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|
(* ========================================================================= *)
(* Proof of some useful AC equivalents like wellordering and Zorn's Lemma. *)
(* *)
(* This is a straight port of the old HOL88 wellorder library. I started to *)
(* clean up the proofs to exploit first order automation, but didn't have *)
(* the patience to persist till the end. Anyway, the proofs work! *)
(* ========================================================================= *)
let PBETA_TAC = CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV);;
let EXPAND_TAC s = FIRST_ASSUM(SUBST1_TAC o SYM o
check((=) s o fst o dest_var o rhs o concl)) THEN BETA_TAC;;
let SUBSET_PRED = prove
(`!P Q. P SUBSET Q <=> !x. P x ==> Q x`,
REWRITE_TAC[SUBSET; IN]);;
let UNIONS_PRED = prove
(`UNIONS P = \x. ?p. P p /\ p x`,
REWRITE_TAC[UNIONS; FUN_EQ_THM; IN_ELIM_THM; IN]);;
(* ======================================================================== *)
(* (1) Definitions and general lemmas. *)
(* ======================================================================== *)
(* ------------------------------------------------------------------------ *)
(* Irreflexive version of an ordering. *)
(* ------------------------------------------------------------------------ *)
let less = new_definition
`(less l)(x,y) <=> (l:A#A->bool)(x,y) /\ ~(x = y)`;;
(* ------------------------------------------------------------------------ *)
(* Field of an uncurried binary relation *)
(* ------------------------------------------------------------------------ *)
let fl = new_definition
`fl(l:A#A->bool) x <=> ?y:A. l(x,y) \/ l(y,x)`;;
(* ------------------------------------------------------------------------ *)
(* Partial order (we infer the domain from the field of the relation) *)
(* ------------------------------------------------------------------------ *)
let poset = new_definition
`poset (l:A#A->bool) <=>
(!x. fl(l) x ==> l(x,x)) /\
(!x y z. l(x,y) /\ l(y,z) ==> l(x,z)) /\
(!x y. l(x,y) /\ l(y,x) ==> (x = y))`;;
(* ------------------------------------------------------------------------ *)
(* Chain in a poset (Defined as a subset of the field, not the ordering) *)
(* ------------------------------------------------------------------------ *)
let chain = new_definition
`chain(l:A#A->bool) P <=> (!x y. P x /\ P y ==> l(x,y) \/ l(y,x))`;;
(* ------------------------------------------------------------------------- *)
(* Total order. *)
(* ------------------------------------------------------------------------- *)
let toset = new_definition
`toset (l:A#A->bool) <=>
poset l /\ !x y. x IN fl(l) /\ y IN fl(l) ==> l(x,y) \/ l(y,x)`;;
(* ------------------------------------------------------------------------ *)
(* Wellorder *)
(* ------------------------------------------------------------------------ *)
let woset = new_definition
`woset (l:A#A->bool) <=>
(!x. fl(l) x ==> l(x,x)) /\
(!x y z. l(x,y) /\ l(y,z) ==> l(x,z)) /\
(!x y. l(x,y) /\ l(y,x) ==> (x = y)) /\
(!x y. fl(l) x /\ fl(l) y ==> l(x,y) \/ l(y,x)) /\
(!P. (!x. P x ==> fl(l) x) /\ (?x. P x) ==>
(?y. P y /\ (!z. P z ==> l(y,z))))`;;
(* ------------------------------------------------------------------------ *)
(* General (reflexive) notion of initial segment. *)
(* ------------------------------------------------------------------------ *)
parse_as_infix("inseg",(12,"right"));;
let inseg = new_definition
`(l:A#A->bool) inseg m <=> !x y. l(x,y) <=> m(x,y) /\ fl(l) y`;;
let INSEG_ANTISYM = prove
(`!l m:A#A->bool. l inseg m /\ m inseg l ==> l = m`,
REWRITE_TAC[FUN_EQ_THM; FORALL_PAIR_THM] THEN
REWRITE_TAC[inseg] THEN MESON_TAC[]);;
let INSEG_REFL = prove
(`!l:A#A->bool. l inseg l`,
REWRITE_TAC[inseg; fl] THEN MESON_TAC[]);;
let INSEG_TRANS = prove
(`!l m n:A#A->bool. l inseg m /\ m inseg n ==> l inseg n`,
REWRITE_TAC[inseg; fl] THEN MESON_TAC[]);;
(* ------------------------------------------------------------------------ *)
(* Specific form of initial segment: `all elements in fl(l) less than a`. *)
(* ------------------------------------------------------------------------ *)
let linseg = new_definition
`linseg (l:A#A->bool) a = \(x,y). l(x,y) /\ (less l)(y,a)`;;
(* ------------------------------------------------------------------------ *)
(* `Ordinals`, i.e. canonical wosets using choice operator. *)
(* ------------------------------------------------------------------------ *)
let ordinal = new_definition
`ordinal(l:A#A->bool) <=>
woset(l) /\ (!x. fl(l) x ==> (x = (@) (\y. ~(less l)(y,x))))`;;
(* ------------------------------------------------------------------------ *)
(* Now useful things about the orderings *)
(* ------------------------------------------------------------------------ *)
let [POSET_REFL; POSET_TRANS; POSET_ANTISYM] =
map (GEN `l:A#A->bool` o DISCH_ALL)
(CONJUNCTS(PURE_ONCE_REWRITE_RULE[poset] (ASSUME `poset (l:A#A->bool)`)));;
let POSET_FLEQ = prove
(`!l:A#A->bool. poset l ==> (!x. fl(l) x <=> l(x,x))`,
MESON_TAC[POSET_REFL; fl]);;
let CHAIN_SUBSET = prove
(`!(l:A#A->bool) P Q. chain(l) P /\ Q SUBSET P ==> chain(l) Q`,
REWRITE_TAC[chain; SUBSET_PRED] THEN MESON_TAC[]);;
let [WOSET_REFL; WOSET_TRANS; WOSET_ANTISYM; WOSET_TOTAL; WOSET_WELL] =
map (GEN `l:A#A->bool` o DISCH_ALL)
(CONJUNCTS(PURE_ONCE_REWRITE_RULE[woset] (ASSUME `woset (l:A#A->bool)`)));;
let WOSET_POSET = prove
(`!l:A#A->bool. woset l ==> poset l`,
GEN_TAC THEN REWRITE_TAC[woset; poset] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[]);;
let WOSET_FLEQ = prove
(`!l:A#A->bool. woset l ==> (!x. fl(l) x <=> l(x,x))`,
MESON_TAC[WOSET_POSET; POSET_FLEQ]);;
let WOSET_TRANS_LESS = prove
(`!l:A#A->bool. woset l ==>
!x y z. (less l)(x,y) /\ l(y,z) ==> (less l)(x,z)`,
REWRITE_TAC[woset; less] THEN MESON_TAC[]);;
(* ------------------------------------------------------------------------ *)
(* Wellfoundedness (in two slightly different senses) and either totality *)
(* or antisymmetry are sufficient for a wellorder. *)
(* ------------------------------------------------------------------------ *)
let WOSET = prove
(`!l:A#A->bool.
woset l <=>
(!x y. l(x,y) /\ l(y,x) ==> (x = y)) /\
(!P. (!x. P x ==> fl(l) x) /\ (?x. P x) ==>
(?y. P y /\ (!z. P z ==> l(y,z))))`,
GEN_TAC THEN REWRITE_TAC[woset] THEN EQ_TAC THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `(!x y z. l(x,y) /\ l(y,z) ==> l(x,z)) /\
(!x:A y. fl(l) x /\ fl(l) y ==> l(x,y) \/ l(y,x))`
MP_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REPEAT STRIP_TAC THENL
[FIRST_ASSUM(MP_TAC o SPEC `\w:A. (w = x) \/ (w = y) \/ (w = z)`) THEN
REWRITE_TAC[fl];
FIRST_ASSUM(MP_TAC o SPEC `\w:A. (w = x) \/ (w = y)`)] THEN
ASM_MESON_TAC[]);;
let WOSET_WF = prove
(`!l:A#A->bool.
