/
ind_types.ml
1657 lines (1503 loc) · 73.5 KB
/
ind_types.ml
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(* ========================================================================= *)
(* Inductive (or free recursive) types. *)
(* *)
(* John Harrison, University of Cambridge Computer Laboratory *)
(* *)
(* (c) Copyright, University of Cambridge 1998 *)
(* (c) Copyright, John Harrison 1998-2007 *)
(* ========================================================================= *)
needs "grobner.ml";;
(* ------------------------------------------------------------------------- *)
(* Abstract left inverses for binary injections (we could construct them...) *)
(* ------------------------------------------------------------------------- *)
let INJ_INVERSE2 = prove
(`!P:A->B->C.
(!x1 y1 x2 y2. (P x1 y1 = P x2 y2) <=> (x1 = x2) /\ (y1 = y2))
==> ?X Y. !x y. (X(P x y) = x) /\ (Y(P x y) = y)`,
GEN_TAC THEN DISCH_TAC THEN
EXISTS_TAC `\z:C. @x:A. ?y:B. P x y = z` THEN
EXISTS_TAC `\z:C. @y:B. ?x:A. P x y = z` THEN
REPEAT GEN_TAC THEN ASM_REWRITE_TAC[BETA_THM] THEN
CONJ_TAC THEN MATCH_MP_TAC SELECT_UNIQUE THEN GEN_TAC THEN BETA_TAC THEN
EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
W(EXISTS_TAC o rand o snd o dest_exists o snd) THEN REFL_TAC);;
(* ------------------------------------------------------------------------- *)
(* Define an injective pairing function on ":num". *)
(* ------------------------------------------------------------------------- *)
let NUMPAIR = new_definition
`NUMPAIR x y = (2 EXP x) * (2 * y + 1)`;;
let NUMPAIR_INJ_LEMMA = prove
(`!x1 y1 x2 y2. (NUMPAIR x1 y1 = NUMPAIR x2 y2) ==> (x1 = x2)`,
REWRITE_TAC[NUMPAIR] THEN REPEAT(INDUCT_TAC THEN GEN_TAC) THEN
ASM_REWRITE_TAC[EXP; GSYM MULT_ASSOC; ARITH; EQ_MULT_LCANCEL;
NOT_SUC; GSYM NOT_SUC; SUC_INJ] THEN
DISCH_THEN(MP_TAC o AP_TERM `EVEN`) THEN
REWRITE_TAC[EVEN_MULT; EVEN_ADD; ARITH]);;
let NUMPAIR_INJ = prove
(`!x1 y1 x2 y2. (NUMPAIR x1 y1 = NUMPAIR x2 y2) <=> (x1 = x2) /\ (y1 = y2)`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(SUBST_ALL_TAC o MATCH_MP NUMPAIR_INJ_LEMMA) THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[NUMPAIR] THEN
REWRITE_TAC[EQ_MULT_LCANCEL; EQ_ADD_RCANCEL; EXP_EQ_0; ARITH]);;
let NUMPAIR_DEST = new_specification
["NUMFST"; "NUMSND"]
(MATCH_MP INJ_INVERSE2 NUMPAIR_INJ);;
(* ------------------------------------------------------------------------- *)
(* Also, an injective map bool->num->num (even easier!) *)
(* ------------------------------------------------------------------------- *)
let NUMSUM = new_definition
`NUMSUM b x = if b then SUC(2 * x) else 2 * x`;;
let NUMSUM_INJ = prove
(`!b1 x1 b2 x2. (NUMSUM b1 x1 = NUMSUM b2 x2) <=> (b1 = b2) /\ (x1 = x2)`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM(MP_TAC o REWRITE_RULE[NUMSUM]) THEN
DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(AP_TERM `EVEN` th)) THEN
REPEAT COND_CASES_TAC THEN REWRITE_TAC[EVEN; EVEN_DOUBLE] THEN
REWRITE_TAC[SUC_INJ; EQ_MULT_LCANCEL; ARITH]);;
let NUMSUM_DEST = new_specification
["NUMLEFT"; "NUMRIGHT"]
(MATCH_MP INJ_INVERSE2 NUMSUM_INJ);;
(* ------------------------------------------------------------------------- *)
(* Injection num->Z, where Z == num->A->bool. *)
(* ------------------------------------------------------------------------- *)
let INJN = new_definition
`INJN (m:num) = \(n:num) (a:A). n = m`;;
let INJN_INJ = prove
(`!n1 n2. (INJN n1 :num->A->bool = INJN n2) <=> (n1 = n2)`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM(MP_TAC o C AP_THM `n1:num` o REWRITE_RULE[INJN]) THEN
DISCH_THEN(MP_TAC o C AP_THM `a:A`) THEN REWRITE_TAC[BETA_THM]);;
(* ------------------------------------------------------------------------- *)
(* Injection A->Z, where Z == num->A->bool. *)
(* ------------------------------------------------------------------------- *)
let INJA = new_definition
`INJA (a:A) = \(n:num) b. b = a`;;
let INJA_INJ = prove
(`!a1 a2. (INJA a1 = INJA a2) <=> (a1:A = a2)`,
REPEAT GEN_TAC THEN REWRITE_TAC[INJA; FUN_EQ_THM] THEN EQ_TAC THENL
[DISCH_THEN(MP_TAC o SPEC `a1:A`) THEN REWRITE_TAC[];
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[]]);;
(* ------------------------------------------------------------------------- *)
(* Injection (num->Z)->Z, where Z == num->A->bool. *)
(* ------------------------------------------------------------------------- *)
let INJF = new_definition
`INJF (f:num->(num->A->bool)) = \n. f (NUMFST n) (NUMSND n)`;;
let INJF_INJ = prove
(`!f1 f2. (INJF f1 :num->A->bool = INJF f2) <=> (f1 = f2)`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[FUN_EQ_THM] THEN
MAP_EVERY X_GEN_TAC [`n:num`; `m:num`; `a:A`] THEN
POP_ASSUM(MP_TAC o REWRITE_RULE[INJF]) THEN
DISCH_THEN(MP_TAC o C AP_THM `a:A` o C AP_THM `NUMPAIR n m`) THEN
REWRITE_TAC[NUMPAIR_DEST]);;
(* ------------------------------------------------------------------------- *)
(* Injection Z->Z->Z, where Z == num->A->bool. *)
(* ------------------------------------------------------------------------- *)
let INJP = new_definition
`INJP f1 f2:num->A->bool =
\n a. if NUMLEFT n then f1 (NUMRIGHT n) a else f2 (NUMRIGHT n) a`;;
let INJP_INJ = prove
(`!(f1:num->A->bool) f1' f2 f2'.
