author | huffman |
Fri, 08 May 2009 16:19:51 -0700 | |
changeset 31076 | 99fe356cbbc2 |
parent 31023 | d027411c9a38 |
child 31160 | 2823f1b6b860 |
permissions | -rw-r--r-- |
23152 | 1 |
(* Title: HOLCF/Tools/domain/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 command. |
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*) |
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val HOLCF_ss = @{simpset}; |
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31005 | 11 |
signature DOMAIN_THEOREMS = |
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sig |
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val theorems: Domain_Library.eq * Domain_Library.eq list -> theory -> thm list * theory; |
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val comp_theorems: bstring * Domain_Library.eq list -> theory -> thm list * theory; |
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end; |
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structure Domain_Theorems :> DOMAIN_THEOREMS = |
31005 | 18 |
struct |
23152 | 19 |
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val quiet_mode = ref false; |
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val trace_domain = ref false; |
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fun message s = if !quiet_mode then () else writeln s; |
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fun trace s = if !trace_domain then tracing s else (); |
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local |
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val adm_impl_admw = @{thm adm_impl_admw}; |
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val adm_all = @{thm adm_all}; |
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val adm_conj = @{thm adm_conj}; |
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val adm_subst = @{thm adm_subst}; |
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val antisym_less_inverse = @{thm below_antisym_inverse}; |
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val beta_cfun = @{thm beta_cfun}; |
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val cfun_arg_cong = @{thm cfun_arg_cong}; |
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val ch2ch_Rep_CFunL = @{thm ch2ch_Rep_CFunL}; |
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val ch2ch_Rep_CFunR = @{thm ch2ch_Rep_CFunR}; |
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val chain_iterate = @{thm chain_iterate}; |
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val compact_ONE = @{thm compact_ONE}; |
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val compact_sinl = @{thm compact_sinl}; |
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val compact_sinr = @{thm compact_sinr}; |
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val compact_spair = @{thm compact_spair}; |
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val compact_up = @{thm compact_up}; |
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val contlub_cfun_arg = @{thm contlub_cfun_arg}; |
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val contlub_cfun_fun = @{thm contlub_cfun_fun}; |
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val fix_def2 = @{thm fix_def2}; |
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val injection_eq = @{thm injection_eq}; |
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val injection_less = @{thm injection_below}; |
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val lub_equal = @{thm lub_equal}; |
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val monofun_cfun_arg = @{thm monofun_cfun_arg}; |
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val retraction_strict = @{thm retraction_strict}; |
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val spair_eq = @{thm spair_eq}; |
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parents:
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diff
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val spair_less = @{thm spair_below}; |
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val sscase1 = @{thm sscase1}; |
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val ssplit1 = @{thm ssplit1}; |
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val strictify1 = @{thm strictify1}; |
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val wfix_ind = @{thm wfix_ind}; |
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val iso_intro = @{thm iso.intro}; |
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val iso_abs_iso = @{thm iso.abs_iso}; |
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val iso_rep_iso = @{thm iso.rep_iso}; |
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val iso_abs_strict = @{thm iso.abs_strict}; |
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val iso_rep_strict = @{thm iso.rep_strict}; |
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val iso_abs_defin' = @{thm iso.abs_defin'}; |
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val iso_rep_defin' = @{thm iso.rep_defin'}; |
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val iso_abs_defined = @{thm iso.abs_defined}; |
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val iso_rep_defined = @{thm iso.rep_defined}; |
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val iso_compact_abs = @{thm iso.compact_abs}; |
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val iso_compact_rep = @{thm iso.compact_rep}; |
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val iso_iso_swap = @{thm iso.iso_swap}; |
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val exh_start = @{thm exh_start}; |
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val ex_defined_iffs = @{thms ex_defined_iffs}; |
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val exh_casedist0 = @{thm exh_casedist0}; |
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val exh_casedists = @{thms exh_casedists}; |
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23152 | 75 |
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open Domain_Library; |
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infixr 0 ===>; |
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infixr 0 ==>; |
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infix 0 == ; |
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infix 1 ===; |
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infix 1 ~= ; |
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infix 1 <<; |
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infix 1 ~<<; |
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infix 9 ` ; |
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infix 9 `% ; |
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infix 9 `%%; |
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infixr 9 oo; |
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(* ----- general proof facilities ------------------------------------------- *) |
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||
24503 | 91 |