woset l <=>
WF(\x y. l(x,y) /\ ~(x = y)) /\
(!x y. fl l x /\ fl l y ==> l(x,y) \/ l(y,x))`,
GEN_TAC THEN
ASM_CASES_TAC `!x y:A. fl l x /\ fl l y ==> l(x,y) \/ l(y,x)` THENL
[ASM_REWRITE_TAC[WOSET]; ASM_REWRITE_TAC[woset]] THEN
ASM_CASES_TAC `!x y:A. l(x,y) /\ l(y,x) ==> x = y` THEN
ASM_REWRITE_TAC[] THENL
[REWRITE_TAC[WF];
DISCH_THEN(MP_TAC o MATCH_MP WF_ANTISYM) THEN ASM_MESON_TAC[]] THEN
EQ_TAC THENL [DISCH_TAC; MATCH_MP_TAC MONO_FORALL THEN ASM_MESON_TAC[]] THEN
X_GEN_TAC `P:A->bool` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `\x:A. P x /\ fl l x`) THEN
REWRITE_TAC[fl] THEN ASM_MESON_TAC[]);;
(* ------------------------------------------------------------------------ *)
(* Misc lemmas. *)
(* ------------------------------------------------------------------------ *)
let PAIRED_EXT = prove
(`!(l:A#B->C) m. (!x y. l(x,y) = m(x,y)) <=> (l = m)`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `p:A#B` THEN
SUBST1_TAC(SYM(SPEC `p:A#B` PAIR)) THEN POP_ASSUM MATCH_ACCEPT_TAC);;
let WOSET_TRANS_LE = prove
(`!l:A#A->bool. woset l ==>
!x y z. l(x,y) /\ (less l)(y,z) ==> (less l)(x,z)`,
REWRITE_TAC[less] THEN MESON_TAC[WOSET_TRANS; WOSET_ANTISYM]);;
let WOSET_WELL_CONTRAPOS = prove
(`!l:A#A->bool. woset l ==>
(!P. (!x. P x ==> fl(l) x) /\ (?x. P x) ==>
(?y. P y /\ (!z. (less l)(z,y) ==> ~P z)))`,
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `P:A->bool` o MATCH_MP WOSET_WELL) THEN
ASM_MESON_TAC[WOSET_TRANS_LE; less]);;
let WOSET_TOTAL_LE = prove
(`!l:A#A->bool. woset l
==> !x y. fl(l) x /\ fl(l) y ==> l(x,y) \/ (less l)(y,x)`,
REWRITE_TAC[less] THEN MESON_TAC[WOSET_REFL; WOSET_TOTAL]);;
let WOSET_TOTAL_LT = prove
(`!l:A#A->bool. woset l ==>
!x y. fl(l) x /\ fl(l) y ==> (x = y) \/ (less l)(x,y) \/ (less l)(y,x)`,
REWRITE_TAC[less] THEN MESON_TAC[WOSET_TOTAL]);;
let ORDINAL_IMP_WOSET = prove
(`!l:A#A->bool. ordinal l ==> woset l`,
SIMP_TAC[ordinal]);;
let FL = prove
(`!l:A#A->bool. fl l = {x:A | ?y. l(x,y) \/ l(y,x)}`,
REWRITE_TAC[FUN_EQ_THM; IN_ELIM_THM; fl]);;
let FL_SUBSET = prove
(`!l r. l SUBSET r ==> fl l SUBSET fl r`,
REWRITE_TAC[SUBSET; IN; fl] THEN MESON_TAC[]);;
let FINITE_FL = prove
(`!l:A#A->bool. FINITE(fl l) <=> FINITE l`,
GEN_TAC THEN REWRITE_TAC[FL] THEN EQ_TAC THENL
[DISCH_THEN(MP_TAC o MATCH_MP FINITE_CROSS o W CONJ) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN
REWRITE_TAC[SUBSET; FORALL_PAIR_THM; IN_CROSS] THEN SET_TAC[];
DISCH_THEN((fun th ->
MP_TAC(ISPEC `FST:A#A->A` th) THEN MP_TAC(ISPEC `SND:A#A->A` th)) o
MATCH_MP FINITE_IMAGE) THEN
REWRITE_TAC[IMP_IMP; GSYM FINITE_UNION] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN
REWRITE_TAC[SUBSET; FORALL_PAIR_THM; IN_CROSS; IN_UNION; IN_IMAGE] THEN
REWRITE_TAC[EXISTS_PAIR_THM; IN_ELIM_THM] THEN REWRITE_TAC[IN] THEN
SET_TAC[]]);;
let WOSET_FINITE_TOSET = prove
(`!l:A#A->bool. toset l /\ FINITE l ==> woset l`,
ONCE_REWRITE_TAC[GSYM FINITE_FL] THEN
SIMP_TAC[toset; WOSET_WF; poset; IN] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC WF_FINITE THEN REWRITE_TAC[] THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; X_GEN_TAC `a:A`] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ]
FINITE_SUBSET)) THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN REWRITE_TAC[fl; IN] THEN
ASM_MESON_TAC[]);;
(* ======================================================================== *)
(* (2) AXIOM OF CHOICE ==> CANTOR-ZERMELO WELLORDERING THEOREM *)
(* ======================================================================== *)
(* ------------------------------------------------------------------------ *)
(* UNIONS of initial segments is an initial segment. *)
(* ------------------------------------------------------------------------ *)
let UNION_FL = prove
(`!P (l:A#A->bool). fl(UNIONS P) x <=> ?l. P l /\ fl(l) x`,
REPEAT GEN_TAC THEN REWRITE_TAC[UNIONS_PRED; fl] THEN MESON_TAC[]);;
let UNION_INSEG = prove
(`!P (l:A#A->bool). (!m. P m ==> m inseg l) ==> (UNIONS P) inseg l`,
REWRITE_TAC[inseg; UNION_FL; UNIONS_PRED] THEN ASM_MESON_TAC[]);;
(* ------------------------------------------------------------------------ *)
(* Initial segment of a woset is a woset. *)
(* ------------------------------------------------------------------------ *)
let INSEG_SUBSET = prove
(`!(l:A#A->bool) m. m inseg l ==> !x y. m(x,y) ==> l(x,y)`,
REPEAT GEN_TAC THEN REWRITE_TAC[inseg] THEN MESON_TAC[]);;
let INSEG_SUBSET_FL = prove
(`!(l:A#A->bool) m. m inseg l ==> !x. fl(m) x ==> fl(l) x`,
REWRITE_TAC[fl] THEN MESON_TAC[INSEG_SUBSET]);;
let INSEG_FL_SUBSET = prove
(`!l m:A#A->bool. l inseg m ==> fl l SUBSET fl m`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP INSEG_SUBSET_FL) THEN
SET_TAC[]);;
let INSEG_WOSET = prove
(`!(l:A#A->bool) m. m inseg l /\ woset l ==> woset m`,
REWRITE_TAC[inseg] THEN REPEAT STRIP_TAC THEN
REWRITE_TAC[WOSET] THEN CONJ_TAC THENL
[ASM_MESON_TAC[WOSET_ANTISYM];
GEN_TAC THEN FIRST_ASSUM(MP_TAC o SPEC_ALL o MATCH_MP WOSET_WELL) THEN
ASM_MESON_TAC[fl]]);;
let INSEG_ORDINAL = prove
(`!l m:A#A->bool. m inseg l /\ ordinal l ==> ordinal m`,
REPEAT GEN_TAC THEN REWRITE_TAC[ordinal] THEN STRIP_TAC THEN
CONJ_TAC THENL [ASM_MESON_TAC[INSEG_WOSET]; ALL_TAC] THEN
X_GEN_TAC `x:A` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:A`) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP INSEG_SUBSET_FL) THEN
ASM_SIMP_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
AP_TERM_TAC THEN ABS_TAC THEN AP_TERM_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[inseg]) THEN ASM_REWRITE_TAC[less]);;
(* ------------------------------------------------------------------------ *)
(* Properties of segments of the `linseg` form. *)
(* ------------------------------------------------------------------------ *)
let LINSEG_INSEG = prove
(`!(l:A#A->bool) a. woset l ==> (linseg l a) inseg l`,
REPEAT STRIP_TAC THEN REWRITE_TAC[inseg; linseg; fl] THEN PBETA_TAC THEN
ASM_MESON_TAC[WOSET_TRANS_LE]);;
let LINSEG_WOSET = prove
(`!(l:A#A->bool) a. woset l ==> woset(linseg l a)`,
MESON_TAC[INSEG_WOSET; LINSEG_INSEG]);;
let LINSEG_FL = prove
(`!(l:A#A->bool) a x. woset l ==> (fl (linseg l a) x <=> (less l)(x,a))`,
REWRITE_TAC[fl; linseg; less] THEN PBETA_TAC THEN
MESON_TAC[WOSET_REFL; WOSET_TRANS; WOSET_ANTISYM; fl]);;
(* ------------------------------------------------------------------------ *)