(INJP f1 f2 = INJP f1' f2') <=> (f1 = f1') /\ (f2 = f2')`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[AND_FORALL_THM] THEN
X_GEN_TAC `n:num` THEN POP_ASSUM(MP_TAC o REWRITE_RULE[INJP]) THEN
DISCH_THEN(MP_TAC o GEN `b:bool` o C AP_THM `NUMSUM b n`) THEN
DISCH_THEN(fun th -> MP_TAC(SPEC `T` th) THEN MP_TAC(SPEC `F` th)) THEN
ASM_SIMP_TAC[NUMSUM_DEST; ETA_AX]);;
(* ------------------------------------------------------------------------- *)
(* Now, set up "constructor" and "bottom" element. *)
(* ------------------------------------------------------------------------- *)
let ZCONSTR = new_definition
`ZCONSTR c i r :num->A->bool
= INJP (INJN (SUC c)) (INJP (INJA i) (INJF r))`;;
let ZBOT = new_definition
`ZBOT = INJP (INJN 0) (@z:num->A->bool. T)`;;
let ZCONSTR_ZBOT = prove
(`!c i r. ~(ZCONSTR c i r :num->A->bool = ZBOT)`,
REWRITE_TAC[ZCONSTR; ZBOT; INJP_INJ; INJN_INJ; NOT_SUC]);;
(* ------------------------------------------------------------------------- *)
(* Carve out an inductively defined set. *)
(* ------------------------------------------------------------------------- *)
let ZRECSPACE_RULES,ZRECSPACE_INDUCT,ZRECSPACE_CASES =
new_inductive_definition
`ZRECSPACE (ZBOT:num->A->bool) /\
(!c i r. (!n. ZRECSPACE (r n)) ==> ZRECSPACE (ZCONSTR c i r))`;;
let recspace_tydef =
new_basic_type_definition "recspace" ("_mk_rec","_dest_rec")
(CONJUNCT1 ZRECSPACE_RULES);;
(* ------------------------------------------------------------------------- *)
(* Define lifted constructors. *)
(* ------------------------------------------------------------------------- *)
let BOTTOM = new_definition
`BOTTOM = _mk_rec (ZBOT:num->A->bool)`;;
let CONSTR = new_definition
`CONSTR c i r :(A)recspace
= _mk_rec (ZCONSTR c i (\n. _dest_rec(r n)))`;;
(* ------------------------------------------------------------------------- *)
(* Some lemmas. *)
(* ------------------------------------------------------------------------- *)
let MK_REC_INJ = prove
(`!x y. (_mk_rec x :(A)recspace = _mk_rec y)
==> (ZRECSPACE x /\ ZRECSPACE y ==> (x = y))`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
REWRITE_TAC[snd recspace_tydef] THEN
DISCH_THEN(fun th -> ONCE_REWRITE_TAC[GSYM th]) THEN
ASM_REWRITE_TAC[]);;
let DEST_REC_INJ = prove
(`!x y. (_dest_rec x = _dest_rec y) <=> (x:(A)recspace = y)`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM(MP_TAC o AP_TERM
`_mk_rec:(num->A->bool)->(A)recspace`) THEN
REWRITE_TAC[fst recspace_tydef]);;
(* ------------------------------------------------------------------------- *)
(* Show that the set is freely inductively generated. *)
(* ------------------------------------------------------------------------- *)
let CONSTR_BOT = prove
(`!c i r. ~(CONSTR c i r :(A)recspace = BOTTOM)`,
REPEAT GEN_TAC THEN REWRITE_TAC[CONSTR; BOTTOM] THEN
DISCH_THEN(MP_TAC o MATCH_MP MK_REC_INJ) THEN
REWRITE_TAC[ZCONSTR_ZBOT; ZRECSPACE_RULES] THEN
MATCH_MP_TAC(CONJUNCT2 ZRECSPACE_RULES) THEN
REWRITE_TAC[fst recspace_tydef; snd recspace_tydef]);;
let CONSTR_INJ = prove
(`!c1 i1 r1 c2 i2 r2. (CONSTR c1 i1 r1 :(A)recspace = CONSTR c2 i2 r2) <=>
(c1 = c2) /\ (i1 = i2) /\ (r1 = r2)`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM(MP_TAC o REWRITE_RULE[CONSTR]) THEN
DISCH_THEN(MP_TAC o MATCH_MP MK_REC_INJ) THEN
W(C SUBGOAL_THEN ASSUME_TAC o funpow 2 lhand o snd) THENL
[CONJ_TAC THEN MATCH_MP_TAC(CONJUNCT2 ZRECSPACE_RULES) THEN
REWRITE_TAC[fst recspace_tydef; snd recspace_tydef];
ASM_REWRITE_TAC[] THEN REWRITE_TAC[ZCONSTR] THEN
REWRITE_TAC[INJP_INJ; INJN_INJ; INJF_INJ; INJA_INJ] THEN
ONCE_REWRITE_TAC[FUN_EQ_THM] THEN BETA_TAC THEN
REWRITE_TAC[SUC_INJ; DEST_REC_INJ]]);;
let CONSTR_IND = prove
(`!P. P(BOTTOM) /\
(!c i r. (!n. P(r n)) ==> P(CONSTR c i r))
==> !x:(A)recspace. P(x)`,
REPEAT STRIP_TAC THEN
MP_TAC(SPEC `\z:num->A->bool. ZRECSPACE(z) /\ P(_mk_rec z)`
ZRECSPACE_INDUCT) THEN
BETA_TAC THEN ASM_REWRITE_TAC[ZRECSPACE_RULES; GSYM BOTTOM] THEN
W(C SUBGOAL_THEN ASSUME_TAC o funpow 2 lhand o snd) THENL
[REPEAT GEN_TAC THEN REWRITE_TAC[FORALL_AND_THM] THEN
REPEAT STRIP_TAC THENL
[MATCH_MP_TAC(CONJUNCT2 ZRECSPACE_RULES) THEN ASM_REWRITE_TAC[];
FIRST_ASSUM(ANTE_RES_THEN MP_TAC) THEN
REWRITE_TAC[CONSTR] THEN
RULE_ASSUM_TAC(REWRITE_RULE[snd recspace_tydef]) THEN
ASM_SIMP_TAC[ETA_AX]];
ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC `_dest_rec (x:(A)recspace)`) THEN
REWRITE_TAC[fst recspace_tydef] THEN
REWRITE_TAC[ITAUT `(a ==> a /\ b) <=> (a ==> b)`] THEN
DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[fst recspace_tydef; snd recspace_tydef]]);;
(* ------------------------------------------------------------------------- *)
(* Now prove the recursion theorem (this subcase is all we need). *)
(* ------------------------------------------------------------------------- *)
let CONSTR_REC = prove
(`!Fn:num->A->(num->(A)recspace)->(num->B)->B.