fun legacy_infer_term thy t = |
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let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy) |
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in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end; |
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fun pg'' thy defs t tacs = |
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let |
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24503 | 97 |
val t' = legacy_infer_term thy t; |
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val asms = Logic.strip_imp_prems t'; |
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val prop = Logic.strip_imp_concl t'; |
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26711 | 100 |
fun tac {prems, context} = |
23152 | 101 |
rewrite_goals_tac defs THEN |
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EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context}) |
23152 | 103 |
in Goal.prove_global thy [] asms prop tac end; |
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fun pg' thy defs t tacsf = |
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let |
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fun tacs {prems, context} = |
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if null prems then tacsf context |
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else cut_facts_tac prems 1 :: tacsf context; |
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in pg'' thy defs t tacs end; |
111 |
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fun case_UU_tac ctxt rews i v = |
5fe899199f85
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InductTacs.case_tac ctxt (v^"=UU") i THEN |
23152 | 114 |
asm_simp_tac (HOLCF_ss addsimps rews) i; |
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val chain_tac = |
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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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29064 | 122 |
val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp} |
23152 | 123 |
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31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
31023
diff
changeset
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val dist_eqI = @{lemma "!!x::'a::po. ~ x << y ==> x ~= y" by (blast dest!: below_antisym_inverse)} |
23152 | 125 |
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in |
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fun theorems (((dname, _), cons) : eq, eqs : eq list) thy = |
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let |
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29402 | 131 |
val _ = message ("Proving isomorphism properties of domain "^dname^" ..."); |
23152 | 132 |
val pg = pg' thy; |
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(* ----- getting the axioms and definitions --------------------------------- *) |
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local |
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fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s); |
23152 | 138 |
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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fun get_def mk_name (con,_) = ga (mk_name con^"_def") dname; |
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val axs_con_def = map (get_def extern_name) cons; |
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val axs_dis_def = map (get_def dis_name) cons; |
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val axs_mat_def = map (get_def mat_name) cons; |
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val axs_pat_def = map (get_def pat_name) cons; |
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val axs_sel_def = |
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let |
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fun def_of_sel sel = ga (sel^"_def") dname; |
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fun def_of_arg arg = Option.map def_of_sel (sel_of arg); |
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fun defs_of_con (_, args) = List.mapPartial def_of_arg args; |
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in |
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26336
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maps defs_of_con cons |
23152 | 154 |
end; |
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val ax_copy_def = ga "copy_def" dname; |
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end; (* local *) |
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(* ----- theorems concerning the isomorphism -------------------------------- *) |
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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]; |
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val abs_strict = ax_rep_iso RS (allI RS retraction_strict); |
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val rep_strict = ax_abs_iso RS (allI RS retraction_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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29402 | 174 |
val _ = trace " Proving beta reduction rules..."; |
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23152 | 176 |
local |
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fun arglist (Const _ $ Abs (s, _, t)) = |
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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 = [] |
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| bound_vars i = Bound (i-1) :: bound_vars (i - 1); |
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in |
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fun appl_of_def def = |
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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 = list_ccomb (con, bound_vars (length vars)); |
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val appl = bind_fun vars (lhs == rhs); |
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val cs = ContProc.cont_thms lam; |
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val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs; |
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27208
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wenzelm
parents:
27153
diff
changeset
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194 |
in pg (def::betas) appl (K [rtac reflexive_thm 1]) end; |
23152 | 195 |
end; |
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29402 | 197 |
val _ = trace "Proving when_appl..."; |
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val when_appl = appl_of_def ax_when_def; |
29402 | 199 |
val _ = trace "Proving con_appls..."; |
23152 | 200 |
val con_appls = map appl_of_def axs_con_def; |
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local |
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fun arg2typ n arg = |
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let val t = TVar (("'a", n), pcpoS) |
