author | kleing |
Fri, 27 May 2005 01:09:44 +0200 | |
changeset 16095 | f6af6b265d20 |
parent 15742 | 64eae3513064 |
child 16224 | 57094b83774e |
permissions | -rw-r--r-- |
2276 | 1 |
(* Title: HOLCF/domain/theorems.ML |
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ID: $Id$ |
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Author: David von Oheimb |
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New proofs/tactics by Brian Huffman |
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Proof generator for domain section. |
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*) |
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structure Domain_Theorems = struct |
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local |
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open Domain_Library; |
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infixr 0 ===>;infixr 0 ==>;infix 0 == ; |
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infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<; |
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infix 9 ` ; infix 9 `% ; infix 9 `%%; infixr 9 oo; |
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(* ----- general proof facilities ------------------------------------------- *) |
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|
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fun inferT sg pre_tm = |
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#1 (Sign.infer_types (Sign.pp sg) sg (K NONE) (K NONE) [] true ([pre_tm],propT)); |
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fun pg'' thy defs t = let val sg = sign_of thy; |
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val ct = Thm.cterm_of sg (inferT sg t); |
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in prove_goalw_cterm defs ct end; |
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fun pg' thy defs t tacsf=pg'' thy defs t (fn [] => tacsf |
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| prems=> (cut_facts_tac prems 1)::tacsf); |
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|
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fun case_UU_tac rews i v = case_tac (v^"=UU") i THEN |
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asm_simp_tac (HOLCF_ss addsimps rews) i; |
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val chain_tac = REPEAT_DETERM o resolve_tac |
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[chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL]; |
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(* ----- general proofs ----------------------------------------------------- *) |
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val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R" |
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(fn prems =>[ |
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resolve_tac prems 1, |
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cut_facts_tac prems 1, |
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fast_tac HOL_cs 1]); |
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val dist_eqI = prove_goal Porder.thy "!!x::'a::po. ~ x << y ==> x ~= y" |
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(fn prems => [ |
|
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(blast_tac (claset() addDs [antisym_less_inverse]) 1)]); |
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(* |
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infixr 0 y; |
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val b = 0; |
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fun _ y t = by t; |
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fun g defs t = let val sg = sign_of thy; |
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val ct = Thm.cterm_of sg (inferT sg t); |
|
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in goalw_cterm defs ct end; |
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*) |
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|
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in |
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||
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fun theorems (((dname,_),cons) : eq, eqs : eq list) thy = |
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let |
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val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ..."); |
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val pg = pg' thy; |
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(* ----- getting the axioms and definitions --------------------------------- *) |
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local fun ga s dn = get_thm thy (dn ^ "." ^ s, NONE) in |
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val ax_abs_iso = ga "abs_iso" dname; |
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val ax_rep_iso = ga "rep_iso" dname; |
|
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val ax_when_def = ga "when_def" dname; |
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val axs_con_def = map (fn (con,_) => ga (extern_name con^"_def") dname) cons; |
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val axs_dis_def = map (fn (con,_) => ga ( dis_name con^"_def") dname) cons; |
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val axs_sel_def = List.concat(map (fn (_,args) => |
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map (fn arg => ga (sel_of arg ^"_def") dname) args) |
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cons); |
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val ax_copy_def = ga "copy_def" dname; |
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1274 | 76 |
end; (* local *) |
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(* ----- theorems concerning the isomorphism -------------------------------- *) |
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|
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val dc_abs = %%:(dname^"_abs"); |