(* Key fact: a proper initial segment is of the special form. *)
(* ------------------------------------------------------------------------ *)
let INSEG_PROPER_SUBSET = prove
(`!(l:A#A->bool) m. m inseg l /\ ~(l = m) ==>
?x y. l(x,y) /\ ~m(x,y)`,
REWRITE_TAC[GSYM PAIRED_EXT] THEN MESON_TAC[INSEG_SUBSET]);;
let INSEG_PROPER_SUBSET_FL = prove
(`!(l:A#A->bool) m. m inseg l /\ ~(l = m) ==>
?a. fl(l) a /\ ~fl(m) a`,
MESON_TAC[INSEG_PROPER_SUBSET; fl; inseg]);;
let INSEG_LINSEG = prove
(`!(l:A#A->bool) m. woset l ==>
(m inseg l <=> (m = l) \/ (?a. fl(l) a /\ (m = linseg l a)))`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `m:A#A->bool = l` THEN
ASM_REWRITE_TAC[] THENL
[REWRITE_TAC[inseg; fl] THEN MESON_TAC[]; ALL_TAC] THEN
EQ_TAC THEN STRIP_TAC THENL [ALL_TAC; ASM_MESON_TAC[LINSEG_INSEG]] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP WOSET_WELL_CONTRAPOS) THEN
DISCH_THEN(MP_TAC o SPEC `\x:A. fl(l) x /\ ~fl(m) x`) THEN REWRITE_TAC[] THEN
REWRITE_TAC[linseg; GSYM PAIRED_EXT] THEN PBETA_TAC THEN
W(C SUBGOAL_THEN (fun t -> REWRITE_TAC[t]) o funpow 2 lhand o snd) THENL
[ASM_MESON_TAC[INSEG_PROPER_SUBSET_FL]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `a:A` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `a:A` THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[INSEG_SUBSET; INSEG_SUBSET_FL; fl;
WOSET_TOTAL_LE; less; inseg]);;
(* ------------------------------------------------------------------------ *)
(* A proper initial segment can be extended by its bounding element. *)
(* ------------------------------------------------------------------------ *)
let EXTEND_FL = prove
(`!(l:A#A->bool) x. woset l ==> (fl (\(x,y). l(x,y) /\ l(y,a)) x <=> l(x,a))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[fl] THEN PBETA_TAC THEN
ASM_MESON_TAC[WOSET_TRANS; WOSET_REFL; fl]);;
let EXTEND_INSEG = prove
(`!(l:A#A->bool) a. woset l /\ fl(l) a ==> (\(x,y). l(x,y) /\ l(y,a)) inseg l`,
REPEAT STRIP_TAC THEN REWRITE_TAC[inseg] THEN PBETA_TAC THEN
REPEAT GEN_TAC THEN IMP_RES_THEN (fun t ->REWRITE_TAC[t]) EXTEND_FL);;
let EXTEND_LINSEG = prove
(`!(l:A#A->bool) a. woset l /\ fl(l) a ==>
(\(x,y). linseg l a (x,y) \/ (y = a) /\ (fl(linseg l a) x \/ (x = a)))
inseg l`,
REPEAT GEN_TAC THEN DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN
MP_TAC (MATCH_MP EXTEND_INSEG th)) THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN
AP_TERM_TAC THEN ONCE_REWRITE_TAC[GSYM PAIRED_EXT] THEN PBETA_TAC THEN
REPEAT GEN_TAC THEN IMP_RES_THEN (fun th -> REWRITE_TAC[th]) LINSEG_FL THEN
REWRITE_TAC[linseg; less] THEN PBETA_TAC THEN ASM_MESON_TAC[WOSET_REFL]);;
(* ------------------------------------------------------------------------ *)
(* Key canonicality property of ordinals. *)
(* ------------------------------------------------------------------------ *)
let ORDINAL_CHAINED_LEMMA = prove
(`!(k:A#A->bool) l m. ordinal(l) /\ ordinal(m)
==> k inseg l /\ k inseg m
==> (k = l) \/ (k = m) \/ ?a. fl(l) a /\ fl(m) a /\
(k = linseg l a) /\
(k = linseg m a)`,
REPEAT GEN_TAC THEN REWRITE_TAC[ordinal] THEN STRIP_TAC THEN
EVERY_ASSUM(fun th -> TRY
(fun g -> REWRITE_TAC[MATCH_MP INSEG_LINSEG th] g)) THEN
ASM_CASES_TAC `k:A#A->bool = l` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `k:A#A->bool = m` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `a:A` STRIP_ASSUME_TAC)
(X_CHOOSE_THEN `b:A` STRIP_ASSUME_TAC)) THEN
EXISTS_TAC `a:A` THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `a:A = b` (fun th -> ASM_MESON_TAC[th]) THEN
FIRST_ASSUM(fun th -> SUBST1_TAC(MATCH_MP th (ASSUME `fl(l) (a:A)`))) THEN
FIRST_ASSUM(fun th -> SUBST1_TAC(MATCH_MP th (ASSUME `fl(m) (b:A)`))) THEN
AP_TERM_TAC THEN ABS_TAC THEN AP_TERM_TAC THEN
UNDISCH_TAC `k = linseg m (b:A)` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[linseg; GSYM PAIRED_EXT] THEN PBETA_TAC THEN
ASM_MESON_TAC[WOSET_REFL; less; fl]);;
let ORDINAL_CHAINED = prove
(`!(l:A#A->bool) m. ordinal(l) /\ ordinal(m) ==> m inseg l \/ l inseg m`,
REPEAT GEN_TAC THEN
DISCH_THEN(fun th -> STRIP_ASSUME_TAC(REWRITE_RULE[ordinal] th) THEN
ASSUME_TAC (MATCH_MP ORDINAL_CHAINED_LEMMA th)) THEN
MP_TAC(SPEC `\k:A#A->bool. k inseg l /\ k inseg m` UNION_INSEG) THEN
DISCH_THEN(fun th ->
MP_TAC(CONJ (SPEC `l:A#A->bool` th) (SPEC `m:A#A->bool` th))) THEN
REWRITE_TAC[TAUT `(a /\ b ==> a) /\ (a /\ b ==> b)`] THEN
DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN
FIRST_ASSUM(REPEAT_TCL DISJ_CASES_THEN MP_TAC o
C MATCH_MP th)) THENL
[ASM_MESON_TAC[]; ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `a:A` STRIP_ASSUME_TAC) THEN
MP_TAC(ASSUME `UNIONS (\k. k inseg l /\ k inseg m) = linseg l (a:A)`) THEN
CONV_TAC CONTRAPOS_CONV THEN DISCH_THEN(K ALL_TAC) THEN
REWRITE_TAC[FUN_EQ_THM] THEN DISCH_THEN(MP_TAC o SPEC `(a:A,a)`) THEN
REWRITE_TAC[linseg] THEN PBETA_TAC THEN REWRITE_TAC[less] THEN
REWRITE_TAC[UNIONS_PRED] THEN EXISTS_TAC
`\(x,y). linseg l a (x,y) \/ (y = a) /\ (fl(linseg l a) x \/ (x = a:A))` THEN
PBETA_TAC THEN REWRITE_TAC[] THEN CONJ_TAC THENL
[ALL_TAC;
UNDISCH_TAC `UNIONS (\k. k inseg l /\ k inseg m) = linseg l (a:A)` THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[]] THEN
MATCH_MP_TAC EXTEND_LINSEG THEN ASM_REWRITE_TAC[]);;
let ORDINAL_FL_UNIQUE = prove
(`!l m:A#A->bool.
ordinal l /\ ordinal m /\ fl l = fl m ==> l = m`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`l:A#A->bool`; `m:A#A->bool`]
ORDINAL_CHAINED) THEN
REWRITE_TAC[inseg; FUN_EQ_THM; FORALL_PAIR_THM] THEN
ASM_MESON_TAC[fl]);;
let ORDINAL_FL_SUBSET = prove
(`!l m:A#A->bool.
ordinal l /\ ordinal m /\ fl l SUBSET fl m ==> l inseg m`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`l:A#A->bool`; `m:A#A->bool`]
ORDINAL_CHAINED) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
MATCH_MP_TAC(MESON[INSEG_REFL] `x = y ==> x inseg y`) THEN
MATCH_MP_TAC ORDINAL_FL_UNIQUE THEN
FIRST_ASSUM(MP_TAC o MATCH_MP INSEG_SUBSET_FL) THEN ASM SET_TAC[]);;
let ORDINAL_FL_SUBSET_EQ = prove
(`!l m:A#A->bool.
ordinal l /\ ordinal m ==> (fl l SUBSET fl m <=> l inseg m)`,
MESON_TAC[ORDINAL_FL_SUBSET; INSEG_FL_SUBSET]);;
(* ------------------------------------------------------------------------ *)
(* Proof that any none-universe ordinal can be extended to its "successor". *)
(* ------------------------------------------------------------------------ *)
let FL_SUC = prove
(`!(l:A#A->bool) a.