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in (n + 1, if is_lazy arg then mk_uT t else t) end; |
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206 |
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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) = |
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210 |
let |
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211 |
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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||
215 |
fun cons2typ n [] = (n,oneT) |
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216 |
| cons2typ n [con] = args2typ n (snd con) |
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| cons2typ n (con::cons) = |
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218 |
let |
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219 |
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 thy (snd (cons2typ 1 cons)); |
|
224 |
end; |
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225 |
||
226 |
local |
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val iso_swap = iso_locale RS iso_iso_swap; |
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fun one_con (con, args) = |
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let |
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val vns = map vname args; |
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val eqn = %:x_name === con_app2 con %: vns; |
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val conj = foldr1 mk_conj (eqn :: map (defined o %:) (nonlazy args)); |
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in Library.foldr mk_ex (vns, conj) end; |
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234 |
||
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1a4167d761ac
tactics: avoid dynamic reference to accidental theory context (via ML_Context.the_context etc.);
wenzelm
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23152
diff
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|
235 |
val conj_assoc = @{thm conj_assoc}; |
23152 | 236 |
val exh = foldr1 mk_disj ((%:x_name === UU) :: map one_con cons); |
237 |
val thm1 = instantiate' [SOME (cons2ctyp cons)] [] exh_start; |
|
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val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1; |
|
25805 | 239 |
val thm3 = rewrite_rule [mk_meta_eq @{thm conj_assoc}] thm2; |
23152 | 240 |
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(* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *) |
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242 |
val tacs = [ |
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243 |
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], |
|
247 |
rtac thm3 1]; |
|
248 |
in |
|
29402 | 249 |
val _ = trace " Proving exhaust..."; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
250 |
val exhaust = pg con_appls (mk_trp exh) (K tacs); |
29402 | 251 |
val _ = trace " Proving casedist..."; |
23152 | 252 |
val casedist = |
253 |
standard (rewrite_rule exh_casedists (exhaust RS exh_casedist0)); |
|
254 |
end; |
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255 |
||
256 |
local |
|
257 |
fun bind_fun t = Library.foldr mk_All (when_funs cons, t); |
|
258 |
fun bound_fun i _ = Bound (length cons - i); |
|
259 |
val when_app = list_ccomb (%%:(dname^"_when"), mapn bound_fun 1 cons); |
|
260 |
in |
|
29402 | 261 |
val _ = trace " Proving when_strict..."; |
23152 | 262 |
val when_strict = |
263 |
let |
|
264 |
val axs = [when_appl, mk_meta_eq rep_strict]; |
|
265 |
val goal = bind_fun (mk_trp (strict when_app)); |
|
266 |
val tacs = [resolve_tac [sscase1, ssplit1, strictify1] 1]; |
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27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
267 |
in pg axs goal (K tacs) end; |
23152 | 268 |
|
29402 | 269 |
val _ = trace " Proving when_apps..."; |
23152 | 270 |
val when_apps = |
271 |
let |
|
272 |
fun one_when n (con,args) = |
|
273 |
let |
|
274 |
val axs = when_appl :: con_appls; |
|
275 |
val goal = bind_fun (lift_defined %: (nonlazy args, |
|
276 |
mk_trp (when_app`(con_app con args) === |
|
277 |
list_ccomb (bound_fun n 0, map %# args)))); |
|
278 |
val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
279 |
in pg axs goal (K tacs) end; |
23152 | 280 |
in mapn one_when 1 cons end; |
281 |
end; |
|
282 |
val when_rews = when_strict :: when_apps; |
|
283 |
||
284 |
(* ----- theorems concerning the constructors, discriminators and selectors - *) |
|
285 |
||
286 |
local |
|
287 |
fun dis_strict (con, _) = |
|
288 |
let |
|
289 |
val goal = mk_trp (strict (%%:(dis_name con))); |
|
27208
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wenzelm
parents:
27153
diff
changeset
|
290 |
in pg axs_dis_def goal (K [rtac when_strict 1]) end; |
23152 | 291 |
|
292 |
fun dis_app c (con, args) = |
|
293 |
let |
|
294 |
val lhs = %%:(dis_name c) ` con_app con args; |
|
26012 | 295 |
val rhs = if con = c then TT else FF; |
23152 | 296 |
val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs)); |
297 |
val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; |
|
27208
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proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
298 |
in pg axs_dis_def goal (K tacs) end; |
23152 | 299 |
|
29402 | 300 |
val _ = trace " Proving dis_apps..."; |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
301 |
val dis_apps = maps (fn (c,_) => map (dis_app c) cons) cons; |
23152 | 302 |
|
303 |
fun dis_defin (con, args) = |
|
304 |
let |
|
305 |
val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name); |
|
306 |
val tacs = |
|
307 |
[rtac casedist 1, |
|
308 |
contr_tac 1, |
|
309 |
DETERM_UNTIL_SOLVED (CHANGED |
|
310 |
(asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
311 |
in pg [] goal (K tacs) end; |
23152 | 312 |
|
29402 | 313 |
val _ = trace " Proving dis_stricts..."; |
23152 | 314 |
val dis_stricts = map dis_strict cons; |
29402 | 315 |
val _ = trace " Proving dis_defins..."; |
23152 | 316 |
val dis_defins = map dis_defin cons; |
317 |
in |
|
318 |
val dis_rews = dis_stricts @ dis_defins @ dis_apps; |
|
319 |
end; |
|
320 |
||
321 |
local |
|
322 |
fun mat_strict (con, _) = |
|
323 |
let |
|
30912
4022298c1a86
change definition of match combinators for fixrec package
huffman
parents:
30911
diff
changeset
|
324 |
val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU); |
4022298c1a86
change definition of match combinators for fixrec package
huffman
parents:
30911
diff
changeset
|
325 |
val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1]; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
326 |
in pg axs_mat_def goal (K tacs) end; |
23152 | 327 |
|
29402 | 328 |
val _ = trace " Proving mat_stricts..."; |
23152 | 329 |
val mat_stricts = map mat_strict cons; |
330 |
||
331 |
fun one_mat c (con, args) = |
|
332 |
let |
|
30912
4022298c1a86
change definition of match combinators for fixrec package
huffman
parents:
30911
diff
changeset
|
333 |
val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs"; |