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val dc_rep = %%:(dname^"_rep"); |
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val dc_copy = %%:(dname^"_copy"); |
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val x_name = "x"; |
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val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso]; |
86 |
val abs_strict = iso_locale RS iso_abs_strict; |
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val rep_strict = iso_locale RS iso_rep_strict; |
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val abs_defin' = iso_locale RS iso_abs_defin'; |
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val rep_defin' = iso_locale RS iso_rep_defin'; |
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val iso_rews = map standard [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict]; |
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(* ----- generating beta reduction rules from definitions-------------------- *) |
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94 |
local |
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95 |
fun k NONE = cont_const | k (SOME x) = x; |
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fun ap NONE NONE = NONE |
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| ap x y = SOME (standard (cont2cont_Rep_CFun OF [k x, k y])); |
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100 |
fun var 0 = [SOME cont_id] |
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| var n = NONE :: var (n-1); |
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fun zip [] [] = [] |
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| zip [] (y::ys) = (ap NONE y ) :: zip [] ys |
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| zip (x::xs) [] = (ap x NONE) :: zip xs [] |
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| zip (x::xs) (y::ys) = (ap x y ) :: zip xs ys |
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fun lam [] = ([], cont_const) |
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| lam (x::ys) = let val x' = k x |
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val Lx = x' RS cont2cont_LAM |
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in (map (fn y => SOME (k y RS Lx)) ys, x') |
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112 |
end; |
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fun term_conts (Bound n) = (var n, []) |
|
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| term_conts (Const _) = ([],[]) |
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| term_conts (Const _ $ Abs (_,_,t)) = let |
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val (cs,ls) = term_conts t |
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val (cs',l) = lam cs |
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in (cs',l::ls) |
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end |
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| term_conts (Const _ $ f $ t) |
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= (zip (fst (term_conts f)) (fst (term_conts t)), []) |
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| term_conts t = let val dummy = prin t in ([],[]) end; |
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fun arglist (Const _ $ Abs (s,_,t)) = let |
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val (vars,body) = arglist t |
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in (s :: vars, body) end |
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| arglist t = ([],t); |
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fun bind_fun vars t = Library.foldr mk_All (vars,t); |
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fun bound_vars 0 = [] | bound_vars i = (Bound (i-1) :: bound_vars (i-1)); |
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in |
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fun appl_of_def def = let |
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val (_ $ con $ lam) = concl_of def |
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val (vars, rhs) = arglist lam |
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val lhs = Library.foldl (op `) (con, bound_vars (length vars)); |
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val appl = bind_fun vars (lhs == rhs) |
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val ([],cs) = term_conts lam |
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val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs |
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in pg (def::betas) appl [rtac reflexive_thm 1] end; |
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end; |
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val when_appl = appl_of_def ax_when_def; |
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val con_appls = map appl_of_def axs_con_def; |
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local |
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fun arg2typ n arg = let val t = TVar (("'a",n),["Pcpo.pcpo"]) |
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in (n+1, if is_lazy arg then mk_uT t else t) end; |
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fun args2typ n [] = (n,oneT) |
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| args2typ n [arg] = arg2typ n arg |
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| args2typ n (arg::args) = let val (n1,t1) = arg2typ n arg; |
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val (n2,t2) = args2typ n1 args |
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in (n2, mk_sprodT (t1, t2)) end; |
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fun cons2typ n [] = (n,oneT) |
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| cons2typ n [con] = args2typ n (snd con) |
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| cons2typ n (con::cons) = let val (n1,t1) = args2typ n (snd con); |
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val (n2,t2) = cons2typ n1 cons |
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in (n2, mk_ssumT (t1, t2)) end; |