fl(\(x,y). l(x,y) \/ (y = a) /\ (fl(l) x \/ (x = a))) x <=>
fl(l) x \/ (x = a)`,
REPEAT GEN_TAC THEN REWRITE_TAC[fl] THEN PBETA_TAC THEN EQ_TAC THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN TRY DISJ1_TAC THEN
ASM_MESON_TAC[]);;
let ORDINAL_SUC = prove
(`!l:A#A->bool. ordinal(l) /\ (?x. ~(fl(l) x)) ==>
ordinal(\(x,y). l(x,y) \/ (y = @y. ~fl(l) y) /\
(fl(l) x \/ (x = @y. ~fl(l) y)))`,
REPEAT GEN_TAC THEN REWRITE_TAC[ordinal] THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
ABBREV_TAC `a:A = @y. ~fl(l) y` THEN
SUBGOAL_THEN `~fl(l:A#A->bool) a` ASSUME_TAC THENL
[EXPAND_TAC "a" THEN CONV_TAC SELECT_CONV THEN
ASM_REWRITE_TAC[]; ALL_TAC] THEN
PBETA_TAC THEN CONJ_TAC THENL
[REWRITE_TAC[WOSET] THEN PBETA_TAC THEN CONJ_TAC THENL
[REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[WOSET_ANTISYM]; ALL_TAC; ALL_TAC] THEN
UNDISCH_TAC `~fl(l:A#A->bool) a` THEN CONV_TAC CONTRAPOS_CONV THEN
DISCH_TAC THEN FIRST_ASSUM(UNDISCH_TAC o check is_eq o concl) THEN
DISCH_THEN SUBST_ALL_TAC THEN REWRITE_TAC[fl] THENL
[EXISTS_TAC `y:A`; EXISTS_TAC `x:A`] THEN
ASM_REWRITE_TAC[];
X_GEN_TAC `P:A->bool` THEN REWRITE_TAC[FL_SUC] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_TAC `w:A`)) THEN
IMP_RES_THEN (MP_TAC o SPEC `\x:A. P x /\ fl(l) x`) WOSET_WELL THEN
BETA_TAC THEN REWRITE_TAC[TAUT `a /\ b ==> b`] THEN
ASM_CASES_TAC `?x:A. P x /\ fl(l) x` THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[];
FIRST_ASSUM(MP_TAC o SPEC `w:A` o
GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_MESON_TAC[]]];
X_GEN_TAC `z:A` THEN REWRITE_TAC[FL_SUC; less] THEN
PBETA_TAC THEN DISCH_THEN DISJ_CASES_TAC THENL
[UNDISCH_TAC `!x:A. fl l x ==> (x = (@y. ~less l (y,x)))` THEN
DISCH_THEN(IMP_RES_THEN MP_TAC) THEN
DISCH_THEN(fun th -> GEN_REWRITE_TAC LAND_CONV [th]) THEN
AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
X_GEN_TAC `y:A` THEN
BETA_TAC THEN REWRITE_TAC[less] THEN AP_TERM_TAC THEN
ASM_CASES_TAC `y:A = z` THEN ASM_REWRITE_TAC[] THEN
EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `fl(l:A#A->bool) z` THEN ASM_REWRITE_TAC[];
UNDISCH_TAC `z:A = a` THEN DISCH_THEN SUBST_ALL_TAC THEN
GEN_REWRITE_TAC LAND_CONV [SYM(ASSUME `(@y:A. ~fl(l) y) = a`)] THEN
AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
X_GEN_TAC `y:A` THEN
BETA_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[] THEN
ASM_CASES_TAC `y:A = a` THEN ASM_REWRITE_TAC[] THEN
EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[fl] THEN EXISTS_TAC `a:A` THEN ASM_REWRITE_TAC[]]]);;
(* ------------------------------------------------------------------------ *)
(* The union of a set of ordinals is an ordinal. *)
(* ------------------------------------------------------------------------ *)
let ORDINAL_UNION = prove
(`!P. (!l:A#A->bool. P l ==> ordinal(l)) ==> ordinal(UNIONS P)`,
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[ordinal; WOSET] THEN
REPEAT CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN REWRITE_TAC[UNIONS_PRED] THEN
BETA_TAC THEN DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `l:A#A->bool` (CONJUNCTS_THEN2 (ANTE_RES_THEN ASSUME_TAC)
ASSUME_TAC))
(X_CHOOSE_THEN `m:A#A->bool` (CONJUNCTS_THEN2 (ANTE_RES_THEN ASSUME_TAC)
ASSUME_TAC))) THEN
MP_TAC(SPECL [`l:A#A->bool`; `m:A#A->bool`] ORDINAL_CHAINED) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN DISJ_CASES_TAC THENL
[MP_TAC(SPEC `l:A#A->bool` WOSET_ANTISYM);
MP_TAC(SPEC `m:A#A->bool` WOSET_ANTISYM)] THEN
RULE_ASSUM_TAC(REWRITE_RULE[ordinal]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN IMP_RES_THEN MATCH_MP_TAC INSEG_SUBSET THEN
ASM_REWRITE_TAC[];
X_GEN_TAC `Q:A->bool` THEN REWRITE_TAC[UNION_FL] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_TAC `a:A`)) THEN
MP_TAC(ASSUME `!x:A. Q x ==> (?l. P l /\ fl l x)`) THEN
DISCH_THEN(IMP_RES_THEN
(X_CHOOSE_THEN `l:A#A->bool` STRIP_ASSUME_TAC)) THEN
IMP_RES_THEN ASSUME_TAC (ASSUME `!l:A#A->bool. P l ==> ordinal l`) THEN
ASSUME_TAC(CONJUNCT1
(REWRITE_RULE[ordinal] (ASSUME `ordinal(l:A#A->bool)`))) THEN
IMP_RES_THEN(MP_TAC o SPEC `\x:A. fl(l) x /\ Q x`) WOSET_WELL THEN
BETA_TAC THEN REWRITE_TAC[TAUT `a /\ b ==> a`] THEN
SUBGOAL_THEN `?x:A. fl(l) x /\ Q x` (fun th -> REWRITE_TAC[th]) THENL
[EXISTS_TAC `a:A` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `b:A` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `b:A` THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `x:A` THEN DISCH_TAC THEN
ANTE_RES_THEN MP_TAC (ASSUME `(Q:A->bool) x`) THEN
DISCH_THEN(X_CHOOSE_THEN `m:A#A->bool` STRIP_ASSUME_TAC) THEN
ANTE_RES_THEN ASSUME_TAC (ASSUME `(P:(A#A->bool)->bool) m`) THEN
MP_TAC(SPECL [`l:A#A->bool`; `m:A#A->bool`] ORDINAL_CHAINED) THEN
ASM_REWRITE_TAC[UNIONS_PRED] THEN BETA_TAC THEN
DISCH_THEN DISJ_CASES_TAC THENL
[EXISTS_TAC `l:A#A->bool` THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
IMP_RES_THEN MATCH_MP_TAC INSEG_SUBSET_FL THEN ASM_REWRITE_TAC[];
EXISTS_TAC `m:A#A->bool` THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o SPECL [`x:A`; `b:A`] o REWRITE_RULE[inseg]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(ASSUME_TAC o SYM) THEN
IMP_RES_THEN (MP_TAC o SPEC `b:A`) INSEG_SUBSET_FL THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(CONJUNCT1(REWRITE_RULE[ordinal]
(ASSUME `ordinal(m:A#A->bool)`))) THEN
DISCH_THEN(MP_TAC o SPECL [`b:A`; `x:A`] o MATCH_MP WOSET_TOTAL) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (DISJ_CASES_THEN MP_TAC) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
IMP_RES_THEN MATCH_MP_TAC INSEG_SUBSET THEN
FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[fl] THEN
EXISTS_TAC `b:A` THEN ASM_REWRITE_TAC[]];
X_GEN_TAC `x:A` THEN REWRITE_TAC[UNION_FL] THEN
DISCH_THEN(X_CHOOSE_THEN `l:A#A->bool` STRIP_ASSUME_TAC) THEN
MP_TAC(ASSUME `!l:A#A->bool. P l ==> ordinal l`) THEN
DISCH_THEN(IMP_RES_THEN MP_TAC) THEN REWRITE_TAC[ordinal] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `x:A`)) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN
REPEAT AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
X_GEN_TAC `y:A` THEN BETA_TAC THEN AP_TERM_TAC THEN
ASM_CASES_TAC `y:A = x` THEN ASM_REWRITE_TAC[less; UNIONS_PRED] THEN
BETA_TAC THEN EQ_TAC THEN DISCH_TAC THENL
[EXISTS_TAC `l:A#A->bool` THEN ASM_REWRITE_TAC[];
FIRST_ASSUM(X_CHOOSE_THEN `m:A#A->bool` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `ordinal(l:A#A->bool) /\ ordinal(m:A#A->bool)` MP_TAC THENL
[CONJ_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
DISCH_THEN(DISJ_CASES_TAC o MATCH_MP ORDINAL_CHAINED)] THENL
[IMP_RES_THEN MATCH_MP_TAC INSEG_SUBSET THEN ASM_REWRITE_TAC[];
RULE_ASSUM_TAC(REWRITE_RULE[inseg]) THEN ASM_REWRITE_TAC[]]]]);;
(* ------------------------------------------------------------------------ *)
(* Consequently, every type can be wellordered (and by an ordinal). *)
(* ------------------------------------------------------------------------ *)
let ORDINAL_UNION_LEMMA = prove
(`!(l:A#A->bool) x. ordinal l ==> fl(l) x ==> fl(UNIONS(ordinal)) x`,
REPEAT STRIP_TAC THEN REWRITE_TAC[UNION_FL] THEN
EXISTS_TAC `l:A#A->bool` THEN ASM_REWRITE_TAC[]);;
let ORDINAL_UP = prove
(`!l:A#A->bool. ordinal(l) ==> (!x. fl(l) x) \/
(?m x. ordinal(m) /\ fl(m) x /\ ~fl(l) x)`,
GEN_TAC THEN DISCH_TAC THEN
REWRITE_TAC[TAUT `a \/ b <=> ~a ==> b`] THEN
GEN_REWRITE_TAC LAND_CONV [NOT_FORALL_THM] THEN DISCH_TAC THEN
MP_TAC(SPEC `l:A#A->bool` ORDINAL_SUC) THEN ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN MAP_EVERY EXISTS_TAC
[`\(x,y). l(x,y) \/ (y = @y:A. ~fl l y) /\ (fl(l) x \/ (x = @y. ~fl l y))`;
`@y. ~fl(l:A#A->bool) y`] THEN
ASM_REWRITE_TAC[FL_SUC] THEN
CONV_TAC SELECT_CONV THEN ASM_REWRITE_TAC[]);;