?f. (!c i r. f (CONSTR c i r) = Fn c i r (\n. f (r n)))`,
REPEAT STRIP_TAC THEN (MP_TAC o prove_inductive_relations_exist)
`(Z:(A)recspace->B->bool) BOTTOM b /\
(!c i r y. (!n. Z (r n) (y n)) ==> Z (CONSTR c i r) (Fn c i r y))` THEN
DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (ASSUME_TAC o GSYM)) THEN
SUBGOAL_THEN `!x. ?!y. (Z:(A)recspace->B->bool) x y` MP_TAC THENL
[W(MP_TAC o PART_MATCH rand CONSTR_IND o snd) THEN
DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THEN REPEAT GEN_TAC THENL
[FIRST_ASSUM(fun t -> GEN_REWRITE_TAC BINDER_CONV [GSYM t]) THEN
REWRITE_TAC[GSYM CONSTR_BOT; EXISTS_UNIQUE_REFL];
DISCH_THEN(MP_TAC o REWRITE_RULE[EXISTS_UNIQUE_THM; FORALL_AND_THM]) THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
DISCH_THEN(MP_TAC o REWRITE_RULE[SKOLEM_THM]) THEN
DISCH_THEN(X_CHOOSE_THEN `y:num->B` ASSUME_TAC) THEN
REWRITE_TAC[EXISTS_UNIQUE_THM] THEN
FIRST_ASSUM(fun th -> CHANGED_TAC(ONCE_REWRITE_TAC[GSYM th])) THEN
CONJ_TAC THENL
[EXISTS_TAC `(Fn:num->A->(num->(A)recspace)->(num->B)->B) c i r y` THEN
REWRITE_TAC[CONSTR_BOT; CONSTR_INJ; GSYM CONJ_ASSOC] THEN
REWRITE_TAC[UNWIND_THM1; RIGHT_EXISTS_AND_THM] THEN
EXISTS_TAC `y:num->B` THEN ASM_REWRITE_TAC[];
REWRITE_TAC[CONSTR_BOT; CONSTR_INJ; GSYM CONJ_ASSOC] THEN
REWRITE_TAC[UNWIND_THM1; RIGHT_EXISTS_AND_THM] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
REPEAT AP_TERM_TAC THEN ONCE_REWRITE_TAC[FUN_EQ_THM] THEN
X_GEN_TAC `w:num` THEN FIRST_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `w:num` THEN ASM_REWRITE_TAC[]]];
REWRITE_TAC[UNIQUE_SKOLEM_ALT] THEN
DISCH_THEN(X_CHOOSE_THEN `fn:(A)recspace->B` (ASSUME_TAC o GSYM)) THEN
EXISTS_TAC `fn:(A)recspace->B` THEN ASM_REWRITE_TAC[] THEN
REPEAT GEN_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN GEN_TAC THEN
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [GSYM th]) THEN
REWRITE_TAC[BETA_THM]]);;
(* ------------------------------------------------------------------------- *)
(* The following is useful for coding up functions casewise. *)
(* ------------------------------------------------------------------------- *)
let FCONS = new_recursive_definition num_RECURSION
`(!a f. FCONS (a:A) f 0 = a) /\
(!a f n. FCONS (a:A) f (SUC n) = f n)`;;
let FCONS_UNDO = prove
(`!f:num->A. f = FCONS (f 0) (f o SUC)`,
GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN
INDUCT_TAC THEN REWRITE_TAC[FCONS; o_THM]);;
let FNIL = new_definition
`FNIL (n:num) = @x:A. T`;;
(* ------------------------------------------------------------------------- *)
(* The initial mutual type definition function, with a type-restricted *)
(* recursion theorem. *)
(* ------------------------------------------------------------------------- *)
let define_type_raw =
(* ----------------------------------------------------------------------- *)
(* Handy utility to produce "SUC o SUC o SUC ..." form of numeral. *)
(* ----------------------------------------------------------------------- *)
let sucivate =
let zero = `0` and suc = `SUC` in
fun n -> funpow n (curry mk_comb suc) zero in
(* ----------------------------------------------------------------------- *)
(* Eliminate local "definitions" in hyps. *)
(* ----------------------------------------------------------------------- *)
let SCRUB_EQUATION eq (th,insts) = (*HA*)
let eq' = itlist subst (map (fun t -> [t]) insts) eq in
let l,r = dest_eq eq' in
(MP (INST [r,l] (DISCH eq' th)) (REFL r),(r,l)::insts) in
(* ----------------------------------------------------------------------- *)
(* Proves existence of model (inductively); use pseudo-constructors. *)
(* *)
(* Returns suitable definitions of constructors in terms of CONSTR, and *)
(* the rule and induction theorems from the inductive relation package. *)
(* ----------------------------------------------------------------------- *)
let justify_inductive_type_model =
let t_tm = `T` and n_tm = `n:num` and beps_tm = `@x:bool. T` in
let rec munion s1 s2 =
if s1 = [] then s2 else
let h1 = hd s1
and s1' = tl s1 in
try let _,s2' = remove (fun h2 -> h2 = h1) s2 in h1::(munion s1' s2')
with Failure _ -> h1::(munion s1' s2) in
fun def ->
let newtys,rights = unzip def in
let tyargls = itlist ((@) o map snd) rights [] in
let alltys = itlist (munion o C subtract newtys) tyargls [] in
let epstms = map (fun ty -> mk_select(mk_var("v",ty),t_tm)) alltys in
let pty =
try end_itlist (fun ty1 ty2 -> mk_type("prod",[ty1;ty2])) alltys
with Failure _ -> bool_ty in
let recty = mk_type("recspace",[pty]) in
let constr = mk_const("CONSTR",[pty,aty]) in
let fcons = mk_const("FCONS",[recty,aty]) in
let bot = mk_const("BOTTOM",[pty,aty]) in
let bottail = mk_abs(n_tm,bot) in
let mk_constructor n (cname,cargs) =
let ttys = map (fun ty -> if mem ty newtys then recty else ty) cargs in
let args = make_args "a" [] ttys in
let rargs,iargs = partition (fun t -> type_of t = recty) args in
let rec mk_injector epstms alltys iargs =
if alltys = [] then [] else
let ty = hd alltys in
try let a,iargs' = remove (fun t -> type_of t = ty) iargs in
a::(mk_injector (tl epstms) (tl alltys) iargs')
with Failure _ ->
(hd epstms)::(mk_injector (tl epstms) (tl alltys) iargs) in