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in |
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fun cons2ctyp cons = ctyp_of (sign_of thy) (snd (cons2typ 1 cons)); |
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end; |
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local |
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val iso_swap = iso_locale RS iso_iso_swap; |
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fun one_con (con,args) = let val vns = map vname args in |
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Library.foldr mk_ex (vns, foldr' mk_conj ((%:x_name === con_app2 con %: vns):: |
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map (defined o %:) (nonlazy args))) end; |
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val exh = foldr' mk_disj ((%:x_name===UU)::map one_con cons); |
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val my_ctyp = cons2ctyp cons; |
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val thm1 = instantiate' [SOME my_ctyp] [] exh_start; |
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val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1; |
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val thm3 = rewrite_rule [mk_meta_eq conj_assoc] thm2; |
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in |
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val exhaust = pg con_appls (mk_trp exh)[ |
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(* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *) |
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rtac disjE 1, |
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etac (rep_defin' RS disjI1) 2, |
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etac disjI2 2, |
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rewrite_goals_tac [mk_meta_eq iso_swap], |
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rtac thm3 1]; |
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val casedist = standard (rewrite_rule exh_casedists (exhaust RS exh_casedist0)); |
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end; |
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local |
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fun bind_fun t = Library.foldr mk_All (when_funs cons,t); |
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fun bound_fun i _ = Bound (length cons - i); |
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val when_app = Library.foldl (op `) (%%:(dname^"_when"), mapn bound_fun 1 cons); |
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in |
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val when_strict = pg [when_appl, mk_meta_eq rep_strict] |
189 |
(bind_fun(mk_trp(strict when_app))) |
|
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[resolve_tac [sscase1,ssplit1,strictify1] 1]; |
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val when_apps = let fun one_when n (con,args) = pg (when_appl :: con_appls) |
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(bind_fun (lift_defined %: (nonlazy args, |
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mk_trp(when_app`(con_app con args) === |
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mk_cRep_CFun(bound_fun n 0,map %# args)))))[ |
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asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1]; |
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in mapn one_when 1 cons end; |
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end; |
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val when_rews = when_strict::when_apps; |
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(* ----- theorems concerning the constructors, discriminators and selectors - *) |
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val dis_rews = let |
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val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp( |
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strict(%%:(dis_name con)))) [ |
15742 | 205 |
rtac when_strict 1]) cons; |
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val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def |
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(lift_defined %: (nonlazy args, |
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208 |
(mk_trp((%%:(dis_name c))`(con_app con args) === |
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209 |
%%:(if con=c then "TT" else "FF"))))) [ |
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asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; |
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in List.concat(map (fn (c,_) => map (one_dis c) cons) cons) end; |
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val dis_defins = map (fn (con,args) => pg [] (defined(%:x_name) ==> |
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defined(%%:(dis_name con)`%x_name)) [ |
4043 | 214 |
rtac casedist 1, |
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contr_tac 1, |
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216 |
DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac |
2033 | 217 |
(HOLCF_ss addsimps dis_apps) 1))]) cons; |
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218 |
in dis_stricts @ dis_defins @ dis_apps end; |
1274 | 219 |
|
15570 | 220 |
val con_stricts = List.concat(map (fn (con,args) => map (fn vn => |
15742 | 221 |
pg con_appls |
2033 | 222 |
(mk_trp(con_app2 con (fn arg => if vname arg = vn |
223 |
then UU else %# arg) args === UU))[ |
|
224 |
asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1] |
|
225 |
) (nonlazy args)) cons); |
|
1274 | 226 |
val con_defins = map (fn (con,args) => pg [] |
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227 |
(lift_defined %: (nonlazy args, |
2033 | 228 |
mk_trp(defined(con_app con args)))) ([ |
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229 |
rtac rev_contrapos 1, |
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eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1, |