let WO_ORDINAL = prove
(`?l:A#A->bool. ordinal(l) /\ !x. fl(l) x`,
EXISTS_TAC `UNIONS (ordinal:(A#A->bool)->bool)` THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[MATCH_MP_TAC ORDINAL_UNION THEN REWRITE_TAC[];
DISCH_THEN(DISJ_CASES_TAC o MATCH_MP ORDINAL_UP) THEN
ASM_REWRITE_TAC[] THEN POP_ASSUM(X_CHOOSE_THEN `m:A#A->bool`
(X_CHOOSE_THEN `x:A` STRIP_ASSUME_TAC)) THEN
IMP_RES_THEN (IMP_RES_THEN MP_TAC) ORDINAL_UNION_LEMMA THEN
ASM_REWRITE_TAC[]]);;
(* ------------------------------------------------------------------------ *)
(* At least -- every set can be wellordered. *)
(* ------------------------------------------------------------------------ *)
let FL_RESTRICT = prove
(`!l. woset l ==>
!P. fl(\(x:A,y). P x /\ P y /\ l(x,y)) x <=> P x /\ fl(l) x`,
REPEAT STRIP_TAC THEN REWRITE_TAC[fl] THEN PBETA_TAC THEN EQ_TAC THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
TRY(EXISTS_TAC `y:A` THEN ASM_REWRITE_TAC[] THEN NO_TAC) THEN
EXISTS_TAC `x:A` THEN ASM_REWRITE_TAC[] THEN
IMP_RES_THEN MATCH_MP_TAC WOSET_REFL THEN
REWRITE_TAC[fl] THEN EXISTS_TAC `y:A` THEN ASM_REWRITE_TAC[]);;
let WO = prove
(`!P. ?l:A#A->bool. woset l /\ (fl(l) = P)`,
GEN_TAC THEN X_CHOOSE_THEN `l:A#A->bool` STRIP_ASSUME_TAC
(REWRITE_RULE[ordinal] WO_ORDINAL) THEN
EXISTS_TAC `\(x:A,y). P x /\ P y /\ l(x,y)` THEN REWRITE_TAC[WOSET] THEN
PBETA_TAC THEN
GEN_REWRITE_TAC RAND_CONV [FUN_EQ_THM] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP FL_RESTRICT th]) THEN
PBETA_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP WOSET_ANTISYM) THEN
ASM_REWRITE_TAC[];
X_GEN_TAC `Q:A->bool` THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP WOSET_WELL) THEN
DISCH_THEN(MP_TAC o SPEC `Q:A->bool`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `y:A` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `y:A` THEN ASM_REWRITE_TAC[] THEN GEN_TAC THEN DISCH_TAC THEN
REPEAT CONJ_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
FIRST_ASSUM ACCEPT_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Moreover, the ordinals themselves are wellordered by "inseg". *)
(* ------------------------------------------------------------------------- *)
let WF_INSEG_WOSET = prove
(`WF(\(x:A#A->bool) y. woset x /\ woset y /\ x inseg y /\ ~(x = y))`,
REWRITE_TAC[WF] THEN X_GEN_TAC `P:(A#A->bool)->bool` THEN
DISCH_TAC THEN
ONCE_REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN
REWRITE_TAC[TAUT `~(p /\ q) <=> p ==> ~q`] THEN
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [NOT_FORALL_THM] THEN
REWRITE_TAC[NOT_IMP] THEN DISCH_TAC THEN
SUBGOAL_THEN `!x:A#A->bool. P x ==> woset x` ASSUME_TAC THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `!y. ?(a:A) z. P y ==> P z /\ fl y a /\ linseg y a = z`
MP_TAC THENL
[X_GEN_TAC `y:A#A->bool` THEN
ASM_CASES_TAC `(P:(A#A->bool)->bool) y` THEN ASM_REWRITE_TAC[] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `y:A#A->bool`)) THEN
ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[INSEG_LINSEG];
ALL_TAC] THEN
REWRITE_TAC[SKOLEM_THM; NOT_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:(A#A->bool)->A`; `l:(A#A->bool)->(A#A->bool)`] THEN
DISCH_TAC THEN FIRST_X_ASSUM(X_CHOOSE_TAC `z:A#A->bool`) THEN
SUBGOAL_THEN `woset(z:A#A->bool)` MP_TAC THENL
[ASM_MESON_TAC[]; REWRITE_TAC[woset]] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o SPEC `{(a:(A#A->bool)->A) x | P x /\ x inseg z}`) THEN
REWRITE_TAC[IN_ELIM_THM] THEN
REWRITE_TAC[TAUT `P /\ x = y <=> x = y /\ P`] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN
REWRITE_TAC[GSYM CONJ_ASSOC] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM; SWAP_FORALL_THM] THEN
REWRITE_TAC[UNWIND_THM2; FORALL_UNWIND_THM2; IMP_CONJ;
RIGHT_FORALL_IMP_THM] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM; SWAP_FORALL_THM] THEN
REWRITE_TAC[UNWIND_THM2; FORALL_UNWIND_THM2; IMP_CONJ;
RIGHT_FORALL_IMP_THM; NOT_IMP] THEN
CONJ_TAC THENL [ASM_MESON_TAC[INSEG_SUBSET_FL]; ALL_TAC] THEN
CONJ_TAC THENL [ASM_MESON_TAC[INSEG_REFL]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `w:A#A->bool` MP_TAC) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o SPEC `(l:(A#A->bool)->(A#A->bool)) w`) THEN
ASM_SIMP_TAC[NOT_IMP] THEN CONJ_TAC THENL
[ASM_MESON_TAC[LINSEG_INSEG; INSEG_TRANS]; DISCH_TAC] THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(SPEC `(l:(A#A->bool)->(A#A->bool)) w` th) THEN
MP_TAC(SPEC `w:A#A->bool` th)) THEN
ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THEN
UNDISCH_TAC `fl ((l:(A#A->bool)->(A#A->bool)) w) (a (l w))` THEN
FIRST_X_ASSUM(fun th ->
GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [SYM th]) THEN
ASM_SIMP_TAC[LINSEG_FL; less] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[inseg]) THEN ASM_MESON_TAC[]);;
let WOSET_INSEG_ORDINAL = prove
(`woset (\(x:A#A->bool,y). ordinal x /\ ordinal y /\ x inseg y)`,
REWRITE_TAC[WOSET_WF; fl] THEN
CONJ_TAC THENL [ALL_TAC; MESON_TAC[ORDINAL_CHAINED]] THEN
MATCH_MP_TAC(MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] WF_SUBSET)
WF_INSEG_WOSET) THEN
SIMP_TAC[ordinal]);;
let SUBWOSET_ISO_INSEG = prove
(`!l s. woset l /\ fl l = (:A)
==> ?f. (!x y. x IN s /\ y IN s ==> (l(f x,f y) <=> l(x,y))) /\
(!x y. y IN IMAGE f s /\ l(x,y) ==> x IN IMAGE f s)`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o last o CONJUNCTS o GEN_REWRITE_RULE I [woset]) THEN
DISCH_THEN(MP_TAC o GEN `s:A->bool` o SPEC `\x:A. x IN s`) THEN
ASM_REWRITE_TAC[UNIV; MEMBER_NOT_EMPTY] THEN
GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `m:(A->bool)->A` THEN DISCH_TAC THEN
SUBGOAL_THEN
`?f:A->A. !x. f(x) = m (UNIV DIFF IMAGE f {u | u IN s /\ less l (u,x)})`
MP_TAC THENL
[FIRST_ASSUM(MP_TAC o CONJUNCT1 o REWRITE_RULE[WOSET_WF]) THEN
DISCH_THEN(MATCH_MP_TAC o MATCH_MP WF_REC) THEN
REWRITE_TAC[less] THEN REPEAT STRIP_TAC THEN
AP_TERM_TAC THEN ASM SET_TAC[];
MATCH_MP_TAC MONO_EXISTS] THEN
X_GEN_TAC `f:A->A` THEN DISCH_THEN(ASSUME_TAC o GSYM) THEN
SUBGOAL_THEN `!x. x IN s ==> (l:A#A->bool)(f x,x)` ASSUME_TAC THENL
[FIRST_ASSUM(MP_TAC o CONJUNCT1 o REWRITE_RULE[WOSET_WF]) THEN
REWRITE_TAC[WF_IND] THEN DISCH_THEN MATCH_MP_TAC THEN
X_GEN_TAC `x:A` THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC
`(:A) DIFF IMAGE (f:A->A) {u | u IN s /\ less l (u,x)}`) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[less; woset] THEN SET_TAC[];
REWRITE_TAC[IN_DIFF; IN_IMAGE; IN_ELIM_THM; IN_UNIV] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MATCH_MP_TAC) THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[less; woset] THEN SET_TAC[]];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP MONO_FORALL o GEN `x:A` o SPEC
`(:A) DIFF IMAGE (f:A->A) {u | u IN s /\ less l (u,x)}`) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[less; woset] THEN SET_TAC[];
FIRST_X_ASSUM(K ALL_TAC o CONV_RULE (BINDER_CONV SYM_CONV))] THEN
REWRITE_TAC[IN_UNIV; IN_IMAGE; IN_DIFF; IN_ELIM_THM; FORALL_AND_THM] THEN
STRIP_TAC THEN
SUBGOAL_THEN
`!x z:A. x IN s /\ less l (z,f x) ==> ?u. u IN s /\ less l (u,x) /\ f u = z`
ASSUME_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[less; woset] THEN SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`!x z:A. x IN s /\ l(z,f x) ==> ?u. u IN s /\ l(u,x) /\ f u = z`
ASSUME_TAC THENL
[REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[less; woset] THEN SET_TAC[];
ALL_TAC] THEN
CONJ_TAC THENL
[SUBGOAL_THEN
`!x y:A. x IN s /\ y IN s /\ less l (x,y) ==> less l (f x,f y)`
MP_TAC THENL
[REPEAT STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`x:A`; `(f:A->A) y`])) THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[less; woset] THEN SET_TAC[];
MATCH_MP_TAC(MESON[]