23152 | 334 |
val rhs = |
335 |
if con = c |
|
30912
4022298c1a86
change definition of match combinators for fixrec package
huffman
parents:
30911
diff
changeset
|
336 |
then list_ccomb (%:"rhs", map %# args) |
26012 | 337 |
else mk_fail; |
23152 | 338 |
val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs)); |
339 |
val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
340 |
in pg axs_mat_def goal (K tacs) end; |
23152 | 341 |
|
29402 | 342 |
val _ = trace " Proving mat_apps..."; |
23152 | 343 |
val mat_apps = |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
344 |
maps (fn (c,_) => map (one_mat c) cons) cons; |
23152 | 345 |
in |
346 |
val mat_rews = mat_stricts @ mat_apps; |
|
347 |
end; |
|
348 |
||
349 |
local |
|
350 |
fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args; |
|
351 |
||
26012 | 352 |
fun pat_lhs (con,args) = mk_branch (list_comb (%%:(pat_name con), ps args)); |
23152 | 353 |
|
26012 | 354 |
fun pat_rhs (con,[]) = mk_return ((%:"rhs") ` HOLogic.unit) |
23152 | 355 |
| pat_rhs (con,args) = |
26012 | 356 |
(mk_branch (mk_ctuple_pat (ps args))) |
23152 | 357 |
`(%:"rhs")`(mk_ctuple (map %# args)); |
358 |
||
359 |
fun pat_strict c = |
|
360 |
let |
|
25132 | 361 |
val axs = @{thm branch_def} :: axs_pat_def; |
23152 | 362 |
val goal = mk_trp (strict (pat_lhs c ` (%:"rhs"))); |
363 |
val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
364 |
in pg axs goal (K tacs) end; |
23152 | 365 |
|
366 |
fun pat_app c (con, args) = |
|
367 |
let |
|
25132 | 368 |
val axs = @{thm branch_def} :: axs_pat_def; |
23152 | 369 |
val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args); |
26012 | 370 |
val rhs = if con = fst c then pat_rhs c else mk_fail; |
23152 | 371 |
val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs)); |
372 |
val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
373 |
in pg axs goal (K tacs) end; |
23152 | 374 |
|
29402 | 375 |
val _ = trace " Proving pat_stricts..."; |
23152 | 376 |
val pat_stricts = map pat_strict cons; |
29402 | 377 |
val _ = trace " Proving pat_apps..."; |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
378 |
val pat_apps = maps (fn c => map (pat_app c) cons) cons; |
23152 | 379 |
in |
380 |
val pat_rews = pat_stricts @ pat_apps; |
|
381 |
end; |
|
382 |
||
383 |
local |
|
384 |
fun con_strict (con, args) = |
|
385 |
let |
|
30911
7809cbaa1b61
domain package: simplify internal proofs of con_rews
huffman
parents:
30829
diff
changeset
|
386 |
val rules = abs_strict :: @{thms con_strict_rules}; |
23152 | 387 |
fun one_strict vn = |
388 |
let |
|
389 |
fun f arg = if vname arg = vn then UU else %# arg; |
|
390 |
val goal = mk_trp (con_app2 con f args === UU); |
|
30911
7809cbaa1b61
domain package: simplify internal proofs of con_rews
huffman
parents:
30829
diff
changeset
|
391 |
val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1]; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
392 |
in pg con_appls goal (K tacs) end; |
23152 | 393 |
in map one_strict (nonlazy args) end; |
394 |
||
395 |
fun con_defin (con, args) = |
|
396 |
let |
|
30913
10b26965a08f
domain package now generates iff rules for definedness of constructors
huffman
parents:
30912
diff
changeset
|
397 |
fun iff_disj (t, []) = HOLogic.mk_not t |
10b26965a08f
domain package now generates iff rules for definedness of constructors
huffman
parents:
30912
diff
changeset
|
398 |
| iff_disj (t, ts) = t === foldr1 HOLogic.mk_disj ts; |
10b26965a08f
domain package now generates iff rules for definedness of constructors
huffman
parents:
30912
diff
changeset
|
399 |
val lhs = con_app con args === UU; |
10b26965a08f
domain package now generates iff rules for definedness of constructors
huffman
parents:
30912
diff
changeset
|
400 |
val rhss = map (fn x => %:x === UU) (nonlazy args); |
10b26965a08f
domain package now generates iff rules for definedness of constructors
huffman
parents:
30912
diff
changeset
|
401 |
val goal = mk_trp (iff_disj (lhs, rhss)); |
10b26965a08f
domain package now generates iff rules for definedness of constructors
huffman
parents:
30912
diff
changeset
|
402 |
val rule1 = iso_locale RS @{thm iso.abs_defined_iff}; |
10b26965a08f
domain package now generates iff rules for definedness of constructors
huffman
parents:
30912
diff
changeset
|
403 |
val rules = rule1 :: @{thms con_defined_iff_rules}; |
10b26965a08f
domain package now generates iff rules for definedness of constructors
huffman
parents:
30912
diff
changeset
|
404 |
val tacs = [simp_tac (HOL_ss addsimps rules) 1]; |
30911
7809cbaa1b61
domain package: simplify internal proofs of con_rews
huffman
parents:
30829
diff
changeset
|
405 |
in pg con_appls goal (K tacs) end; |
23152 | 406 |
in |
29402 | 407 |
val _ = trace " Proving con_stricts..."; |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
408 |
val con_stricts = maps con_strict cons; |
30911
7809cbaa1b61
domain package: simplify internal proofs of con_rews
huffman
parents:
30829
diff
changeset
|
409 |
val _ = trace " Proving con_defins..."; |
23152 | 410 |
val con_defins = map con_defin cons; |
411 |
val con_rews = con_stricts @ con_defins; |
|
412 |
end; |
|
413 |
||
414 |
local |
|
415 |
val rules = |
|
416 |
[compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE]; |
|
417 |
fun con_compact (con, args) = |
|
418 |
let |
|
26012 | 419 |
val concl = mk_trp (mk_compact (con_app con args)); |
420 |
val goal = lift (fn x => mk_compact (%#x)) (args, concl); |
|
23152 | 421 |
val tacs = [ |
422 |
rtac (iso_locale RS iso_compact_abs) 1, |
|
423 |
REPEAT (resolve_tac rules 1 ORELSE atac 1)]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
424 |
in pg con_appls goal (K tacs) end; |
23152 | 425 |
in |
29402 | 426 |
val _ = trace " Proving con_compacts..."; |
23152 | 427 |
val con_compacts = map con_compact cons; |
428 |
end; |
|
429 |
||
430 |
local |
|
431 |
fun one_sel sel = |
|
432 |
pg axs_sel_def (mk_trp (strict (%%:sel))) |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
433 |
(K [simp_tac (HOLCF_ss addsimps when_rews) 1]); |
23152 | 434 |
|
435 |
fun sel_strict (_, args) = |
|
436 |
List.mapPartial (Option.map one_sel o sel_of) args; |
|
437 |
in |
|
29402 | 438 |
val _ = trace " Proving sel_stricts..."; |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
439 |
val sel_stricts = maps sel_strict cons; |
23152 | 440 |
end; |
441 |
||
442 |
local |
|
443 |
fun sel_app_same c n sel (con, args) = |
|
444 |
let |
|
445 |
val nlas = nonlazy args; |
|
446 |
val vns = map vname args; |
|
447 |
val vnn = List.nth (vns, n); |
|
448 |
val nlas' = List.filter (fn v => v <> vnn) nlas; |
|
449 |
val lhs = (%%:sel)`(con_app con args); |
|
450 |
val goal = lift_defined %: (nlas', mk_trp (lhs === %:vnn)); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
451 |
fun tacs1 ctxt = |
23152 | 452 |
if vnn mem nlas |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
453 |
then [case_UU_tac ctxt (when_rews @ con_stricts) 1 vnn] |
23152 | 454 |
else []; |
455 |
val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