231 |
asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons; |
|
1274 | 232 |
val con_rews = con_stricts @ con_defins; |
233 |
||
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234 |
val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%:sel))) [ |
2033 | 235 |
simp_tac (HOLCF_ss addsimps when_rews) 1]; |
15570 | 236 |
in List.concat(map (fn (_,args) =>map (fn arg => one_sel (sel_of arg)) args) cons) end; |
1274 | 237 |
val sel_apps = let fun one_sel c n sel = map (fn (con,args) => |
2033 | 238 |
let val nlas = nonlazy args; |
239 |
val vns = map vname args; |
|
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|
240 |
in pg axs_sel_def (lift_defined %: |
15570 | 241 |
(List.filter (fn v => con=c andalso (v<>List.nth(vns,n))) nlas, |
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|
242 |
mk_trp((%%:sel)`(con_app con args) === |
15570 | 243 |
(if con=c then %:(List.nth(vns,n)) else UU)))) |
2033 | 244 |
( (if con=c then [] |
245 |
else map(case_UU_tac(when_rews@con_stricts)1) nlas) |
|
15570 | 246 |
@(if con=c andalso ((List.nth(vns,n)) mem nlas) |
2033 | 247 |
then[case_UU_tac (when_rews @ con_stricts) 1 |
15570 | 248 |
(List.nth(vns,n))] else []) |
2033 | 249 |
@ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons; |
15570 | 250 |
in List.concat(map (fn (c,args) => |
251 |
List.concat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end; |
|
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|
252 |
val sel_defins = if length cons=1 then map (fn arg => pg [](defined(%:x_name)==> |
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|
253 |
defined(%%:(sel_of arg)`%x_name)) [ |
4043 | 254 |
rtac casedist 1, |
2033 | 255 |
contr_tac 1, |
8149
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|
256 |
DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac |
2033 | 257 |
(HOLCF_ss addsimps sel_apps) 1))]) |
258 |
(filter_out is_lazy (snd(hd cons))) else []; |
|
1274 | 259 |
val sel_rews = sel_stricts @ sel_defins @ sel_apps; |
260 |
||
261 |
val distincts_le = let |
|
262 |
fun dist (con1, args1) (con2, args2) = pg [] |
|
11531
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|
263 |
(lift_defined %: ((nonlazy args1), |
2033 | 264 |
(mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([ |
2445
51993fea433f
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oheimb
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diff
changeset
|
265 |
rtac rev_contrapos 1, |
2033 | 266 |
eres_inst_tac[("fo",dis_name con1)] monofun_cfun_arg 1] |
267 |
@map(case_UU_tac (con_stricts @ dis_rews)1)(nonlazy args2) |
|
268 |
@[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]); |
|
1274 | 269 |
fun distinct (con1,args1) (con2,args2) = |
2267 | 270 |
let val arg1 = (con1, args1) |
271 |
val arg2 = (con2, |
|
272 |
ListPair.map (fn (arg,vn) => upd_vname (K vn) arg) |
|
273 |
(args2, variantlist(map vname args2,map vname args1))) |
|
2033 | 274 |
in [dist arg1 arg2, dist arg2 arg1] end; |
1274 | 275 |
fun distincts [] = [] |
276 |
| distincts (c::cs) = (map (distinct c) cs) :: distincts cs; |
|
277 |
in distincts cons end; |
|
15570 | 278 |
val dist_les = List.concat (List.concat distincts_le); |
4043 | 279 |
val dist_eqs = let |
1274 | 280 |
fun distinct (_,args1) ((_,args2),leqs) = let |
2033 | 281 |
val (le1,le2) = (hd leqs, hd(tl leqs)); |
282 |
val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in |
|
283 |
if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else |
|
284 |
if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else |
|
285 |
[eq1, eq2] end; |
|
4221 | 286 |
open BasisLibrary (*restore original List*) |
1274 | 287 |
fun distincts [] = [] |
4062 | 288 |
| distincts ((c,leqs)::cs) = List.concat |
2267 | 289 |
(ListPair.map (distinct c) ((map #1 cs),leqs)) @ |
290 |
distincts cs; |
|
12037 | 291 |
in map standard (distincts (cons~~distincts_le)) end; |
1274 | 292 |
|
293 |
local |
|
294 |
fun pgterm rel con args = let |
|
2033 | 295 |
fun append s = upd_vname(fn v => v^s); |
296 |
val (largs,rargs) = (args, map (append "'") args); |
|
297 |
in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===> |
|
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|
298 |
lift_defined %: ((nonlazy largs),lift_defined %: ((nonlazy rargs), |
2033 | 299 |
mk_trp (foldr' mk_conj |
2267 | 300 |
(ListPair.map rel |
301 |
(map %# largs, map %# rargs)))))) end; |
|
15570 | 302 |
val cons' = List.filter (fn (_,args) => args<>[]) cons; |
1274 | 303 |
in |
304 |
val inverts = map (fn (con,args) => |
|
15570 | 305 |
pgterm (op <<) con args (List.concat(map (fn arg => [ |
2033 | 306 |
TRY(rtac conjI 1), |
307 |
dres_inst_tac [("fo",sel_of arg)] monofun_cfun_arg 1, |
|
308 |
asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1] |
|
309 |
) args))) cons'; |
|
1274 | 310 |
val injects = map (fn ((con,args),inv_thm) => |
2033 | 311 |
pgterm (op ===) con args [ |
312 |
etac (antisym_less_inverse RS conjE) 1, |
|
313 |
dtac inv_thm 1, REPEAT(atac 1), |
|
314 |
dtac inv_thm 1, REPEAT(atac 1), |
|
315 |
TRY(safe_tac HOL_cs), |
|
316 |
REPEAT(rtac antisym_less 1 ORELSE atac 1)] ) |
|
317 |
(cons'~~inverts); |
|
1274 | 318 |
end; |
319 |
||
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|
320 |
(* ----- theorems concerning one induction step ----------------------------- *) |
1274 | 321 |
|
1637
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Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
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|
322 |
val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [ |
2033 | 323 |
asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict, |
324 |
cfst_strict,csnd_strict]) 1]; |
|
1637
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Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
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changeset
|
325 |
val copy_apps = map (fn (con,args) => pg [ax_copy_def] |
11531
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Renamed functions % and %% to avoid clash with syntax for proof terms.
