemma]
(REWRITE_RULE [AND_False_lemma;OR_False_lemma] (SPECL
[(`r:'a->bool`);(`False:'a->bool`);
(`p:'a->bool`);(`q:'a->bool`);
(`Pr:('a->'a)list`)] ENSURES_thm4))))));;
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