let iarg =
try end_itlist (curry mk_pair) (mk_injector epstms alltys iargs)
with Failure _ -> beps_tm in
let rarg = itlist (mk_binop fcons) rargs bottail in
let conty = itlist mk_fun_ty (map type_of args) recty in
let condef = list_mk_comb(constr,[sucivate n; iarg; rarg]) in
mk_eq(mk_var(cname,conty),list_mk_abs(args,condef)) in
let rec mk_constructors n rights =
if rights = [] then [] else
(mk_constructor n (hd rights))::(mk_constructors (n + 1) (tl rights)) in
let condefs = mk_constructors 0 (itlist (@) rights []) in
let conths = map ASSUME condefs in
let predty = mk_fun_ty recty bool_ty in
let edefs = itlist (fun (x,l) acc -> map (fun t -> x,t) l @ acc) def [] in
let idefs = map2 (fun (r,(_,atys)) def -> (r,atys),def) edefs condefs in
let mk_rule ((r,a),condef) =
let left,right = dest_eq condef in
let args,bod = strip_abs right in
let lapp = list_mk_comb(left,args) in
let conds = itlist2
(fun arg argty sofar ->
if mem argty newtys then
mk_comb(mk_var(dest_vartype argty,predty),arg)::sofar
else sofar) args a [] in
let conc = mk_comb(mk_var(dest_vartype r,predty),lapp) in
let rule = if conds = [] then conc
else mk_imp(list_mk_conj conds,conc) in
list_mk_forall(args,rule) in
let rules = list_mk_conj (map mk_rule idefs) in
let th0 = derive_nonschematic_inductive_relations rules in
let th1 = prove_monotonicity_hyps th0 in
let th2a,th2bc = CONJ_PAIR th1 in
let th2b = CONJUNCT1 th2bc in
conths,th2a,th2b in
(* ----------------------------------------------------------------------- *)
(* Shows that the predicates defined by the rules are all nonempty. *)
(* (This could be done much more efficiently/cleverly, but it's OK.) *)
(* ----------------------------------------------------------------------- *)
let prove_model_inhabitation rth =
let srules = map SPEC_ALL (CONJUNCTS rth) in
let imps,bases = partition (is_imp o concl) srules in
let concs = map concl bases @ map (rand o concl) imps in
let preds = setify (map (repeat rator) concs) in
let rec exhaust_inhabitations ths sofar =
let dunnit = setify(map (fst o strip_comb o concl) sofar) in
let useful = filter
(fun th -> not (mem (fst(strip_comb(rand(concl th)))) dunnit)) ths in
if useful = [] then sofar else
let follow_horn thm =
let preds = map (fst o strip_comb) (conjuncts(lhand(concl thm))) in
let asms = map
(fun p -> find (fun th -> fst(strip_comb(concl th)) = p) sofar)
preds in
MATCH_MP thm (end_itlist CONJ asms) in
let newth = tryfind follow_horn useful in
exhaust_inhabitations ths (newth::sofar) in
let ithms = exhaust_inhabitations imps bases in
let exths = map
(fun p -> find (fun th -> fst(strip_comb(concl th)) = p) ithms) preds in
exths in
(* ----------------------------------------------------------------------- *)
(* Makes a type definition for one of the defined subsets. *)
(* ----------------------------------------------------------------------- *)
let define_inductive_type cdefs exth =
let extm = concl exth in
let epred = fst(strip_comb extm) in
let ename = fst(dest_var epred) in
let th1 = ASSUME (find (fun eq -> lhand eq = epred) (hyp exth)) in
let th2 = TRANS th1 (SUBS_CONV cdefs (rand(concl th1))) in
let th3 = EQ_MP (AP_THM th2 (rand extm)) exth in
let th4,_ = itlist SCRUB_EQUATION (hyp th3) (th3,[]) in
let mkname = "_mk_"^ename and destname = "_dest_"^ename in
let bij1,bij2 = new_basic_type_definition ename (mkname,destname) th4 in
let bij2a = AP_THM th2 (rand(rand(concl bij2))) in
let bij2b = TRANS bij2a bij2 in
bij1,bij2b in
(* ----------------------------------------------------------------------- *)
(* Defines a type constructor corresponding to current pseudo-constructor. *)
(* ----------------------------------------------------------------------- *)
let define_inductive_type_constructor defs consindex th =
let avs,bod = strip_forall(concl th) in
let asms,conc =
if is_imp bod then conjuncts(lhand bod),rand bod else [],bod in
let asmlist = map dest_comb asms in
let cpred,cterm = dest_comb conc in
let oldcon,oldargs = strip_comb cterm in
let modify_arg v =
try let dest = snd(assoc (rev_assoc v asmlist) consindex) in
let ty' = hd(snd(dest_type(type_of dest))) in
let v' = mk_var(fst(dest_var v),ty') in
mk_comb(dest,v'),v'
with Failure _ -> v,v in
let newrights,newargs = unzip(map modify_arg oldargs) in
let retmk = fst(assoc cpred consindex) in
let defbod = mk_comb(retmk,list_mk_comb(oldcon,newrights)) in
let defrt = list_mk_abs(newargs,defbod) in
let expth = find (fun th -> lhand(concl th) = oldcon) defs in
let rexpth = SUBS_CONV [expth] defrt in
let deflf = mk_var(fst(dest_var oldcon),type_of defrt) in
let defth = new_definition(mk_eq(deflf,rand(concl rexpth))) in
TRANS defth (SYM rexpth) in
(* ----------------------------------------------------------------------- *)
(* Instantiate the induction theorem on the representatives to transfer *)
(* it to the new type(s). Uses "\x. rep-pred(x) /\ P(mk x)" for "P". *)
(* ----------------------------------------------------------------------- *)
let instantiate_induction_theorem consindex ith =
let avs,bod = strip_forall(concl ith) in
let corlist = map((repeat rator F_F repeat rator) o dest_imp o body o rand)