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parents:
10835
diff
changeset
|
326 |
(lift_defined %: (nonlazy_rec args, |
2033 | 327 |
mk_trp(dc_copy`%"f"`(con_app con args) === |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
328 |
(con_app2 con (app_rec_arg (cproj (%:"f") eqs)) args)))) |
2033 | 329 |
(map (case_UU_tac (abs_strict::when_strict::con_stricts) |
330 |
1 o vname) |
|
15570 | 331 |
(List.filter (fn a => not (is_rec a orelse is_lazy a)) args) |
2033 | 332 |
@[asm_simp_tac (HOLCF_ss addsimps when_apps) 1, |
15742 | 333 |
simp_tac (HOLCF_ss addsimps con_appls) 1]))cons; |
1637
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Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
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diff
changeset
|
334 |
val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU` |
2033 | 335 |
(con_app con args) ===UU)) |
1637
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Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
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diff
changeset
|
336 |
(let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews |
2033 | 337 |
in map (case_UU_tac rews 1) (nonlazy args) @ [ |
338 |
asm_simp_tac (HOLCF_ss addsimps rews) 1] end)) |
|
15570 | 339 |
(List.filter (fn (_,args)=>exists is_nonlazy_rec args) cons); |
1274 | 340 |
val copy_rews = copy_strict::copy_apps @ copy_stricts; |
4043 | 341 |
in thy |> Theory.add_path (Sign.base_name dname) |
8438 | 342 |
|> (#1 o (PureThy.add_thmss (map Thm.no_attributes [ |
4043 | 343 |
("iso_rews" , iso_rews ), |
344 |
("exhaust" , [exhaust] ), |
|
345 |
("casedist" , [casedist]), |
|
346 |
("when_rews", when_rews ), |
|
347 |
("con_rews", con_rews), |
|
348 |
("sel_rews", sel_rews), |
|
349 |
("dis_rews", dis_rews), |
|
350 |
("dist_les", dist_les), |
|
351 |
("dist_eqs", dist_eqs), |
|
352 |
("inverts" , inverts ), |
|
353 |
("injects" , injects ), |
|
8438 | 354 |
("copy_rews", copy_rews)]))) |
12037 | 355 |
|> Theory.parent_path |> rpair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @ |
356 |
dist_les @ dist_eqs @ copy_rews) |
|
1274 | 357 |
end; (* let *) |
358 |
||
4043 | 359 |
fun comp_theorems (comp_dnam, eqs: eq list) thy = |
1274 | 360 |
let |
4008 | 361 |
val dnames = map (fst o fst) eqs; |
362 |
val conss = map snd eqs; |
|
363 |
val comp_dname = Sign.full_name (sign_of thy) comp_dnam; |
|
364 |
||
12037 | 365 |
val d = writeln("Proving induction properties of domain "^comp_dname^" ..."); |
1274 | 366 |
val pg = pg' thy; |
367 |
||
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
368 |
(* ----- getting the composite axiom and definitions ------------------------ *) |
1274 | 369 |
|
15531 | 370 |
local fun ga s dn = get_thm thy (dn ^ "." ^ s, NONE) in |
4043 | 371 |
val axs_reach = map (ga "reach" ) dnames; |
372 |
val axs_take_def = map (ga "take_def" ) dnames; |
|
373 |
val axs_finite_def = map (ga "finite_def") dnames; |
|
374 |
val ax_copy2_def = ga "copy_def" comp_dnam; |
|
375 |
val ax_bisim_def = ga "bisim_def" comp_dnam; |
|
376 |
end; (* local *) |
|
377 |
||
15531 | 378 |
local fun gt s dn = get_thm thy (dn ^ "." ^ s, NONE); |
379 |
fun gts s dn = get_thms thy (dn ^ "." ^ s, NONE) in |
|
4043 | 380 |
val cases = map (gt "casedist" ) dnames; |
15570 | 381 |
val con_rews = List.concat (map (gts "con_rews" ) dnames); |
382 |
val copy_rews = List.concat (map (gts "copy_rews") dnames); |
|
1274 | 383 |
end; (* local *) |
384 |
||
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
385 |
fun dc_take dn = %%:(dn^"_take"); |
1274 | 386 |
val x_name = idx_name dnames "x"; |
387 |
val P_name = idx_name dnames "P"; |
|
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
388 |
val n_eqs = length eqs; |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
389 |
|
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
390 |
(* ----- theorems concerning finite approximation and finite induction ------ *) |
1274 | 391 |
|
392 |
local |
|
4098 | 393 |
val iterate_Cprod_ss = simpset_of Fix.thy |
2033 | 394 |
addsimps [cfst_strict, csnd_strict]addsimps Cprod_rews; |
1274 | 395 |
val copy_con_rews = copy_rews @ con_rews; |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
396 |
val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def; |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
397 |
val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=> |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
398 |
strict(dc_take dn $ %:"n")) eqs))) ([ |
13454 | 399 |
induct_tac "n" 1, |
4030 | 400 |