456 |
in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end; |
23152 | 457 |
|
458 |
fun sel_app_diff c n sel (con, args) = |
|
459 |
let |
|
460 |
val nlas = nonlazy args; |
|
461 |
val goal = mk_trp (%%:sel ` con_app con args === UU); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
462 |
fun tacs1 ctxt = map (case_UU_tac ctxt (when_rews @ con_stricts) 1) nlas; |
23152 | 463 |
val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
464 |
in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end; |
23152 | 465 |
|
466 |
fun sel_app c n sel (con, args) = |
|
467 |
if con = c |
|
468 |
then sel_app_same c n sel (con, args) |
|
469 |
else sel_app_diff c n sel (con, args); |
|
470 |
||
471 |
fun one_sel c n sel = map (sel_app c n sel) cons; |
|
472 |
fun one_sel' c n arg = Option.map (one_sel c n) (sel_of arg); |
|
473 |
fun one_con (c, args) = |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
474 |
flat (map_filter I (mapn (one_sel' c) 0 args)); |
23152 | 475 |
in |
29402 | 476 |
val _ = trace " Proving sel_apps..."; |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
477 |
val sel_apps = maps one_con cons; |
23152 | 478 |
end; |
479 |
||
480 |
local |
|
481 |
fun sel_defin sel = |
|
482 |
let |
|
483 |
val goal = defined (%:x_name) ==> defined (%%:sel`%x_name); |
|
484 |
val tacs = [ |
|
485 |
rtac casedist 1, |
|
486 |
contr_tac 1, |
|
487 |
DETERM_UNTIL_SOLVED (CHANGED |
|
488 |
(asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
489 |
in pg [] goal (K tacs) end; |
23152 | 490 |
in |
29402 | 491 |
val _ = trace " Proving sel_defins..."; |
23152 | 492 |
val sel_defins = |
493 |
if length cons = 1 |
|
494 |
then List.mapPartial (fn arg => Option.map sel_defin (sel_of arg)) |
|
495 |
(filter_out is_lazy (snd (hd cons))) |
|
496 |
else []; |
|
497 |
end; |
|
498 |
||
499 |
val sel_rews = sel_stricts @ sel_defins @ sel_apps; |
|
500 |
||
29402 | 501 |
val _ = trace " Proving dist_les..."; |
23152 | 502 |
val distincts_le = |
503 |
let |
|
504 |
fun dist (con1, args1) (con2, args2) = |
|
505 |
let |
|
506 |
val goal = lift_defined %: (nonlazy args1, |
|
507 |
mk_trp (con_app con1 args1 ~<< con_app con2 args2)); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
508 |
fun tacs ctxt = [ |
25805 | 509 |
rtac @{thm rev_contrapos} 1, |
27239 | 510 |
eres_inst_tac ctxt [(("f", 0), dis_name con1)] monofun_cfun_arg 1] |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
511 |
@ map (case_UU_tac ctxt (con_stricts @ dis_rews) 1) (nonlazy args2) |
23152 | 512 |
@ [asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]; |
513 |
in pg [] goal tacs end; |
|
514 |
||
515 |
fun distinct (con1, args1) (con2, args2) = |
|
516 |
let |
|
517 |
val arg1 = (con1, args1); |
|
518 |
val arg2 = |
|
519 |
(con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg) |
|
520 |
(args2, Name.variant_list (map vname args1) (map vname args2))); |
|
521 |
in [dist arg1 arg2, dist arg2 arg1] end; |
|
522 |
fun distincts [] = [] |
|
523 |
| distincts (c::cs) = (map (distinct c) cs) :: distincts cs; |
|
524 |
in distincts cons end; |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
525 |
val dist_les = flat (flat distincts_le); |
29402 | 526 |
|
527 |
val _ = trace " Proving dist_eqs..."; |
|
23152 | 528 |
val dist_eqs = |
529 |
let |
|
530 |
fun distinct (_,args1) ((_,args2), leqs) = |
|
531 |
let |
|
532 |
val (le1,le2) = (hd leqs, hd(tl leqs)); |
|
533 |
val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) |
|
534 |
in |
|
535 |
if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else |
|
536 |
if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else |
|
537 |
[eq1, eq2] |
|
538 |
end; |
|
539 |
fun distincts [] = [] |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
540 |
| distincts ((c,leqs)::cs) = flat |
23152 | 541 |
(ListPair.map (distinct c) ((map #1 cs),leqs)) @ |
542 |
distincts cs; |
|
543 |
in map standard (distincts (cons ~~ distincts_le)) end; |
|
544 |
||
545 |
local |
|
546 |
fun pgterm rel con args = |
|
547 |
let |
|
548 |
fun append s = upd_vname (fn v => v^s); |
|
549 |
val (largs, rargs) = (args, map (append "'") args); |
|
550 |
val concl = |
|
551 |
foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs)); |
|
552 |
val prem = rel (con_app con largs, con_app con rargs); |
|
553 |
val sargs = case largs of [_] => [] | _ => nonlazy args; |
|
554 |
val prop = lift_defined %: (sargs, mk_trp (prem === concl)); |
|
555 |
in pg con_appls prop end; |
|
556 |
val cons' = List.filter (fn (_,args) => args<>[]) cons; |
|
557 |
in |
|
29402 | 558 |
val _ = trace " Proving inverts..."; |
23152 | 559 |
val inverts = |
560 |
let |
|
561 |
val abs_less = ax_abs_iso RS (allI RS injection_less); |
|
562 |
val tacs = |
|
563 |
[asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
564 |
in map (fn (con, args) => pgterm (op <<) con args (K tacs)) cons' end; |
23152 | 565 |
|
29402 | 566 |
val _ = trace " Proving injects..."; |
23152 | 567 |
val injects = |
568 |
let |
|
569 |
val abs_eq = ax_abs_iso RS (allI RS injection_eq); |
|
570 |
val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
571 |
in map (fn (con, args) => pgterm (op ===) con args (K tacs)) cons' end; |
23152 | 572 |
end; |
573 |
||
574 |
(* ----- theorems concerning one induction step ----------------------------- *) |
|
575 |
||
576 |
val copy_strict = |
|
577 |
let |
|
578 |
val goal = mk_trp (strict (dc_copy `% "f")); |
|
579 |
val tacs = [asm_simp_tac (HOLCF_ss addsimps [abs_strict, when_strict]) 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
580 |
in pg [ax_copy_def] goal (K tacs) end; |
23152 | 581 |
|
582 |
local |
|
583 |
fun copy_app (con, args) = |
|
584 |
let |
|
585 |
val lhs = dc_copy`%"f"`(con_app con args); |
|
586 |
val rhs = con_app2 con (app_rec_arg (cproj (%:"f") eqs)) args; |
|
587 |
val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs)); |
|
588 |
val args' = List.filter (fn a => not (is_rec a orelse is_lazy a)) args; |
|
589 |
val stricts = abs_strict::when_strict::con_stricts; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
590 |
fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args'; |
23152 | 591 |
val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_apps) 1]; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
592 |
in pg [ax_copy_def] goal (fn ctxt => (tacs1 ctxt @ tacs2)) end; |
23152 | 593 |
in |
29402 | 594 |
val _ = trace " Proving copy_apps..."; |
23152 | 595 |
val copy_apps = map copy_app cons; |
596 |
end; |
|
597 |
||
598 |
local |
|
599 |
fun one_strict (con, args) = |
|
600 |
let |
|
601 |
val goal = mk_trp (dc_copy`UU`(con_app con args) === UU); |
|
602 |
val rews = copy_strict :: copy_apps @ con_rews; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
603 |
fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @ |
23152 | 604 |
[asm_simp_tac (HOLCF_ss addsimps rews) 1]; |
605 |
in pg [] goal tacs end; |
|
606 |
||
607 |
fun has_nonlazy_rec (_, args) = exists is_nonlazy_rec args; |
|
608 |
in |
|
29402 | 609 |