(conjuncts(rand bod)) in
let consindex' = map (fun v -> let w = rev_assoc v corlist in
w,assoc w consindex) avs in
let recty = (hd o snd o dest_type o type_of o fst o snd o hd) consindex in
let newtys = map (hd o snd o dest_type o type_of o snd o snd) consindex' in
let ptypes = map (C mk_fun_ty bool_ty) newtys in
let preds = make_args "P" [] ptypes in
let args = make_args "x" [] (map (K recty) preds) in
let lambs = map2 (fun (r,(m,d)) (p,a) ->
mk_abs(a,mk_conj(mk_comb(r,a),mk_comb(p,mk_comb(m,a)))))
consindex' (zip preds args) in
SPECL lambs ith in
(* ----------------------------------------------------------------------- *)
(* Reduce a single clause of the postulated induction theorem (old_ver) ba *)
(* to the kind wanted for the new type (new_ver); |- new_ver ==> old_ver *)
(* ----------------------------------------------------------------------- *)
let pullback_induction_clause tybijpairs conthms =
let PRERULE = GEN_REWRITE_RULE (funpow 3 RAND_CONV) (map SYM conthms) in
let IPRULE = SYM o GEN_REWRITE_RULE I (map snd tybijpairs) in
fun rthm tm ->
let avs,bimp = strip_forall tm in
if is_imp bimp then
let ant,con = dest_imp bimp in
let ths = map (CONV_RULE BETA_CONV) (CONJUNCTS (ASSUME ant)) in
let tths,pths = unzip (map CONJ_PAIR ths) in
let tth = MATCH_MP (SPEC_ALL rthm) (end_itlist CONJ tths) in
let mths = map IPRULE (tth::tths) in
let conth1 = BETA_CONV con in
let contm1 = rand(concl conth1) in
let conth2 = TRANS conth1
(AP_TERM (rator contm1) (SUBS_CONV (tl mths) (rand contm1))) in
let conth3 = PRERULE conth2 in
let lctms = map concl pths in
let asmin = mk_imp(list_mk_conj lctms,rand(rand(concl conth3))) in
let argsin = map rand (conjuncts(lhand asmin)) in
let argsgen =
map (fun tm -> mk_var(fst(dest_var(rand tm)),type_of tm)) argsin in
let asmgen = subst (zip argsgen argsin) asmin in
let asmquant =
list_mk_forall(snd(strip_comb(rand(rand asmgen))),asmgen) in
let th1 = INST (zip argsin argsgen) (SPEC_ALL (ASSUME asmquant)) in
let th2 = MP th1 (end_itlist CONJ pths) in
let th3 = EQ_MP (SYM conth3) (CONJ tth th2) in
DISCH asmquant (GENL avs (DISCH ant th3))
else
let con = bimp in
let conth2 = BETA_CONV con in
let tth = PART_MATCH I rthm (lhand(rand(concl conth2))) in
let conth3 = PRERULE conth2 in
let asmgen = rand(rand(concl conth3)) in
let asmquant = list_mk_forall(snd(strip_comb(rand asmgen)),asmgen) in
let th2 = SPEC_ALL (ASSUME asmquant) in
let th3 = EQ_MP (SYM conth3) (CONJ tth th2) in
DISCH asmquant (GENL avs th3) in
(* ----------------------------------------------------------------------- *)
(* Finish off a consequence of the induction theorem. *)
(* ----------------------------------------------------------------------- *)
let finish_induction_conclusion consindex tybijpairs =
let tybij1,tybij2 = unzip tybijpairs in
let PRERULE =
GEN_REWRITE_RULE (LAND_CONV o LAND_CONV o RAND_CONV) tybij1 o
GEN_REWRITE_RULE LAND_CONV tybij2
and FINRULE = GEN_REWRITE_RULE RAND_CONV tybij1 in
fun th ->
let av,bimp = dest_forall(concl th) in
let pv = lhand(body(rator(rand bimp))) in
let p,v = dest_comb pv in
let mk,dest = assoc p consindex in
let ty = hd(snd(dest_type(type_of dest))) in
let v' = mk_var(fst(dest_var v),ty) in
let dv = mk_comb(dest,v') in
let th1 = PRERULE (SPEC dv th) in
let th2 = MP th1 (REFL (rand(lhand(concl th1)))) in
let th3 = CONV_RULE BETA_CONV th2 in
GEN v' (FINRULE (CONJUNCT2 th3)) in
(* ----------------------------------------------------------------------- *)
(* Derive the induction theorem. *)
(* ----------------------------------------------------------------------- *)
let derive_induction_theorem consindex tybijpairs conthms iith rth =
let bths = map2
(pullback_induction_clause tybijpairs conthms)
(CONJUNCTS rth) (conjuncts(lhand(concl iith))) in
let asm = list_mk_conj(map (lhand o concl) bths) in
let ths = map2 MP bths (CONJUNCTS (ASSUME asm)) in
let th1 = MP iith (end_itlist CONJ ths) in
let th2 = end_itlist CONJ (map
(finish_induction_conclusion consindex tybijpairs) (CONJUNCTS th1)) in
let th3 = DISCH asm th2 in
let preds = map (rator o body o rand) (conjuncts(rand(concl th3))) in
let th4 = GENL preds th3 in
let pasms = filter (C mem (map fst consindex) o lhand) (hyp th4) in
let th5 = itlist DISCH pasms th4 in
let th6,_ = itlist SCRUB_EQUATION (hyp th5) (th5,[]) in
let th7 = UNDISCH_ALL th6 in
fst (itlist SCRUB_EQUATION (hyp th7) (th7,[])) in
(* ----------------------------------------------------------------------- *)
(* Create the recursive functions and eliminate pseudo-constructors. *)
(* (These are kept just long enough to derive the key property.) *)
(* ----------------------------------------------------------------------- *)
let create_recursive_functions tybijpairs consindex conthms rth =
let domtys = map (hd o snd o dest_type o type_of o snd o snd) consindex in
let recty = (hd o snd o dest_type o type_of o fst o snd o hd) consindex in
let ranty = mk_vartype "Z" in
let fn = mk_var("fn",mk_fun_ty recty ranty)
and fns = make_args "fn" [] (map (C mk_fun_ty ranty) domtys) in
let args = make_args "a" [] domtys in
let rights = map2 (fun (_,(_,d)) a -> mk_abs(a,mk_comb(fn,mk_comb(d,a))))