simp_tac iterate_Cprod_ss 1, |
2033 | 401 |
asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]); |
1274 | 402 |
val take_stricts' = rewrite_rule copy_take_defs take_stricts; |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
403 |
val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%:"0") |
2033 | 404 |
`%x_name n === UU))[ |
405 |
simp_tac iterate_Cprod_ss 1]) 1 dnames; |
|
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
406 |
val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1; |
1274 | 407 |
val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj |
15570 | 408 |
(List.concat(map (fn ((dn,_),cons) => map (fn (con,args) => Library.foldr mk_all |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
409 |
(map vname args,(dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args) === |
15570 | 410 |
con_app2 con (app_rec_arg (fn n=>dc_take (List.nth(dnames,n))$ %:"n")) |
2033 | 411 |
args)) cons) eqs)))) ([ |
412 |
simp_tac iterate_Cprod_ss 1, |
|
13454 | 413 |
induct_tac "n" 1, |
2033 | 414 |
simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1, |
415 |
asm_full_simp_tac (HOLCF_ss addsimps |
|
15570 | 416 |
(List.filter (has_fewer_prems 1) copy_rews)) 1, |
2033 | 417 |
TRY(safe_tac HOL_cs)] @ |
15570 | 418 |
(List.concat(map (fn ((dn,_),cons) => map (fn (con,args) => |
2033 | 419 |
if nonlazy_rec args = [] then all_tac else |
420 |
EVERY(map c_UU_tac (nonlazy_rec args)) THEN |
|
421 |
asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1 |
|
422 |
) cons) eqs))); |
|
1274 | 423 |
in |
12037 | 424 |
val take_rews = map standard (atomize take_stricts @ take_0s @ atomize take_apps); |
1274 | 425 |
end; (* local *) |
426 |
||
427 |
local |
|
15570 | 428 |
fun one_con p (con,args) = Library.foldr mk_All (map vname args, |
2033 | 429 |
lift_defined (bound_arg (map vname args)) (nonlazy args, |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
430 |
lift (fn arg => %:(P_name (1+rec_of arg)) $ bound_arg args arg) |
15570 | 431 |
(List.filter is_rec args,mk_trp(%:p $ con_app2 con (bound_arg args) args)))); |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
432 |
fun one_eq ((p,cons),concl) = (mk_trp(%:p $ UU) ===> |
15570 | 433 |
Library.foldr (op ===>) (map (one_con p) cons,concl)); |
434 |
fun ind_term concf = Library.foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss, |
|
2033 | 435 |
mk_trp(foldr' mk_conj (mapn concf 1 dnames))); |
1274 | 436 |
val take_ss = HOL_ss addsimps take_rews; |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
437 |
fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i) |
2033 | 438 |
1 dnames); |
15570 | 439 |
fun ind_prems_tac prems = EVERY(List.concat (map (fn cons => ( |
2033 | 440 |
resolve_tac prems 1 :: |
15570 | 441 |
List.concat (map (fn (_,args) => |
2033 | 442 |
resolve_tac prems 1 :: |
443 |
map (K(atac 1)) (nonlazy args) @ |
|
15570 | 444 |
map (K(atac 1)) (List.filter is_rec args)) |
2033 | 445 |
cons))) conss)); |
1274 | 446 |
local |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
447 |
(* check whether every/exists constructor of the n-th part of the equation: |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
448 |
it has a possibly indirectly recursive argument that isn't/is possibly |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
449 |
indirectly lazy *) |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
450 |
fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => |
2033 | 451 |
is_rec arg andalso not(rec_of arg mem ns) andalso |
452 |
((rec_of arg = n andalso nfn(lazy_rec orelse is_lazy arg)) orelse |
|
453 |
rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) |
|
15570 | 454 |
(lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg)))) |
2033 | 455 |
) o snd) cons; |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
456 |
fun all_rec_to ns = rec_to forall not all_rec_to ns; |
4030 | 457 |
fun warn (n,cons) = if all_rec_to [] false (n,cons) then (warning |
15570 | 458 |
("domain "^List.nth(dnames,n)^" is empty!"); true) else false; |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
459 |
fun lazy_rec_to ns = rec_to exists Id lazy_rec_to ns; |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
460 |
|
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
461 |
in val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs; |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
462 |
val is_emptys = map warn n__eqs; |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
463 |
val is_finite = forall (not o lazy_rec_to [] false) n__eqs; |
1274 | 464 |