val _ = trace " Proving copy_stricts..."; |
23152 | 610 |
val copy_stricts = map one_strict (List.filter has_nonlazy_rec cons); |
611 |
end; |
|
612 |
||
613 |
val copy_rews = copy_strict :: copy_apps @ copy_stricts; |
|
614 |
||
615 |
in |
|
616 |
thy |
|
30364
577edc39b501
moved basic algebra of long names from structure NameSpace to Long_Name;
wenzelm
parents:
30280
diff
changeset
|
617 |
|> Sign.add_path (Long_Name.base_name dname) |
31004 | 618 |
|> snd o PureThy.add_thmss [ |
619 |
((Binding.name "iso_rews" , iso_rews ), [Simplifier.simp_add]), |
|
620 |
((Binding.name "exhaust" , [exhaust] ), []), |
|
621 |
((Binding.name "casedist" , [casedist] ), [Induct.cases_type dname]), |
|
622 |
((Binding.name "when_rews" , when_rews ), [Simplifier.simp_add]), |
|
623 |
((Binding.name "compacts" , con_compacts), [Simplifier.simp_add]), |
|
624 |
((Binding.name "con_rews" , con_rews ), [Simplifier.simp_add]), |
|
625 |
((Binding.name "sel_rews" , sel_rews ), [Simplifier.simp_add]), |
|
626 |
((Binding.name "dis_rews" , dis_rews ), [Simplifier.simp_add]), |
|
627 |
((Binding.name "pat_rews" , pat_rews ), [Simplifier.simp_add]), |
|
628 |
((Binding.name "dist_les" , dist_les ), [Simplifier.simp_add]), |
|
629 |
((Binding.name "dist_eqs" , dist_eqs ), [Simplifier.simp_add]), |
|
630 |
((Binding.name "inverts" , inverts ), [Simplifier.simp_add]), |
|
631 |
((Binding.name "injects" , injects ), [Simplifier.simp_add]), |
|
632 |
((Binding.name "copy_rews" , copy_rews ), [Simplifier.simp_add]), |
|
633 |
((Binding.name "match_rews", mat_rews ), [Simplifier.simp_add])] |
|
24712
64ed05609568
proper Sign operations instead of Theory aliases;
wenzelm
parents:
24503
diff
changeset
|
634 |
|> Sign.parent_path |
28536 | 635 |
|> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @ |
23152 | 636 |
pat_rews @ dist_les @ dist_eqs @ copy_rews) |
637 |
end; (* let *) |
|
638 |
||
639 |
fun comp_theorems (comp_dnam, eqs: eq list) thy = |
|
640 |
let |
|
27232 | 641 |
val global_ctxt = ProofContext.init thy; |
642 |
||
23152 | 643 |
val dnames = map (fst o fst) eqs; |
644 |
val conss = map snd eqs; |
|
28965 | 645 |
val comp_dname = Sign.full_bname thy comp_dnam; |
23152 | 646 |
|
29402 | 647 |
val _ = message ("Proving induction properties of domain "^comp_dname^" ..."); |
23152 | 648 |
val pg = pg' thy; |
649 |
||
650 |
(* ----- getting the composite axiom and definitions ------------------------ *) |
|
651 |
||
652 |
local |
|
26343
0dd2eab7b296
simplified get_thm(s): back to plain name argument;
wenzelm
parents:
26342
diff
changeset
|
653 |
fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s); |
23152 | 654 |
in |
655 |
val axs_reach = map (ga "reach" ) dnames; |
|
656 |
val axs_take_def = map (ga "take_def" ) dnames; |
|
657 |
val axs_finite_def = map (ga "finite_def") dnames; |
|
658 |
val ax_copy2_def = ga "copy_def" comp_dnam; |
|
659 |
val ax_bisim_def = ga "bisim_def" comp_dnam; |
|
660 |
end; |
|
661 |
||
662 |
local |
|
26343
0dd2eab7b296
simplified get_thm(s): back to plain name argument;
wenzelm
parents:
26342
diff
changeset
|
663 |
fun gt s dn = PureThy.get_thm thy (dn ^ "." ^ s); |
0dd2eab7b296
simplified get_thm(s): back to plain name argument;
wenzelm
parents:
26342
diff
changeset
|
664 |
fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s); |
23152 | 665 |
in |
666 |
val cases = map (gt "casedist" ) dnames; |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
667 |
val con_rews = maps (gts "con_rews" ) dnames; |
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
668 |
val copy_rews = maps (gts "copy_rews") dnames; |
23152 | 669 |
end; |
670 |
||
671 |
fun dc_take dn = %%:(dn^"_take"); |
|
672 |
val x_name = idx_name dnames "x"; |
|
673 |
val P_name = idx_name dnames "P"; |
|
674 |
val n_eqs = length eqs; |
|
675 |
||
676 |
(* ----- theorems concerning finite approximation and finite induction ------ *) |
|
677 |
||
678 |
local |
|
26342 | 679 |
val iterate_Cprod_ss = simpset_of @{theory Fix}; |
23152 | 680 |
val copy_con_rews = copy_rews @ con_rews; |
681 |
val copy_take_defs = |
|
682 |
(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def; |
|
29402 | 683 |
val _ = trace " Proving take_stricts..."; |
23152 | 684 |
val take_stricts = |
685 |
let |
|
686 |
fun one_eq ((dn, args), _) = strict (dc_take dn $ %:"n"); |
|
687 |
val goal = mk_trp (foldr1 mk_conj (map one_eq eqs)); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
688 |
fun tacs ctxt = [ |
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
689 |
InductTacs.induct_tac ctxt [[SOME "n"]] 1, |
23152 | 690 |
simp_tac iterate_Cprod_ss 1, |
691 |
asm_simp_tac (iterate_Cprod_ss addsimps copy_rews) 1]; |
|
692 |
in pg copy_take_defs goal tacs end; |
|
693 |
||
694 |
val take_stricts' = rewrite_rule copy_take_defs take_stricts; |
|
695 |
fun take_0 n dn = |
|
696 |
let |
|
697 |
val goal = mk_trp ((dc_take dn $ %%:"HOL.zero") `% x_name n === UU); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
698 |
in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end; |
23152 | 699 |
val take_0s = mapn take_0 1 dnames; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
700 |
fun c_UU_tac ctxt = case_UU_tac ctxt (take_stricts'::copy_con_rews) 1; |
29402 | 701 |
val _ = trace " Proving take_apps..."; |
23152 | 702 |
val take_apps = |
703 |
let |
|
704 |
fun mk_eqn dn (con, args) = |
|
705 |
let |
|
706 |
fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n"; |
|
707 |
val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args); |
|
708 |
val rhs = con_app2 con (app_rec_arg mk_take) args; |
|
709 |
in Library.foldr mk_all (map vname args, lhs === rhs) end; |
|
710 |
fun mk_eqns ((dn, _), cons) = map (mk_eqn dn) cons; |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
711 |
val goal = mk_trp (foldr1 mk_conj (maps mk_eqns eqs)); |
23152 | 712 |
val simps = List.filter (has_fewer_prems 1) copy_rews; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
713 |
fun con_tac ctxt (con, args) = |
23152 | 714 |
if nonlazy_rec args = [] |
715 |
then all_tac |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
716 |
else EVERY (map (c_UU_tac ctxt) (nonlazy_rec args)) THEN |
23152 | 717 |
asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
718 |
fun eq_tacs ctxt ((dn, _), cons) = map (con_tac ctxt) cons; |
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
719 |
fun tacs ctxt = |
23152 | 720 |
simp_tac iterate_Cprod_ss 1 :: |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
721 |
InductTacs.induct_tac ctxt [[SOME "n"]] 1 :: |
23152 | 722 |
simp_tac (iterate_Cprod_ss addsimps copy_con_rews) 1 :: |
723 |
asm_full_simp_tac (HOLCF_ss addsimps simps) 1 :: |
|
724 |
TRY (safe_tac HOL_cs) :: |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
725 |
maps (eq_tacs ctxt) eqs; |
23152 | 726 |
in pg copy_take_defs goal tacs end; |
727 |
in |
|
728 |
val take_rews = map standard |
|
27232 | 729 |
(atomize global_ctxt take_stricts @ take_0s @ atomize global_ctxt take_apps); |
23152 | 730 |
end; (* local *) |
731 |
||
732 |
local |