consindex args in
let eqs = map2 (curry mk_eq) fns rights in
let fdefs = map ASSUME eqs in
let fxths1 = map (fun th1 -> tryfind (fun th2 -> MK_COMB(th2,th1)) fdefs)
conthms in
let fxths2 = map (fun th -> TRANS th (BETA_CONV (rand(concl th)))) fxths1 in
let mk_tybijcons (th1,th2) =
let th3 = INST [rand(lhand(concl th1)),rand(lhand(concl th2))] th2 in
let th4 = AP_TERM (rator(lhand(rand(concl th2)))) th1 in
EQ_MP (SYM th3) th4 in
let SCONV = GEN_REWRITE_CONV I (map mk_tybijcons tybijpairs)
and ERULE = GEN_REWRITE_RULE I (map snd tybijpairs) in
let simplify_fxthm rthm fxth =
let pat = funpow 4 rand (concl fxth) in
if is_imp(repeat (snd o dest_forall) (concl rthm)) then
let th1 = PART_MATCH (rand o rand) rthm pat in
let tms1 = conjuncts(lhand(concl th1)) in
let ths2 = map (fun t -> EQ_MP (SYM(SCONV t)) TRUTH) tms1 in
ERULE (MP th1 (end_itlist CONJ ths2))
else
ERULE (PART_MATCH rand rthm pat) in
let fxths3 = map2 simplify_fxthm (CONJUNCTS rth) fxths2 in
let fxths4 = map2 (fun th1 -> TRANS th1 o AP_TERM fn) fxths2 fxths3 in
let cleanup_fxthm cth fxth =
let tms = snd(strip_comb(rand(rand(concl fxth)))) in
let kth = RIGHT_BETAS tms (ASSUME (hd(hyp cth))) in
TRANS fxth (AP_TERM fn kth) in
let fxth5 = end_itlist CONJ (map2 cleanup_fxthm conthms fxths4) in
let pasms = filter (C mem (map fst consindex) o lhand) (hyp fxth5) in
let fxth6 = itlist DISCH pasms fxth5 in
let fxth7,_ =
itlist SCRUB_EQUATION (itlist (union o hyp) conthms []) (fxth6,[]) in
let fxth8 = UNDISCH_ALL fxth7 in
fst (itlist SCRUB_EQUATION (subtract (hyp fxth8) eqs) (fxth8,[])) in
(* ----------------------------------------------------------------------- *)
(* Create a function for recursion clause. *)
(* ----------------------------------------------------------------------- *)
let create_recursion_iso_constructor =
let s = `s:num->Z` in
let zty = `:Z` in
let numty = `:num` in
let rec extract_arg tup v =
if v = tup then REFL tup else
let t1,t2 = dest_pair tup in
let PAIR_th = ISPECL [t1;t2] (if free_in v t1 then FST else SND) in
let tup' = rand(concl PAIR_th) in
if tup' = v then PAIR_th else
let th = extract_arg (rand(concl PAIR_th)) v in
SUBS[SYM PAIR_th] th in
fun consindex ->
let recty = hd(snd(dest_type(type_of(fst(hd consindex))))) in
let domty = hd(snd(dest_type recty)) in
let i = mk_var("i",domty)
and r = mk_var("r",mk_fun_ty numty recty) in
let mks = map (fst o snd) consindex in
let mkindex = map (fun t -> hd(tl(snd(dest_type(type_of t)))),t) mks in
fun cth ->
let artms = snd(strip_comb(rand(rand(concl cth)))) in
let artys = mapfilter (type_of o rand) artms in
let args,bod = strip_abs(rand(hd(hyp cth))) in
let ccitm,rtm = dest_comb bod in
let cctm,itm = dest_comb ccitm in
let rargs,iargs = partition (C free_in rtm) args in
let xths = map (extract_arg itm) iargs in
let cargs' = map (subst [i,itm] o lhand o concl) xths in
let indices = map sucivate (0--(length rargs - 1)) in
let rindexed = map (curry mk_comb r) indices in
let rargs' = map2 (fun a rx -> mk_comb(assoc a mkindex,rx))
artys rindexed in
let sargs' = map (curry mk_comb s) indices in
let allargs = cargs'@ rargs' @ sargs' in
let funty = itlist (mk_fun_ty o type_of) allargs zty in
let funname = fst(dest_const(repeat rator (lhand(concl cth))))^"'" in
let funarg = mk_var(funname,funty) in
list_mk_abs([i;r;s],list_mk_comb(funarg,allargs)) in
(* ----------------------------------------------------------------------- *)
(* Derive the recursion theorem. *)
(* ----------------------------------------------------------------------- *)
let derive_recursion_theorem =
let CCONV = funpow 3 RATOR_CONV (REPEATC (GEN_REWRITE_CONV I [FCONS])) in
fun tybijpairs consindex conthms rath ->
let isocons = map (create_recursion_iso_constructor consindex) conthms in
let ty = type_of(hd isocons) in
let fcons = mk_const("FCONS",[ty,aty])
and fnil = mk_const("FNIL",[ty,aty]) in
let bigfun = itlist (mk_binop fcons) isocons fnil in
let eth = ISPEC bigfun CONSTR_REC in
let fn = rator(rand(hd(conjuncts(concl rath)))) in
let betm = let v,bod = dest_abs(rand(concl eth)) in vsubst[fn,v] bod in
let LCONV = REWR_CONV (ASSUME betm) in
let fnths =
map (fun t -> RIGHT_BETAS [bndvar(rand t)] (ASSUME t)) (hyp rath) in
let SIMPER = PURE_REWRITE_RULE
(map SYM fnths @ map fst tybijpairs @ [FST; SND; FCONS; BETA_THM]) in
let hackdown_rath th =
let ltm,rtm = dest_eq(concl th) in
let wargs = snd(strip_comb(rand ltm)) in
let th1 = TRANS th (LCONV rtm) in
let th2 = TRANS th1 (CCONV (rand(concl th1))) in
let th3 = TRANS th2 (funpow 2 RATOR_CONV BETA_CONV (rand(concl th2))) in
let th4 = TRANS th3 (RATOR_CONV BETA_CONV (rand(concl th3))) in
let th5 = TRANS th4 (BETA_CONV (rand(concl th4))) in
GENL wargs (SIMPER th5) in
let rthm = end_itlist CONJ (map hackdown_rath (CONJUNCTS rath)) in
let seqs =
let unseqs = filter is_eq (hyp rthm) in
let tys = map (hd o snd o dest_type o type_of o snd o snd) consindex in
map (fun ty -> find
(fun t -> hd(snd(dest_type(type_of(lhand t)))) = ty) unseqs) tys in
let rethm = itlist EXISTS_EQUATION seqs rthm in
let fethm = CHOOSE(fn,eth) rethm in
let pcons = map (repeat rator o rand o repeat (snd o dest_forall))