end; |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
465 |
in (* local *) |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
466 |
val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %:(P_name n)$ |
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
467 |
(dc_take dn $ %:"n" `%(x_name n)))) (fn prems => [ |
2033 | 468 |
quant_tac 1, |
2445
51993fea433f
removed Holcfb.thy and Holcfb.ML, moving classical3 to HOL.ML as classical2
oheimb
parents:
2276
diff
changeset
|
469 |
simp_tac HOL_ss 1, |
13454 | 470 |
induct_tac "n" 1, |
2033 | 471 |
simp_tac (take_ss addsimps prems) 1, |
472 |
TRY(safe_tac HOL_cs)] |
|
15570 | 473 |
@ List.concat(map (fn (cons,cases) => [ |
2033 | 474 |
res_inst_tac [("x","x")] cases 1, |
475 |
asm_simp_tac (take_ss addsimps prems) 1] |
|
15570 | 476 |
@ List.concat(map (fn (con,args) => |
2033 | 477 |
asm_simp_tac take_ss 1 :: |
478 |
map (fn arg => |
|
479 |
case_UU_tac (prems@con_rews) 1 ( |
|
15570 | 480 |
List.nth(dnames,rec_of arg)^"_take n$"^vname arg)) |
481 |
(List.filter is_nonlazy_rec args) @ [ |
|
2033 | 482 |
resolve_tac prems 1] @ |
483 |
map (K (atac 1)) (nonlazy args) @ |
|
15570 | 484 |
map (K (etac spec 1)) (List.filter is_rec args)) |
2033 | 485 |
cons)) |
4043 | 486 |
(conss~~cases))); |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
487 |
|
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
488 |
val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n", |
2033 | 489 |
mk_trp(dc_take dn $ Bound 0 `%(x_name n) === |
490 |
dc_take dn $ Bound 0 `%(x_name n^"'"))) |
|
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
491 |
===> mk_trp(%:(x_name n) === %:(x_name n^"'"))) (fn prems => [ |
2033 | 492 |
res_inst_tac[("t",x_name n )](ax_reach RS subst) 1, |
493 |
res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1, |
|
494 |
stac fix_def2 1, |
|
495 |
REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1 |
|
496 |
THEN chain_tac 1)), |
|
497 |
stac contlub_cfun_fun 1, |
|
498 |
stac contlub_cfun_fun 2, |
|
499 |
rtac lub_equal 3, |
|
500 |
chain_tac 1, |
|
501 |
rtac allI 1, |
|
502 |
resolve_tac prems 1])) 1 (dnames~~axs_reach); |
|
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
503 |
|
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
504 |
(* ----- theorems concerning finiteness and induction ----------------------- *) |
1274 | 505 |
|
506 |
val (finites,ind) = if is_finite then |
|
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
507 |
let |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
508 |
fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x"); |
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
509 |
val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%:"x")) ===> |
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
510 |
mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %:"x" === UU), |
2033 | 511 |
take_enough dn)) ===> mk_trp(take_enough dn)) [ |
512 |
etac disjE 1, |
|
513 |
etac notE 1, |
|
514 |
resolve_tac take_lemmas 1, |
|
515 |
asm_simp_tac take_ss 1, |
|
516 |
atac 1]) dnames; |
|
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
517 |
val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn |
2033 | 518 |
(fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args), |
519 |
mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU, |
|
520 |
dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([ |
|
521 |
rtac allI 1, |
|
13454 | 522 |
induct_tac "n" 1, |
2033 | 523 |
simp_tac take_ss 1, |
524 |
TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @ |
|
15570 | 525 |
List.concat(mapn (fn n => fn (cons,cases) => [ |
2033 | 526 |
simp_tac take_ss 1, |
527 |
rtac allI 1, |
|
528 |
res_inst_tac [("x",x_name n)] cases 1, |
|
529 |
asm_simp_tac take_ss 1] @ |
|
15570 | 530 |
List.concat(map (fn (con,args) => |
2033 | 531 |
asm_simp_tac take_ss 1 :: |
15570 | 532 |
List.concat(map (fn vn => [ |
2033 | 533 |
eres_inst_tac [("x",vn)] all_dupE 1, |
534 |
etac disjE 1, |
|
535 |
asm_simp_tac (HOL_ss addsimps con_rews) 1, |
|
536 |
asm_simp_tac take_ss 1]) |
|
537 |
(nonlazy_rec args))) |
|
538 |
cons)) |
|
4043 | 539 |
1 (conss~~cases))); |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
540 |
val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp( |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
541 |
%%:(dn^"_finite") $ %:"x"))[ |
2033 | 542 |
case_UU_tac take_rews 1 "x", |
543 |
eresolve_tac finite_lemmas1a 1, |
|
544 |
step_tac HOL_cs 1, |
|
545 |
step_tac HOL_cs 1, |
|
546 |
cut_facts_tac [l1b] 1, |
|
547 |
fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b); |