|
733 |
fun one_con p (con,args) = |
|
734 |
let |
|
735 |
fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg; |
|
736 |
val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args); |
|
737 |
val t2 = lift ind_hyp (List.filter is_rec args, t1); |
|
738 |
val t3 = lift_defined (bound_arg (map vname args)) (nonlazy args, t2); |
|
739 |
in Library.foldr mk_All (map vname args, t3) end; |
|
740 |
||
741 |
fun one_eq ((p, cons), concl) = |
|
742 |
mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl); |
|
743 |
||
744 |
fun ind_term concf = Library.foldr one_eq |
|
745 |
(mapn (fn n => fn x => (P_name n, x)) 1 conss, |
|
746 |
mk_trp (foldr1 mk_conj (mapn concf 1 dnames))); |
|
747 |
val take_ss = HOL_ss addsimps take_rews; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
748 |
fun quant_tac ctxt i = EVERY |
27239 | 749 |
(mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames); |
23152 | 750 |
|
751 |
fun ind_prems_tac prems = EVERY |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
752 |
(maps (fn cons => |
23152 | 753 |
(resolve_tac prems 1 :: |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
754 |
maps (fn (_,args) => |
23152 | 755 |
resolve_tac prems 1 :: |
756 |
map (K(atac 1)) (nonlazy args) @ |
|
757 |
map (K(atac 1)) (List.filter is_rec args)) |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
758 |
cons)) |
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
759 |
conss); |
23152 | 760 |
local |
761 |
(* check whether every/exists constructor of the n-th part of the equation: |
|
762 |
it has a possibly indirectly recursive argument that isn't/is possibly |
|
763 |
indirectly lazy *) |
|
764 |
fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => |
|
765 |
is_rec arg andalso not(rec_of arg mem ns) andalso |
|
766 |
((rec_of arg = n andalso nfn(lazy_rec orelse is_lazy arg)) orelse |
|
767 |
rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) |
|
768 |
(lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg)))) |
|
769 |
) o snd) cons; |
|
770 |
fun all_rec_to ns = rec_to forall not all_rec_to ns; |
|
771 |
fun warn (n,cons) = |
|
772 |
if all_rec_to [] false (n,cons) |
|
773 |
then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true) |
|
774 |
else false; |
|
775 |
fun lazy_rec_to ns = rec_to exists I lazy_rec_to ns; |
|
776 |
||
777 |
in |
|
778 |
val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs; |
|
779 |
val is_emptys = map warn n__eqs; |
|
780 |
val is_finite = forall (not o lazy_rec_to [] false) n__eqs; |
|
781 |
end; |
|
782 |
in (* local *) |
|
29402 | 783 |
val _ = trace " Proving finite_ind..."; |
23152 | 784 |
val finite_ind = |
785 |
let |
|
786 |
fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n)); |
|
787 |
val goal = ind_term concf; |
|
788 |
||
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
789 |
fun tacf {prems, context} = |
23152 | 790 |
let |
791 |
val tacs1 = [ |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
792 |
quant_tac context 1, |
23152 | 793 |
simp_tac HOL_ss 1, |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
794 |
InductTacs.induct_tac context [[SOME "n"]] 1, |
23152 | 795 |
simp_tac (take_ss addsimps prems) 1, |
796 |
TRY (safe_tac HOL_cs)]; |
|
797 |
fun arg_tac arg = |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
798 |
case_UU_tac context (prems @ con_rews) 1 |
23152 | 799 |
(List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg); |
800 |
fun con_tacs (con, args) = |
|
801 |
asm_simp_tac take_ss 1 :: |
|
802 |
map arg_tac (List.filter is_nonlazy_rec args) @ |
|
803 |
[resolve_tac prems 1] @ |
|
804 |
map (K (atac 1)) (nonlazy args) @ |
|
805 |
map (K (etac spec 1)) (List.filter is_rec args); |
|
806 |
fun cases_tacs (cons, cases) = |
|
27239 | 807 |
res_inst_tac context [(("x", 0), "x")] cases 1 :: |
23152 | 808 |
asm_simp_tac (take_ss addsimps prems) 1 :: |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
809 |
maps con_tacs cons; |
23152 | 810 |
in |
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
811 |
tacs1 @ maps cases_tacs (conss ~~ cases) |
23152 | 812 |
end; |
813 |
in pg'' thy [] goal tacf end; |
|
814 |
||
29402 | 815 |
val _ = trace " Proving take_lemmas..."; |
23152 | 816 |
val take_lemmas = |
817 |
let |
|
818 |
fun take_lemma n (dn, ax_reach) = |
|
819 |
let |
|
820 |
val lhs = dc_take dn $ Bound 0 `%(x_name n); |
|
821 |
val rhs = dc_take dn $ Bound 0 `%(x_name n^"'"); |
|
822 |
val concl = mk_trp (%:(x_name n) === %:(x_name n^"'")); |
|
823 |
val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
824 |
fun tacf {prems, context} = [ |
27239 | 825 |
res_inst_tac context [(("t", 0), x_name n )] (ax_reach RS subst) 1, |
826 |
res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1, |
|
23152 | 827 |
stac fix_def2 1, |
828 |
REPEAT (CHANGED |
|
829 |
(rtac (contlub_cfun_arg RS ssubst) 1 THEN chain_tac 1)), |
|
830 |
stac contlub_cfun_fun 1, |
|
831 |
stac contlub_cfun_fun 2, |
|
832 |
rtac lub_equal 3, |
|
833 |
chain_tac 1, |
|
834 |
rtac allI 1, |
|
835 |
resolve_tac prems 1]; |
|
836 |
in pg'' thy axs_take_def goal tacf end; |
|
837 |
in mapn take_lemma 1 (dnames ~~ axs_reach) end; |
|
838 |
||
839 |
(* ----- theorems concerning finiteness and induction ----------------------- *) |
|
840 |
||
29402 | 841 |
val _ = trace " Proving finites, ind..."; |
23152 | 842 |
val (finites, ind) = |
843 |
if is_finite |
|
844 |
then (* finite case *) |
|
845 |
let |
|
846 |
fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x"); |
|
847 |
fun dname_lemma dn = |
|
848 |
let |
|
849 |
val prem1 = mk_trp (defined (%:"x")); |
|
850 |
val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU); |
|
851 |
val prem2 = mk_trp (mk_disj (disj1, take_enough dn)); |
|
852 |
val concl = mk_trp (take_enough dn); |
|
853 |
val goal = prem1 ===> prem2 ===> concl; |
|
854 |
val tacs = [ |
|
855 |
etac disjE 1, |
|
856 |
etac notE 1, |
|
857 |
resolve_tac take_lemmas 1, |
|
858 |
asm_simp_tac take_ss 1, |
|
859 |
atac 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
860 |
in pg [] goal (K tacs) end; |
23152 | 861 |
val finite_lemmas1a = map dname_lemma dnames; |
862 |
||
863 |
val finite_lemma1b = |
|
864 |
let |
|
865 |
fun mk_eqn n ((dn, args), _) = |
|
866 |
let |
|
867 |
val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU; |
|
868 |
val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0; |
|
869 |
in |
|
870 |
mk_constrainall |
|
871 |
(x_name n, Type (dn,args), mk_disj (disj1, disj2)) |
|
872 |
end; |
|
873 |
val goal = |
|
874 |
mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs))); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
875 |
fun arg_tacs ctxt vn = [ |
27239 | 876 |
eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1, |
23152 | 877 |
etac disjE 1, |
878 |
asm_simp_tac (HOL_ss addsimps con_rews) 1, |
|
879 |
asm_simp_tac take_ss 1]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
880 |
fun con_tacs ctxt (con, args) = |
23152 | 881 |
asm_simp_tac take_ss 1 :: |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