(conjuncts(concl rthm)) in
GENL pcons fethm in
(* ----------------------------------------------------------------------- *)
(* Basic function: returns induction and recursion separately. No parser. *)
(* ----------------------------------------------------------------------- *)
fun def ->
let defs,rth,ith = justify_inductive_type_model def in
let neths = prove_model_inhabitation rth in
let tybijpairs = map (define_inductive_type defs) neths in
let preds = map (repeat rator o concl) neths in
let mkdests = map
(fun (th,_) -> let tm = lhand(concl th) in rator tm,rator(rand tm))
tybijpairs in
let consindex = zip preds mkdests in
let condefs = map (define_inductive_type_constructor defs consindex)
(CONJUNCTS rth) in
let conthms = map
(fun th -> let args = fst(strip_abs(rand(concl th))) in
RIGHT_BETAS args th) condefs in
let iith = instantiate_induction_theorem consindex ith in
let fth = derive_induction_theorem consindex tybijpairs conthms iith rth in
let rath = create_recursive_functions tybijpairs consindex conthms rth in
let kth = derive_recursion_theorem tybijpairs consindex conthms rath in
fth,kth;;
(* ------------------------------------------------------------------------- *)
(* Parser to present a nice interface a la Melham. *)
(* ------------------------------------------------------------------------- *)
let parse_inductive_type_specification =
let parse_type_loc src =
let pty,rst = parse_pretype src in
type_of_pretype pty,rst in
let parse_type_conapp src =
let cn,sps =
match src with (Ident cn)::sps -> cn,sps
| _ -> fail() in
let tys,rst = many parse_type_loc sps in
(cn,tys),rst in
let parse_type_clause src =
let tn,sps =
match src with (Ident tn)::sps -> tn,sps
| _ -> fail() in
let tys,rst = (a (Ident "=") ++ listof parse_type_conapp (a (Resword "|"))
"type definition clauses"
>> snd) sps in
(mk_vartype tn,tys),rst in
let parse_type_definition =
listof parse_type_clause (a (Resword ";")) "type definition" in
fun s ->
let spec,rst = (parse_type_definition o lex o explode) s in
if rst = [] then spec
else failwith "parse_inductive_type_specification: junk after def";;
(* ------------------------------------------------------------------------- *)
(* Use this temporary version to define the sum type. *)
(* ------------------------------------------------------------------------- *)
export_theory "sum-def";;
let (sum_INDUCT,sum_RECURSION) =
let (induct,recursion) =
define_type_raw
(parse_inductive_type_specification "sum = INL A | INR B") in
let th1 = prove
(`!(p : A + B -> bool).
(!a. p (INL a)) /\ (!b. p (INR b)) ==> (!x. p x)`,
MATCH_ACCEPT_TAC induct)
and th2 = prove
(`!f g. ?(fn : A + B -> C).
(!a. fn (INL a) = f a) /\ (!b. fn (INR b) = g b)`,
MATCH_ACCEPT_TAC recursion) in
(th1,th2);;
export_thm sum_INDUCT;;
export_thm sum_RECURSION;;
let OUTL = new_recursive_definition sum_RECURSION
`!a. OUTL ((INL a) : A + B) = a`;;
export_thm OUTL;;
let OUTR = new_recursive_definition sum_RECURSION
`!b. OUTR ((INR b) : A + B) = b`;;
export_thm OUTR;;
let (ISL_INL,ISL_INR) =
let def = new_recursive_definition sum_RECURSION
`(!a. ISL ((INL a) : A + B) = T) /\
(!b. ISL ((INR b) : A + B) = F)` in
CONJ_PAIR (REWRITE_RULE [] def);;
export_thm ISL_INL;;
export_thm ISL_INR;;
let ISL_def = CONJ ISL_INL ISL_INR;;
let (ISR_INL,ISR_INR) =
let def = new_recursive_definition sum_RECURSION
`(!a. ISR ((INL a) : A + B) = F) /\
(!b. ISR ((INR b) : A + B) = T)` in
CONJ_PAIR (REWRITE_RULE [] def);;
export_thm ISR_INL;;
export_thm ISR_INR;;
let ISR_def = CONJ ISR_INL ISR_INR;;
let (case_sum_left,case_sum_right) =
let def = new_recursive_definition sum_RECURSION
`(!(f : A -> C) (g : B -> C) a. case_sum f g ((INL a) : A + B) = f a) /\
(!(f : A -> C) (g : B -> C) b. case_sum f g ((INR b) : A + B) = g b)` in
CONJ_PAIR def;;
export_thm case_sum_left;;
export_thm case_sum_right;;
let case_sum_def = CONJ case_sum_left case_sum_right;;
(* ------------------------------------------------------------------------- *)
(* Generalize the recursion theorem to multiple domain types. *)
(* (We needed to use a single type to justify it via a proforma theorem.) *)
(* *)
(* NB! Before this is called nontrivially (i.e. more than one new type) *)
(* the type constructor ":sum", used internally, must have been defined. *)
(* ------------------------------------------------------------------------- *)
let define_type_raw =
let generalize_recursion_theorem =
let ELIM_OUTCOMBS = GEN_REWRITE_RULE TOP_DEPTH_CONV [OUTL; OUTR] in
let rec mk_sum tys =
let k = length tys in
if k = 1 then hd tys else
let tys1,tys2 = chop_list (k / 2) tys in
mk_type("sum",[mk_sum tys1; mk_sum tys2]) in
let mk_inls =
let rec mk_inls ty =
if is_vartype ty then [mk_var("x",ty)] else
let _,[ty1;ty2] = dest_type ty in
let inls1 = mk_inls ty1
and inls2 = mk_inls ty2 in
let inl = mk_const("INL",[ty1,aty; ty2,bty])
and inr = mk_const("INR",[ty1,aty; ty2,bty]) in
map (curry mk_comb inl) inls1 @ map (curry mk_comb inr) inls2 in
fun ty -> let bods = mk_inls ty in
map (fun t -> mk_abs(find_term is_var t,t)) bods in
let mk_outls =
let rec mk_inls sof ty =
if is_vartype ty then [sof] else