`(!x y. P x y /\ P y x ==> Q x y)
==> (!x y. P x y) ==> (!x y. Q x y)`) THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[less; woset] THEN SET_TAC[]];
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[less; woset] THEN SET_TAC[]]);;
(* ======================================================================== *)
(* (3) CANTOR-ZERMELO WELL-ORDERING THEOREM ==> HAUSDORFF MAXIMAL PRINCIPLE *)
(* ======================================================================== *)
let HP = prove
(`!l:A#A->bool. poset l ==>
?P. chain(l) P /\ !Q. chain(l) Q /\ P SUBSET Q ==> (Q = P)`,
GEN_TAC THEN DISCH_TAC THEN
X_CHOOSE_THEN `w:A#A->bool` MP_TAC (SPEC `\x:A. T` WO) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
GEN_REWRITE_TAC LAND_CONV [FUN_EQ_THM] THEN BETA_TAC THEN
REWRITE_TAC[] THEN DISCH_TAC THEN
IMP_RES_THEN (MP_TAC o SPEC `\x:A. fl(l) x`) WOSET_WELL THEN
BETA_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `?x:A. fl(l) x` THEN ASM_REWRITE_TAC[] THENL
[DISCH_THEN(X_CHOOSE_THEN `b:A` STRIP_ASSUME_TAC);
FIRST_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
EXISTS_TAC `\x:A. F` THEN REWRITE_TAC[chain; SUBSET_PRED] THEN
GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [FUN_EQ_THM] THEN
CONV_TAC CONTRAPOS_CONV THEN REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `u:A` MP_TAC o
GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
REWRITE_TAC[] THEN DISCH_TAC THEN
DISCH_THEN(MP_TAC o SPECL [`u:A`; `u:A`]) THEN
IMP_RES_THEN(ASSUME_TAC o GSYM) POSET_FLEQ THEN ASM_REWRITE_TAC[]] THEN
SUBGOAL_THEN `?f. !x. f x = if fl(l) x /\
(!y. less w (y,x) ==> l (x,f y) \/ l (f y,x))
then (x:A) else b`
(CHOOSE_TAC o GSYM) THENL
[SUBGOAL_THEN `WF(\x:A y. (less w)(x,y))` MP_TAC THENL
[REWRITE_TAC[WF] THEN GEN_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC_ALL o MATCH_MP WOSET_WELL) THEN
ASM_REWRITE_TAC[less] THEN ASM_MESON_TAC[WOSET_ANTISYM];
DISCH_THEN(MATCH_MP_TAC o MATCH_MP WF_REC) THEN
REWRITE_TAC[] THEN REPEAT GEN_TAC THEN
REPEAT COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[]]; ALL_TAC] THEN
IMP_RES_THEN(IMP_RES_THEN ASSUME_TAC) POSET_REFL THEN
SUBGOAL_THEN `(f:A->A) b = b` ASSUME_TAC THENL
[FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `b:A`) THEN
REWRITE_TAC[COND_ID] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `!x:A. fl(l:A#A->bool) (f x)` ASSUME_TAC THENL
[GEN_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `x:A`) THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
FIRST_ASSUM(ANTE_RES_THEN (ASSUME_TAC o GEN_ALL) o SPEC_ALL) THEN
SUBGOAL_THEN `!x:A. (l:A#A->bool)(b,f x) \/ l(f x,b)` ASSUME_TAC THENL
[GEN_TAC THEN MP_TAC(SPEC `x:A` (ASSUME `!x:A. (w:A#A->bool)(b,f x)`)) THEN
FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `x:A`) THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `x:A = b` THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
SUBGOAL_THEN `(less w)(b:A,x)` MP_TAC THENL
[ASM_REWRITE_TAC[less] THEN CONV_TAC(RAND_CONV SYM_CONV) THEN
FIRST_ASSUM ACCEPT_TAC; ALL_TAC] THEN
DISCH_THEN(fun th -> FIRST_ASSUM(MP_TAC o C MATCH_MP th o CONJUNCT2)) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN DISJ_CASES_TAC THEN
ASM_REWRITE_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `!x y. l((f:A->A) x,f y) \/ l(f y,f x)` ASSUME_TAC THENL
[REPEAT GEN_TAC THEN
IMP_RES_THEN(MP_TAC o SPECL [`x:A`; `y:A`]) WOSET_TOTAL_LT THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC) THENL
[ASM_REWRITE_TAC[] THEN IMP_RES_THEN MATCH_MP_TAC POSET_REFL;
ONCE_REWRITE_TAC[DISJ_SYM] THEN
FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `y:A`);
FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `x:A`)] THEN
TRY COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(IMP_RES_THEN ACCEPT_TAC o CONJUNCT2); ALL_TAC] THEN
EXISTS_TAC `\y:A. ?x:A. y = f(x)` THEN
SUBGOAL_THEN `chain(l:A#A->bool)(\y. ?x:A. y = f x)` ASSUME_TAC THENL
[REWRITE_TAC[chain] THEN BETA_TAC THEN REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN(CHOOSE_THEN SUBST1_TAC)); ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `Q:A->bool` THEN STRIP_TAC THEN
GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `z:A` THEN EQ_TAC THENL
[DISCH_TAC THEN BETA_TAC THEN EXISTS_TAC `z:A` THEN
FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `z:A`) THEN
SUBGOAL_THEN `fl(l:A#A->bool) z /\
!y. (less w)(y,z) ==> l(z,f y) \/ l(f y,z)`
(fun th -> REWRITE_TAC[th]) THEN CONJ_TAC THENL
[UNDISCH_TAC `chain(l:A#A->bool) Q` THEN REWRITE_TAC[chain] THEN
DISCH_THEN(MP_TAC o SPECL [`z:A`; `z:A`]) THEN ASM_REWRITE_TAC[fl] THEN
DISCH_TAC THEN EXISTS_TAC `z:A` THEN ASM_REWRITE_TAC[];
X_GEN_TAC `y:A` THEN DISCH_TAC THEN
UNDISCH_TAC `chain(l:A#A->bool) Q` THEN REWRITE_TAC[chain] THEN
DISCH_THEN(MP_TAC o SPECL [`z:A`; `(f:A->A) y`]) THEN
DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET_PRED]) THEN
BETA_TAC THEN EXISTS_TAC `y:A` THEN REFL_TAC];
SPEC_TAC(`z:A`,`z:A`) THEN ASM_REWRITE_TAC[GSYM SUBSET_PRED]]);;
(* ======================================================================== *)
(* (4) HAUSDORFF MAXIMAL PRINCIPLE ==> ZORN'S LEMMA *)
(* ======================================================================== *)
let ZL = prove
(`!l:A#A->bool. poset l /\
(!P. chain(l) P ==> (?y. fl(l) y /\ !x. P x ==> l(x,y))) ==>
?y. fl(l) y /\ !x. l(y,x) ==> (y = x)`,
GEN_TAC THEN STRIP_TAC THEN
FIRST_ASSUM(X_CHOOSE_THEN `M:A->bool` STRIP_ASSUME_TAC o MATCH_MP HP) THEN
UNDISCH_TAC `!P. chain(l:A#A->bool) P
==> (?y. fl(l) y /\ !x. P x ==> l(x,y))` THEN
DISCH_THEN(MP_TAC o SPEC `M:A->bool`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `m:A` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `m:A` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `z:A` THEN
DISCH_TAC THEN GEN_REWRITE_TAC I [TAUT `a <=> ~ ~a`] THEN DISCH_TAC THEN
SUBGOAL_THEN `chain(l) (\x:A. M x \/ (x = z))` MP_TAC THENL
[REWRITE_TAC[chain] THEN BETA_TAC THEN REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN DISJ_CASES_TAC) THEN
ASM_REWRITE_TAC[] THENL
[UNDISCH_TAC `chain(l:A#A->bool) M` THEN REWRITE_TAC[chain] THEN
DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
DISJ1_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP POSET_TRANS) THEN
EXISTS_TAC `m:A` THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM MATCH_MP_TAC THEN FIRST_ASSUM ACCEPT_TAC;
DISJ2_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP POSET_TRANS) THEN
EXISTS_TAC `m:A` THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM MATCH_MP_TAC THEN FIRST_ASSUM ACCEPT_TAC;
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP POSET_REFL) THEN
REWRITE_TAC[fl] THEN EXISTS_TAC `m:A` THEN ASM_REWRITE_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN `M SUBSET (\x:A. M x \/ (x = z))` MP_TAC THENL
[REWRITE_TAC[SUBSET_PRED] THEN GEN_TAC THEN BETA_TAC THEN
DISCH_THEN(fun th -> REWRITE_TAC[th]); ALL_TAC] THEN
GEN_REWRITE_TAC I [TAUT `(a ==> b ==> c) <=> (b /\ a ==> c)`] THEN
DISCH_THEN(fun th -> FIRST_ASSUM(MP_TAC o C MATCH_MP th)) THEN
REWRITE_TAC[] THEN GEN_REWRITE_TAC RAND_CONV [FUN_EQ_THM] THEN
DISCH_THEN(MP_TAC o SPEC `z:A`) THEN BETA_TAC THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(fun th -> FIRST_ASSUM(ASSUME_TAC o C MATCH_MP th)) THEN
FIRST_ASSUM(MP_TAC o SPECL [`m:A`; `z:A`] o MATCH_MP POSET_ANTISYM) THEN
ASM_REWRITE_TAC[]);;
(* ======================================================================== *)
(* (5) ZORN'S LEMMA ==> KURATOWSKI'S LEMMA *)
(* ======================================================================== *)
let KL_POSET_LEMMA = prove
(`poset (\(c1,c2). C SUBSET c1 /\ c1 SUBSET c2 /\ chain(l:A#A->bool) c2)`,