882 |
maps (arg_tacs ctxt) (nonlazy_rec args); |
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
883 |
fun foo_tacs ctxt n (cons, cases) = |
23152 | 884 |
simp_tac take_ss 1 :: |
885 |
rtac allI 1 :: |
|
27239 | 886 |
res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 :: |
23152 | 887 |
asm_simp_tac take_ss 1 :: |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
888 |
maps (con_tacs ctxt) cons; |
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
889 |
fun tacs ctxt = |
23152 | 890 |
rtac allI 1 :: |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
891 |
InductTacs.induct_tac ctxt [[SOME "n"]] 1 :: |
23152 | 892 |
simp_tac take_ss 1 :: |
893 |
TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) :: |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
894 |
flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases)); |
23152 | 895 |
in pg [] goal tacs end; |
896 |
||
897 |
fun one_finite (dn, l1b) = |
|
898 |
let |
|
899 |
val goal = mk_trp (%%:(dn^"_finite") $ %:"x"); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
900 |
fun tacs ctxt = [ |
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
901 |
case_UU_tac ctxt take_rews 1 "x", |
23152 | 902 |
eresolve_tac finite_lemmas1a 1, |
903 |
step_tac HOL_cs 1, |
|
904 |
step_tac HOL_cs 1, |
|
905 |
cut_facts_tac [l1b] 1, |
|
906 |
fast_tac HOL_cs 1]; |
|
907 |
in pg axs_finite_def goal tacs end; |
|
908 |
||
27232 | 909 |
val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b); |
23152 | 910 |
val ind = |
911 |
let |
|
912 |
fun concf n dn = %:(P_name n) $ %:(x_name n); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
913 |
fun tacf {prems, context} = |
23152 | 914 |
let |
915 |
fun finite_tacs (finite, fin_ind) = [ |
|
916 |
rtac(rewrite_rule axs_finite_def finite RS exE)1, |
|
917 |
etac subst 1, |
|
918 |
rtac fin_ind 1, |
|
919 |
ind_prems_tac prems]; |
|
920 |
in |
|
921 |
TRY (safe_tac HOL_cs) :: |
|
27232 | 922 |
maps finite_tacs (finites ~~ atomize global_ctxt finite_ind) |
23152 | 923 |
end; |
924 |
in pg'' thy [] (ind_term concf) tacf end; |
|
925 |
in (finites, ind) end (* let *) |
|
926 |
||
927 |
else (* infinite case *) |
|
928 |
let |
|
929 |
fun one_finite n dn = |
|
27239 | 930 |
read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle; |
23152 | 931 |
val finites = mapn one_finite 1 dnames; |
932 |
||
933 |
val goal = |
|
934 |
let |
|
26012 | 935 |
fun one_adm n _ = mk_trp (mk_adm (%:(P_name n))); |
23152 | 936 |
fun concf n dn = %:(P_name n) $ %:(x_name n); |
937 |
in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
938 |
fun tacf {prems, context} = |
23152 | 939 |
map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [ |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
940 |
quant_tac context 1, |
23152 | 941 |
rtac (adm_impl_admw RS wfix_ind) 1, |
25895
0eaadfa8889e
converted adm_all and adm_ball to rule_format; cleaned up
huffman
parents:
25805
diff
changeset
|
942 |
REPEAT_DETERM (rtac adm_all 1), |
23152 | 943 |
REPEAT_DETERM ( |
944 |
TRY (rtac adm_conj 1) THEN |
|
945 |
rtac adm_subst 1 THEN |
|
946 |
cont_tacR 1 THEN resolve_tac prems 1), |
|
947 |
strip_tac 1, |
|
948 |
rtac (rewrite_rule axs_take_def finite_ind) 1, |
|
949 |
ind_prems_tac prems]; |
|
950 |
val ind = (pg'' thy [] goal tacf |
|
951 |
handle ERROR _ => |
|
952 |
(warning "Cannot prove infinite induction rule"; refl)); |
|
953 |
in (finites, ind) end; |
|
954 |
end; (* local *) |
|
955 |
||
956 |
(* ----- theorem concerning coinduction ------------------------------------- *) |
|
957 |
||
958 |
local |
|
959 |
val xs = mapn (fn n => K (x_name n)) 1 dnames; |
|
960 |
fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1); |
|
961 |
val take_ss = HOL_ss addsimps take_rews; |
|
962 |
val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")")); |
|
29402 | 963 |
val _ = trace " Proving coind_lemma..."; |
23152 | 964 |
val coind_lemma = |
965 |
let |
|
966 |
fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1; |
|
967 |
fun mk_eqn n dn = |
|
968 |
(dc_take dn $ %:"n" ` bnd_arg n 0) === |
|
969 |
(dc_take dn $ %:"n" ` bnd_arg n 1); |
|
970 |
fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t)); |
|
971 |
val goal = |
|
972 |
mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R", |
|
973 |
Library.foldr mk_all2 (xs, |
|
974 |
Library.foldr mk_imp (mapn mk_prj 0 dnames, |
|
975 |
foldr1 mk_conj (mapn mk_eqn 0 dnames))))); |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
976 |
fun x_tacs ctxt n x = [ |
23152 | 977 |
rotate_tac (n+1) 1, |
978 |
etac all2E 1, |
|
27239 | 979 |
eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1, |
23152 | 980 |
TRY (safe_tac HOL_cs), |
981 |
REPEAT (CHANGED (asm_simp_tac take_ss 1))]; |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
982 |
fun tacs ctxt = [ |
23152 | 983 |
rtac impI 1, |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
984 |
InductTacs.induct_tac ctxt [[SOME "n"]] 1, |
23152 | 985 |
simp_tac take_ss 1, |
986 |
safe_tac HOL_cs] @ |
|
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
987 |
flat (mapn (x_tacs ctxt) 0 xs); |
23152 | 988 |
in pg [ax_bisim_def] goal tacs end; |
989 |
in |
|
29402 | 990 |
val _ = trace " Proving coind..."; |
23152 | 991 |
val coind = |
992 |
let |
|
993 |
fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'")); |
|
994 |
fun mk_eqn x = %:x === %:(x^"'"); |
|
995 |
val goal = |
|
996 |
mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===> |
|
997 |
Logic.list_implies (mapn mk_prj 0 xs, |
|
998 |
mk_trp (foldr1 mk_conj (map mk_eqn xs))); |
|
999 |
val tacs = |
|
1000 |
TRY (safe_tac HOL_cs) :: |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
1001 |
maps (fn take_lemma => [ |
23152 | 1002 |
rtac take_lemma 1, |
1003 |
cut_facts_tac [coind_lemma] 1, |
|
1004 |
fast_tac HOL_cs 1]) |
|
26336
a0e2b706ce73
renamed datatype thmref to Facts.ref, tuned interfaces;
wenzelm
parents:
26012
diff
changeset
|
1005 |
take_lemmas; |
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
27153
diff
changeset
|
1006 |
in pg [] goal (K tacs) end; |
23152 | 1007 |
end; (* local *) |
1008 |
||
30829 | 1009 |
val inducts = ProjectRule.projections (ProofContext.init thy) ind; |
1010 |
fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]); |
|
1011 |
||
24712
64ed05609568
proper Sign operations instead of Theory aliases;
wenzelm
parents:
24503
diff
changeset
|
1012 |
in thy |> Sign.add_path comp_dnam |
31004 | 1013 |
|> snd o PureThy.add_thmss [ |
1014 |
((Binding.name "take_rews" , take_rews ), [Simplifier.simp_add]), |
|
1015 |
((Binding.name "take_lemmas", take_lemmas ), []), |
|
1016 |
((Binding.name "finites" , finites ), []), |
|
1017 |
((Binding.name "finite_ind" , [finite_ind]), []), |
|
1018 |
((Binding.name "ind" , [ind] ), []), |
|
1019 |
((Binding.name "coind" , [coind] ), [])] |
|
1020 |
|> snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)) |
|
28536 | 1021 |
|> Sign.parent_path |> pair take_rews |
23152 | 1022 |
end; (* let *) |
1023 |
end; (* local *) |
|
1024 |
end; (* struct *) |