let _,[ty1;ty2] = dest_type ty in
let outl = mk_const("OUTL",[ty1,aty; ty2,bty])
and outr = mk_const("OUTR",[ty1,aty; ty2,bty]) in
mk_inls (mk_comb(outl,sof)) ty1 @ mk_inls (mk_comb(outr,sof)) ty2 in
fun ty -> let x = mk_var("x",ty) in
map (curry mk_abs x) (mk_inls x ty) in
let mk_newfun fn outl =
let s,ty = dest_var fn in
let dty = hd(snd(dest_type ty)) in
let x = mk_var("x",dty) in
let y,bod = dest_abs outl in
let r = mk_abs(x,vsubst[mk_comb(fn,x),y] bod) in
let l = mk_var(s,type_of r) in
let th1 = ASSUME (mk_eq(l,r)) in
RIGHT_BETAS [x] th1 in
fun th ->
let avs,ebod = strip_forall(concl th) in
let evs,bod = strip_exists ebod in
let n = length evs in
if n = 1 then th else
let tys = map (fun i -> mk_vartype ("Z"^(string_of_int i)))
(0--(n - 1)) in
let sty = mk_sum tys in
let inls = mk_inls sty
and outls = mk_outls sty in
let zty = type_of(rand(snd(strip_forall(hd(conjuncts bod))))) in
let ith = INST_TYPE [sty,zty] th in
let avs,ebod = strip_forall(concl ith) in
let evs,bod = strip_exists ebod in
let fns' = map2 mk_newfun evs outls in
let fnalist = zip evs (map (rator o lhs o concl) fns')
and inlalist = zip evs inls
and outlalist = zip evs outls in
let hack_clause tm =
let avs,bod = strip_forall tm in
let l,r = dest_eq bod in
let fn,args = strip_comb r in
let pargs = map
(fun a -> let g = genvar(type_of a) in
if is_var a then g,g else
let outl = assoc (rator a) outlalist in
mk_comb(outl,g),g) args in
let args',args'' = unzip pargs in
let inl = assoc (rator l) inlalist in
let rty = hd(snd(dest_type(type_of inl))) in
let nty = itlist (mk_fun_ty o type_of) args' rty in
let fn' = mk_var(fst(dest_var fn),nty) in
let r' = list_mk_abs(args'',mk_comb(inl,list_mk_comb(fn',args'))) in
r',fn in
let defs = map hack_clause (conjuncts bod) in
let jth = BETA_RULE (SPECL (map fst defs) ith) in
let bth = ASSUME (snd(strip_exists(concl jth))) in
let finish_clause th =
let avs,bod = strip_forall (concl th) in
let outl = assoc (rator (lhand bod)) outlalist in
GENL avs (BETA_RULE (AP_TERM outl (SPECL avs th))) in
let cth = end_itlist CONJ (map finish_clause (CONJUNCTS bth)) in
let dth = ELIM_OUTCOMBS cth in
let eth = GEN_REWRITE_RULE ONCE_DEPTH_CONV (map SYM fns') dth in
let fth = itlist SIMPLE_EXISTS (map snd fnalist) eth in
let dtms = map (hd o hyp) fns' in
let gth = itlist (fun e th -> let l,r = dest_eq e in
MP (INST [r,l] (DISCH e th)) (REFL r)) dtms fth in
let hth = PROVE_HYP jth (itlist SIMPLE_CHOOSE evs gth) in
let xvs = map (fst o strip_comb o rand o snd o strip_forall)
(conjuncts(concl eth)) in
GENL xvs hth in
fun def -> let ith,rth = define_type_raw def in
ith,generalize_recursion_theorem rth;;
(* ------------------------------------------------------------------------- *)
(* Set up options and lists. *)
(* ------------------------------------------------------------------------- *)
export_theory "option-def";;
let option_INDUCT,option_RECURSION =
let (induct,recursion) =
define_type_raw
(parse_inductive_type_specification "option = NONE | SOME A") in
let th1 = prove
(`!(p : A option -> bool). p NONE /\ (!a. p (SOME a)) ==> (!x. p x)`,
MATCH_ACCEPT_TAC induct)
and th2 = prove
(`!b f. ?(fn : A option -> B). fn NONE = b /\ !a. fn (SOME a) = f a`,
MATCH_ACCEPT_TAC recursion) in
(th1,th2);;
export_thm option_INDUCT;;
export_thm option_RECURSION;;
export_theory "list-def";;
let list_INDUCT,list_RECURSION =
let (induct,recursion) =
define_type_raw
(parse_inductive_type_specification "list = NIL | CONS A list") in
let th1 = prove
(`!(p : A list -> bool).
p [] /\ (!h t. p t ==> p (CONS h t)) ==> (!l. p l)`,
MATCH_ACCEPT_TAC induct)
and th2 = prove
(`!b f. ?(fn : A list -> B).
fn [] = b /\ !h t. fn (CONS h t) = f h t (fn t)`,
MATCH_ACCEPT_TAC recursion) in
(th1,th2);;
export_thm list_INDUCT;;
export_thm list_RECURSION;;
(* ------------------------------------------------------------------------- *)
(* Tools for proving injectivity and distinctness of constructors. *)
(* ------------------------------------------------------------------------- *)
let prove_constructors_injective =
let DEPAIR = GEN_REWRITE_RULE TOP_SWEEP_CONV [PAIR_EQ] in
let prove_distinctness ax pat =
let f,args = strip_comb pat in
let rt = end_itlist (curry mk_pair) args in
let ty = mk_fun_ty (type_of pat) (type_of rt) in
let fn = genvar ty in
let dtm = mk_eq(mk_comb(fn,pat),rt) in
let eth = prove_recursive_functions_exist ax (list_mk_forall(args,dtm)) in
let args' = variants args args in
let atm = mk_eq(pat,list_mk_comb(f,args')) in
let ath = ASSUME atm in
let bth = AP_TERM fn ath in
let cth1 = SPECL args (ASSUME(snd(dest_exists(concl eth)))) in
let cth2 = INST (zip args' args) cth1 in
let pth = TRANS (TRANS (SYM cth1) bth) cth2 in
let qth = DEPAIR pth in
let qtm = concl qth in
let rth = rev_itlist (C(curry MK_COMB)) (CONJUNCTS(ASSUME qtm)) (REFL f) in
let tth = IMP_ANTISYM_RULE (DISCH atm qth) (DISCH qtm rth) in
let uth = GENL args (GENL args' tth) in
PROVE_HYP eth (SIMPLE_CHOOSE fn uth) in
fun ax ->
let cls = conjuncts(snd(strip_exists(snd(strip_forall(concl ax))))) in
let pats = map (rand o lhand o snd o strip_forall) cls in
end_itlist CONJ (mapfilter (prove_distinctness ax) pats);;