REWRITE_TAC[poset] THEN PBETA_TAC THEN REPEAT CONJ_TAC THENL
[X_GEN_TAC `P:A->bool` THEN REWRITE_TAC[fl] THEN PBETA_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `Q:A->bool` STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[SUBSET_REFL] THENL
[MATCH_MP_TAC CHAIN_SUBSET; MATCH_MP_TAC SUBSET_TRANS];
GEN_TAC THEN X_GEN_TAC `Q:A->bool` THEN GEN_TAC THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS;
REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM] THEN
TRY(EXISTS_TAC `Q:A->bool`) THEN ASM_REWRITE_TAC[]);;
let KL = prove
(`!l:A#A->bool. poset l ==>
!C. chain(l) C ==>
?P. (chain(l) P /\ C SUBSET P) /\
(!R. chain(l) R /\ P SUBSET R ==> (R = P))`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPEC `\(c1,c2). C SUBSET c1 /\ c1 SUBSET c2 /\
chain(l:A#A->bool) c2` ZL) THEN PBETA_TAC THEN
REWRITE_TAC[KL_POSET_LEMMA; MATCH_MP POSET_FLEQ KL_POSET_LEMMA] THEN
PBETA_TAC THEN
W(C SUBGOAL_THEN (fun t ->REWRITE_TAC[t]) o
funpow 2 (fst o dest_imp) o snd) THENL
[X_GEN_TAC `P:(A->bool)->bool` THEN GEN_REWRITE_TAC LAND_CONV [chain] THEN
PBETA_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `?D:A->bool. P D` THENL
[EXISTS_TAC `UNIONS(P) :A->bool` THEN REWRITE_TAC[SUBSET_REFL] THEN
FIRST_ASSUM(X_CHOOSE_TAC `D:A->bool`) THEN
FIRST_ASSUM(MP_TAC o SPECL [`D:A->bool`; `D:A->bool`]) THEN
REWRITE_TAC[ASSUME `(P:(A->bool)->bool) D`; SUBSET_REFL] THEN
STRIP_TAC THEN
MATCH_MP_TAC(TAUT `a /\ b /\ (a /\ b ==> c) ==> (a /\ b) /\ c`) THEN
REPEAT CONJ_TAC THENL
[REWRITE_TAC[UNIONS_PRED; SUBSET_PRED] THEN REPEAT STRIP_TAC THEN
BETA_TAC THEN EXISTS_TAC `D:A->bool` THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET_PRED]) THEN
ASM_REWRITE_TAC[];
REWRITE_TAC[chain; UNIONS_PRED] THEN
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN
BETA_TAC THEN DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_TAC `A:A->bool`) (X_CHOOSE_TAC `B:A->bool`)) THEN
FIRST_ASSUM(UNDISCH_TAC o check is_forall o concl) THEN
DISCH_THEN(MP_TAC o SPECL [`A:A->bool`; `B:A->bool`]) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THENL
[UNDISCH_TAC `chain(l:A#A->bool) B`;
UNDISCH_TAC `chain(l:A#A->bool) A`] THEN
REWRITE_TAC[chain] THEN DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET_PRED]) THEN
ASM_REWRITE_TAC[];
STRIP_TAC THEN X_GEN_TAC `X:A->bool` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPECL [`X:A->bool`; `X:A->bool`]) THEN
REWRITE_TAC[] THEN DISCH_THEN(IMP_RES_THEN STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[UNIONS_PRED; SUBSET_PRED] THEN
REPEAT STRIP_TAC THEN BETA_TAC THEN EXISTS_TAC `X:A->bool` THEN
ASM_REWRITE_TAC[]];
EXISTS_TAC `C:A->bool` THEN
FIRST_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
ASM_REWRITE_TAC[SUBSET_REFL]];
DISCH_THEN(X_CHOOSE_THEN `D:A->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `D:A->bool` THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]]);;
(* ------------------------------------------------------------------------- *)
(* Special case of Zorn's Lemma for restriction of subset lattice. *)
(* ------------------------------------------------------------------------- *)
let POSET_RESTRICTED_SUBSET = prove
(`!P. poset(\(x,y). P(x) /\ P(y) /\ x SUBSET y)`,
GEN_TAC THEN REWRITE_TAC[poset; fl] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[SUBSET; EXTENSION] THEN MESON_TAC[]);;
let FL_RESTRICTED_SUBSET = prove
(`!P. fl(\(x,y). P(x) /\ P(y) /\ x SUBSET y) = P`,
REWRITE_TAC[fl; FORALL_PAIR_THM; FUN_EQ_THM] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN MESON_TAC[SUBSET_REFL]);;
let ZL_SUBSETS = prove
(`!P. (!c. (!x. x IN c ==> P x) /\
(!x y. x IN c /\ y IN c ==> x SUBSET y \/ y SUBSET x)
==> ?z. P z /\ (!x. x IN c ==> x SUBSET z))
==> ?a:A->bool. P a /\ (!x. P x /\ a SUBSET x ==> (a = x))`,
GEN_TAC THEN
MP_TAC(ISPEC `\(x,y). P(x:A->bool) /\ P(y) /\ x SUBSET y` ZL) THEN
REWRITE_TAC[POSET_RESTRICTED_SUBSET; FL_RESTRICTED_SUBSET] THEN
REWRITE_TAC[chain] THEN
CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[IN] THEN MATCH_MP_TAC MONO_IMP THEN CONJ_TAC THENL
[MATCH_MP_TAC MONO_FORALL; ALL_TAC] THEN
MESON_TAC[]);;
let ZL_SUBSETS_UNIONS = prove
(`!P. (!c. (!x. x IN c ==> P x) /\
(!x y. x IN c /\ y IN c ==> x SUBSET y \/ y SUBSET x)
==> P(UNIONS c))
==> ?a:A->bool. P a /\ (!x. P x /\ a SUBSET x ==> (a = x))`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC ZL_SUBSETS THEN
REPEAT STRIP_TAC THEN EXISTS_TAC `UNIONS(c:(A->bool)->bool)` THEN
ASM_MESON_TAC[SUBSET; IN_UNIONS]);;
let ZL_SUBSETS_UNIONS_NONEMPTY = prove
(`!P. (?x. P x) /\
(!c. (?x. x IN c) /\
(!x. x IN c ==> P x) /\
(!x y. x IN c /\ y IN c ==> x SUBSET y \/ y SUBSET x)
==> P(UNIONS c))
==> ?a:A->bool. P a /\ (!x. P x /\ a SUBSET x ==> (a = x))`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC ZL_SUBSETS THEN
REPEAT STRIP_TAC THEN ASM_CASES_TAC `?x:A->bool. x IN c` THENL
[EXISTS_TAC `UNIONS(c:(A->bool)->bool)` THEN
ASM_SIMP_TAC[] THEN MESON_TAC[SUBSET; IN_UNIONS];
ASM_MESON_TAC[]]);;
(* ------------------------------------------------------------------------- *)
(* A form of Tukey's lemma. *)
(* ------------------------------------------------------------------------- *)
let TUKEY = prove
(`!s:(A->bool)->bool.
~(s = {}) /\
(!t. (!c. FINITE c /\ c SUBSET t ==> c IN s) <=> t IN s)
==> ?u. u IN s /\ !v. v IN s /\ u SUBSET v ==> u = v`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC ZL_SUBSETS_UNIONS_NONEMPTY THEN
CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
X_GEN_TAC `c:(A->bool)->bool` THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [GSYM th]) THEN
SUBGOAL_THEN
`!d. FINITE d ==> d SUBSET UNIONS c ==> ?e:A->bool. e IN c /\ d SUBSET e`
MP_TAC THENL [ALL_TAC; MATCH_MP_TAC MONO_FORALL THEN ASM SET_TAC[]] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
REWRITE_TAC[INSERT_SUBSET] THEN ASM SET_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Also the order extension theorem, using Abian's proof. *)
(* ------------------------------------------------------------------------- *)
let OEP = prove
(`!p:A#A->bool. poset p ==> ?t. toset t /\ fl(t) = fl(p) /\ p SUBSET t`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPEC `fl(p:A#A->bool)` WO) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `w:A#A->bool` STRIP_ASSUME_TAC) THEN
ABBREV_TAC
`t = \(x:A,y:A). fl p x /\ fl p y /\
(x = y \/
?i. fl p i /\
(!j. w(j,i) /\ ~(j = i) ==> (p(j,x) <=> p(j,y))) /\
~p(i,x) /\ p(i,y))` THEN
EXISTS_TAC `t:A#A->bool` THEN
SUBGOAL_THEN
`!x:A y:A. fl p x /\ fl p y /\ ~(x = y)
==> ?i. fl p i /\
(!j:A. w(j,i) /\ ~(j = i) ==> (p(j,x) <=> p(j,y))) /\
~(p(i,x) <=> p(i,y))`
(LABEL_TAC "*") THENL
[REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [woset]) THEN ASM_SIMP_TAC[] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o SPEC `\i:A. fl p i /\ ~(p(i,x) <=> p(i,y))`) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [poset]) THEN ASM_MESON_TAC[];
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `i:A` THEN
ASM_MESON_TAC[fl]];
ALL_TAC] THEN
SUBGOAL_THEN `!x:A y:A. p(x,y) ==> t(x,y)` ASSUME_TAC THENL
[EXPAND_TAC "t" THEN REWRITE_TAC[] THEN
REPEAT GEN_TAC THEN STRIP_TAC THEN
REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[fl]; ALL_TAC]) THEN
ASM_CASES_TAC `x:A = y` THENL [ASM_MESON_TAC[fl]; ALL_TAC] THEN
REMOVE_THEN "*" (MP_TAC o SPECL [`x:A`; `y:A`]) THEN ASM_SIMP_TAC[] THEN
ANTS_TAC THENL [ASM_MESON_TAC[fl]; MATCH_MP_TAC MONO_EXISTS] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [poset]) THEN ASM_MESON_TAC[];
ALL_TAC] THEN
ASM_REWRITE_TAC[SUBSET; FORALL_PAIR_THM; IN] THEN
MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL
[MATCH_MP_TAC SUBSET_ANTISYM THEN