|
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
548 |
in |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
549 |
(finites, |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
550 |
pg'' thy[](ind_term (fn n => fn dn => %:(P_name n) $ %:(x_name n)))(fn prems => |
2033 | 551 |
TRY(safe_tac HOL_cs) :: |
15570 | 552 |
List.concat (map (fn (finite,fin_ind) => [ |
2033 | 553 |
rtac(rewrite_rule axs_finite_def finite RS exE)1, |
554 |
etac subst 1, |
|
555 |
rtac fin_ind 1, |
|
556 |
ind_prems_tac prems]) |
|
557 |
(finites~~(atomize finite_ind)) )) |
|
1274 | 558 |
) end (* let *) else |
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
559 |
(mapn (fn n => fn dn => read_instantiate_sg (sign_of thy) |
2033 | 560 |
[("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames, |
15570 | 561 |
pg'' thy [] (Library.foldr (op ===>) (mapn (fn n => K(mk_trp(%%:"adm" $ %:(P_name n)))) |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
562 |
1 dnames, ind_term (fn n => fn dn => %:(P_name n) $ %:(x_name n)))) |
2033 | 563 |
(fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1) |
564 |
axs_reach @ [ |
|
565 |
quant_tac 1, |
|
566 |
rtac (adm_impl_admw RS wfix_ind) 1, |
|
4030 | 567 |
REPEAT_DETERM(rtac adm_all2 1), |
568 |
REPEAT_DETERM(TRY(rtac adm_conj 1) THEN |
|
569 |
rtac adm_subst 1 THEN |
|
2033 | 570 |
cont_tacR 1 THEN resolve_tac prems 1), |
571 |
strip_tac 1, |
|
572 |
rtac (rewrite_rule axs_take_def finite_ind) 1, |
|
573 |
ind_prems_tac prems]) |
|
4755 | 574 |
handle ERROR => (warning "Cannot prove infinite induction rule"; refl)) |
1274 | 575 |
end; (* local *) |
576 |
||
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
577 |
(* ----- theorem concerning coinduction ------------------------------------- *) |
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
578 |
|
1274 | 579 |
local |
580 |
val xs = mapn (fn n => K (x_name n)) 1 dnames; |
|
1637
b8a8ae2e5de1
Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
oheimb
parents:
1512
diff
changeset
|
581 |
fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1); |
1274 | 582 |
val take_ss = HOL_ss addsimps take_rews; |
4755 | 583 |
val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")")); |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
584 |
val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%:(comp_dname^"_bisim") $ %:"R", |
15570 | 585 |
Library.foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs, |
586 |
Library.foldr mk_imp (mapn (fn n => K(proj (%:"R") eqs n $ |
|
2033 | 587 |
bnd_arg n 0 $ bnd_arg n 1)) 0 dnames, |
588 |
foldr' mk_conj (mapn (fn n => fn dn => |
|
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
589 |
(dc_take dn $ %:"n" `bnd_arg n 0 === |
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
590 |
(dc_take dn $ %:"n" `bnd_arg n 1)))0 dnames)))))) |
2033 | 591 |
([ rtac impI 1, |
13454 | 592 |
induct_tac "n" 1, |
2033 | 593 |
simp_tac take_ss 1, |
594 |
safe_tac HOL_cs] @ |
|
15570 | 595 |
List.concat(mapn (fn n => fn x => [ |
2033 | 596 |
rotate_tac (n+1) 1, |
597 |
etac all2E 1, |
|
4755 | 598 |
eres_inst_tac [("P1", sproj "R" eqs n^ |
2033 | 599 |
" "^x^" "^x^"'")](mp RS disjE) 1, |
600 |
TRY(safe_tac HOL_cs), |
|
601 |
REPEAT(CHANGED(asm_simp_tac take_ss 1))]) |
|
602 |
0 xs)); |
|
1274 | 603 |
in |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
604 |
val coind = pg [] (mk_trp(%%:(comp_dname^"_bisim") $ %:"R") ===> |
15570 | 605 |
Library.foldr (op ===>) (mapn (fn n => fn x => |
11531
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
606 |
mk_trp(proj (%:"R") eqs n $ %:x $ %:(x^"'"))) 0 xs, |
d038246a62f2
Renamed functions % and %% to avoid clash with syntax for proof terms.
berghofe
parents:
10835
diff
changeset
|
607 |
mk_trp(foldr' mk_conj (map (fn x => %:x === %:(x^"'")) xs)))) ([ |
2033 | 608 |
TRY(safe_tac HOL_cs)] @ |
15570 | 609 |
List.concat(map (fn take_lemma => [ |
2033 | 610 |
rtac take_lemma 1, |
611 |
cut_facts_tac [coind_lemma] 1, |
|
612 |
fast_tac HOL_cs 1]) |
|
613 |
take_lemmas)); |
|
1274 | 614 |
end; (* local *) |
615 |
||
4043 | 616 |
in thy |> Theory.add_path comp_dnam |
8438 | 617 |
|> (#1 o (PureThy.add_thmss (map Thm.no_attributes [ |
4043 | 618 |
("take_rews" , take_rews ), |
619 |
("take_lemmas", take_lemmas), |
|
620 |
("finites" , finites ), |
|
621 |
("finite_ind", [finite_ind]), |
|
622 |
("ind" , [ind ]), |
|
8438 | 623 |
("coind" , [coind ])]))) |
12037 | 624 |
|> Theory.parent_path |> rpair take_rews |
1274 | 625 |
end; (* let *) |
626 |
end; (* local *) |
|
627 |
end; (* struct *) |