1 (* ========================================================================= *)
3 (* ========================================================================= *)
5 let MINIMIZING_CHOICE = prove
6 (`!(m:A->num) s. (?x. P x) ==> ?a. P a /\ !b. P b ==> m(a) <= m(b)`,
7 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM NOT_LT] THEN
8 MP_TAC(ISPEC `\n. ?x. P x /\ (m:A->num) x = n` num_WOP) THEN
11 (* ------------------------------------------------------------------------- *)
12 (* The Nash-Williams minimal bad sequence argument for some predicate `bad` *)
13 (* that is a "safety property" in the Lamport/Alpern/Schneider sense. *)
14 (* ------------------------------------------------------------------------- *)
16 let MINIMAL_BAD_SEQUENCE = prove
17 (`!(bad:(num->A)->bool) (m:A->num).
18 (!x. ~bad x ==> ?n. !y. (!k. k < n ==> y k = x k) ==> ~bad y) /\
21 !z n. bad z /\ (!k. k < n ==> z k = y k) ==> m(y n) <= m(z n)`,
22 REPEAT GEN_TAC THEN DISCH_TAC THEN SUBGOAL_THEN
23 `?x. !n. (x:num->A) n =
24 @a. (?y. bad y /\ (!k. k < n ==> y k = x k) /\ y n = a) /\
25 !z. bad z /\ (!k. k < n ==> z k = x k)
26 ==> (m:A->num)(a) <= m(z n)`
27 STRIP_ASSUME_TAC THENL
28 [MATCH_MP_TAC(MATCH_MP WF_REC WF_num) THEN SIMP_TAC[]; ALL_TAC] THEN
30 `!n. (?y:num->A. bad y /\ (!k. k < n ==> y k = x k) /\ y n = x n) /\
31 !z. bad z /\ (!k. k < n ==> z k = x k) ==> m(x n):num <= m(z n)`
32 ASSUME_TAC THENL [ALL_TAC; EXISTS_TAC `x:num->A` THEN ASM_MESON_TAC[]] THEN
33 MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN
34 FIRST_X_ASSUM(fun th -> DISCH_TAC THEN SUBST1_TAC(SPEC `n:num` th)) THEN
35 CONV_TAC SELECT_CONV THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
36 REWRITE_TAC[TAUT `(p /\ q /\ r) /\ s <=> r /\ p /\ q /\ s`] THEN
37 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM1] THEN
38 REWRITE_TAC[CONJ_ASSOC] THEN MATCH_MP_TAC MINIMIZING_CHOICE THEN
39 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN SPEC_TAC(`n:num`,`n:num`) THEN
40 INDUCT_TAC THEN SIMP_TAC[LT] THEN MESON_TAC[]);;
42 (* ------------------------------------------------------------------------- *)
43 (* Dickson's Lemma itself. *)
44 (* ------------------------------------------------------------------------- *)
47 (`!n x:num->num->num. ?i j. i < j /\ (!k. k < n ==> x i k <= x j k)`,
49 `bad = \n x:num->num->num. !i j. i < j ==> ?k. k < n /\ x j k < x i k` THEN
50 SUBGOAL_THEN `!n:num x:num->num->num. ~(bad n x)` MP_TAC THENL
51 [ALL_TAC; EXPAND_TAC "bad" THEN MESON_TAC[NOT_LT]] THEN
52 INDUCT_TAC THENL [EXPAND_TAC "bad" THEN MESON_TAC[LT]; ALL_TAC] THEN
53 REWRITE_TAC[GSYM NOT_EXISTS_THM] THEN DISCH_TAC THEN
55 `?x. bad (SUC n) (x:num->num->num) /\
56 !y j. bad (SUC n) y /\ (!i. i < j ==> y i = x i)
58 STRIP_ASSUME_TAC THENL
59 [MATCH_MP_TAC MINIMAL_BAD_SEQUENCE THEN ASM_REWRITE_TAC[] THEN
60 X_GEN_TAC `x:num->num->num` THEN EXPAND_TAC "bad" THEN
61 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN
62 MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN
63 EXISTS_TAC `SUC j` THEN X_GEN_TAC `y:num->num->num` THEN
64 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [LT_SUC_LE] THEN
65 REWRITE_TAC[LE_LT] THEN STRIP_TAC THEN
66 MAP_EVERY EXISTS_TAC [`i:num`; `j:num`] THEN ASM_MESON_TAC[];
67 SUBGOAL_THEN `~(bad (n:num) (x:num->num->num))` MP_TAC THENL
68 [ASM_MESON_TAC[]; EXPAND_TAC "bad" THEN REWRITE_TAC[]] THEN
69 MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN DISCH_TAC THEN
70 MP_TAC(ASSUME `bad (SUC n) (x:num->num->num):bool`) THEN
71 EXPAND_TAC "bad" THEN REWRITE_TAC[] THEN
72 DISCH_THEN(MP_TAC o SPECL [`i:num`; `j:num`]) THEN
73 ASM_REWRITE_TAC[LT] THEN MATCH_MP_TAC MONO_EXISTS THEN
74 X_GEN_TAC `k:num` THEN STRIP_TAC THEN ASM_REWRITE_TAC[LT_REFL] THEN
75 FIRST_X_ASSUM SUBST_ALL_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC
76 `\k. if k < i then (x:num->num->num) k else x (j + k - i)`) THEN
77 DISCH_THEN(MP_TAC o SPEC `i:num`) THEN
78 ASM_REWRITE_TAC[LT_REFL; SUB_REFL; ADD_CLAUSES; NOT_IMP; NOT_LE] THEN
79 SIMP_TAC[] THEN UNDISCH_TAC `bad (SUC n) (x:num->num->num):bool` THEN
80 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN EXPAND_TAC "bad" THEN
81 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN
82 REPEAT GEN_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
83 ASM_MESON_TAC[LT_TRANS; ARITH_RULE
84 `(a:num < i /\ ~(b < i) /\ i < j ==> a < j + b - i) /\
85 (~(a < i) /\ a < b /\ i < j ==> j + a - i < j + b - i)`]]);;