ASM_REWRITE_TAC[SUBSET; FORALL_PAIR_THM; IN] THEN
EXPAND_TAC "t" THEN REWRITE_TAC[fl] THEN ASM_MESON_TAC[];
DISCH_TAC THEN ASM_REWRITE_TAC[toset; poset]] THEN
EXPAND_TAC "t" THEN REWRITE_TAC[] THEN
FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [poset]) THEN
REPEAT CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`x:A`; `y:A`; `z:A`] THEN
ASM_CASES_TAC `x:A = z` THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[] THEN
ASM_CASES_TAC `y:A = z` THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[] THEN
ASM_CASES_TAC `y:A = x` THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[] THEN
ASM_CASES_TAC `fl p (x:A)` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `fl p (y:A)` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `fl p (z:A)` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `m:A` STRIP_ASSUME_TAC)
(X_CHOOSE_THEN `n:A` STRIP_ASSUME_TAC)) THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [woset]) THEN
REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o SPECL [`m:A`; `n:A`] o CONJUNCT1) THEN
ANTS_TAC THENL [ASM_MESON_TAC[fl]; ALL_TAC] THEN STRIP_TAC THENL
[EXISTS_TAC `m:A`; EXISTS_TAC `n:A`] THEN ASM_MESON_TAC[];
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN
ASM_CASES_TAC `y:A = x` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `fl p (x:A)` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `fl p (y:A)` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `m:A` STRIP_ASSUME_TAC)
(X_CHOOSE_THEN `n:A` STRIP_ASSUME_TAC)) THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [woset]) THEN
REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o SPECL [`m:A`; `n:A`] o CONJUNCT1) THEN
ASM_MESON_TAC[];
MAP_EVERY X_GEN_TAC [`x:A`; `y:A`] THEN
ASM_CASES_TAC `y:A = x` THEN ASM_REWRITE_TAC[IN] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
REMOVE_THEN "*" (MP_TAC o SPECL [`x:A`; `y:A`]) THEN
ASM_REWRITE_TAC[OR_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
MESON_TAC[]]);;
(* ------------------------------------------------------------------------- *)
(* Every toset contains a cofinal woset. *)
(* ------------------------------------------------------------------------- *)
let TOSET_COFINAL_WOSET = prove
(`!l. toset l
==> ?w. w SUBSET l /\ woset w /\
!x:A. x IN fl l ==> ?y. y IN fl w /\ l(x,y)`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `fl l:A->bool = {}` THENL
[ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
EXISTS_TAC `(\(x,y). F):A#A->bool` THEN
ASM_REWRITE_TAC[woset; FORALL_PAIR_THM; fl; SUBSET] THEN
REWRITE_TAC[IN] THEN MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `?r. ordinal r /\ fl r = (:A->bool)` STRIP_ASSUME_TAC THENL
[REWRITE_TAC[EXTENSION; IN_UNIV] THEN REWRITE_TAC[IN; WO_ORDINAL];
ALL_TAC] THEN
SUBGOAL_THEN
`?f. !w. f w = if ?x:A. x IN fl l /\
!v:A->bool. r(v,w) /\ ~(v = w) ==> ~l(x,f v)
then @x:A. x IN fl l /\
!v. r(v,w) /\ ~(v = w) ==> ~l(x,f v)
else @x:A. x IN fl l`
STRIP_ASSUME_TAC THENL
[FIRST_ASSUM(MP_TAC o MATCH_MP ORDINAL_IMP_WOSET) THEN
REWRITE_TAC[WOSET_WF] THEN
DISCH_THEN(MATCH_MP_TAC o MATCH_MP WF_REC o CONJUNCT1) THEN
REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(MESON[]
`(p <=> p') /\ x' = x
==> (if p then x else a) = (if p' then x' else a)`) THEN
CONJ_TAC THENL [ALL_TAC; AP_TERM_TAC THEN ABS_TAC] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`!w. (f:(A->bool)->A) w IN fl l`
ASSUME_TAC THENL
[GEN_TAC THEN ONCE_ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
CONV_TAC SELECT_CONV THEN ASM_REWRITE_TAC[MEMBER_NOT_EMPTY];
ALL_TAC] THEN
SUBGOAL_THEN
`?w:A->bool. (!x. x IN fl l ==> ?v. r(v,w) /\ ~(v = w) /\ l(x:A,f v)) /\
!z. (!x. x IN fl l ==> ?v. r(v,z) /\ ~(v = z) /\ l(x,f v))
==> r(w,z)`
STRIP_ASSUME_TAC THENL
[FIRST_ASSUM(MATCH_MP_TAC o last o CONJUNCTS o REWRITE_RULE[woset] o
MATCH_MP ORDINAL_IMP_WOSET) THEN
CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
ONCE_REWRITE_TAC[MESON[] `(?w. P w) <=> ~(!w. ~P w)`] THEN
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [NOT_FORALL_THM] THEN
REWRITE_TAC[NOT_IMP; NOT_EXISTS_THM; TAUT
`~(p /\ q /\ r) <=> (p /\ q ==> ~r)`] THEN
DISCH_TAC THEN
SUBGOAL_THEN `!v w:A->bool. f v:A = f w ==> v = w` MP_TAC THENL
[FIRST_ASSUM(MP_TAC o el 3 o CONJUNCTS o REWRITE_RULE[woset] o
MATCH_MP ORDINAL_IMP_WOSET) THEN
FIRST_X_ASSUM SUBST1_TAC THEN REWRITE_TAC[UNIV] THEN
MATCH_MP_TAC(MESON[]
`(!x y. P x y ==> P y x) /\ (!x y. R x y ==> P x y)
==> (!x y. R x y \/ R y x) ==> (!x y. P x y)`) THEN
CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
REPEAT GEN_TAC THEN DISCH_TAC THEN
GEN_REWRITE_TAC I [GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
FIRST_X_ASSUM(fun t -> GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [t]) THEN
ASM_SIMP_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP (MESON[]
`(y = @x. P x) ==> (?x. P x) ==> P y`)) THEN
ASM_SIMP_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[toset]) THEN
ASM_MESON_TAC[];
REWRITE_TAC[INJECTIVE_LEFT_INVERSE; NOT_EXISTS_THM] THEN
X_GEN_TAC `g:A->(A->bool)` THEN
DISCH_THEN(MP_TAC o SPEC `\x:A. ~(g x x)`) THEN
REWRITE_TAC[FUN_EQ_THM] THEN MESON_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN
`!x y. r(x,y) /\ r(y,w) /\ ~(y = w) ==> l((f:(A->bool)->A) x,f y)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
MATCH_MP_TAC(MESON[]
`(l(a,b) \/ l(b,a)) /\ (~(b = a) ==> ~l(b,a)) ==> l(a,b)`) THEN
CONJ_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[toset]) THEN ASM SET_TAC[];
DISCH_THEN(ASSUME_TAC o MATCH_MP (MESON[]
`~(f x = f y) ==> ~(x = y)`))] THEN
FIRST_X_ASSUM(fun th ->
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV) [th]) THEN
COND_CASES_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o CONJUNCT2 o SELECT_RULE) THEN
ASM_REWRITE_TAC[];
FIRST_ASSUM(MP_TAC o REWRITE_RULE[woset] o
MATCH_MP ORDINAL_IMP_WOSET) THEN
ASM_MESON_TAC[]];
ALL_TAC] THEN
EXISTS_TAC `\(x,y). x IN IMAGE (f:(A->bool)->A) {v | r(v,w) /\ ~(v = w)} /\
y IN IMAGE (f:(A->bool)->A) {v | r(v,w) /\ ~(v = w)} /\
l(x,y)` THEN
SUBGOAL_THEN
`fl(\(x,y). x IN IMAGE (f:(A->bool)->A) {v | r(v,w) /\ ~(v = w)} /\
y IN IMAGE (f:(A->bool)->A) {v | r(v,w) /\ ~(v = w)} /\
l(x,y)) =
IMAGE f {v | r(v,w) /\ ~(v = w)}`
ASSUME_TAC THENL
[GEN_REWRITE_TAC I [EXTENSION] THEN
REWRITE_TAC[IN_ELIM_THM; IN_IMAGE] THEN
REWRITE_TAC[IN; fl] THEN
RULE_ASSUM_TAC(REWRITE_RULE[toset]) THEN ASM_MESON_TAC[];
ALL_TAC] THEN
REPEAT CONJ_TAC THENL
[SIMP_TAC[SUBSET; FORALL_PAIR_THM; IN];
ASM_REWRITE_TAC[REWRITE_RULE[SET_RULE `fl l x <=> x IN fl l`] woset];
ASM_REWRITE_TAC[] THEN REWRITE_TAC[EXISTS_IN_IMAGE] THEN
ASM SET_TAC[]] THEN
FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [toset]) THEN
FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [poset]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[SET_RULE `fl l x <=> x IN fl l`]) THEN
REWRITE_TAC[CONJ_ASSOC] THEN
CONJ_TAC THENL [ASM SET_TAC[]; SIMP_TAC[]] THEN
X_GEN_TAC `P:A->bool` THEN REWRITE_TAC[] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP ORDINAL_IMP_WOSET) THEN
REWRITE_TAC[woset] THEN
DISCH_THEN(MP_TAC o SPEC
`\x:A->bool. (P:A->bool) (f x)` o last o CONJUNCTS) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `z:A->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(f:(A->bool)->A) z` THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`(!z. P z ==> z IN IMAGE f s)
==> (!x. x IN s /\ P(f x) ==> Q(f x))
==> !y. P y ==> Q y`)) THEN
X_GEN_TAC `y:A->bool` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN
CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_ASSUM ACCEPT_TAC;
ASM SET_TAC[]]);;
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