author | huffman |
Sat, 13 Mar 2010 16:48:57 -0800 | |
changeset 35776 | b0bc15d8ad3b |
parent 35775 | 9b7e2e17be69 |
child 35777 | bcc77916b7b9 |
permissions | -rw-r--r-- |
32126 | 1 |
(* Title: HOLCF/Tools/Domain/domain_theorems.ML |
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Author: David von Oheimb |
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Author: Brian Huffman |
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Proof generator for domain command. |
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*) |
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||
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val HOLCF_ss = @{simpset}; |
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signature DOMAIN_THEOREMS = |
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sig |
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val theorems: |
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Domain_Library.eq * Domain_Library.eq list -> |
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binding -> |
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(binding * (bool * binding option * typ) list * mixfix) list -> |
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Domain_Take_Proofs.iso_info -> |
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Domain_Take_Proofs.take_induct_info -> |
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theory -> thm list * theory; |
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val comp_theorems : |
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binding * Domain_Library.eq list -> |
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Domain_Take_Proofs.take_induct_info -> |
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theory -> thm list * theory |
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val quiet_mode: bool Unsynchronized.ref; |
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val trace_domain: bool Unsynchronized.ref; |
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end; |
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structure Domain_Theorems :> DOMAIN_THEOREMS = |
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struct |
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val quiet_mode = Unsynchronized.ref false; |
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val trace_domain = Unsynchronized.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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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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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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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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fun tac {prems, context} = |
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rewrite_goals_tac defs THEN |
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EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context}) |
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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; |
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(* FIXME!!!!!!!!! *) |
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(* We should NEVER re-parse variable names as strings! *) |
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(* The names can conflict with existing constants or other syntax! *) |
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fun case_UU_tac ctxt rews i v = |
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InductTacs.case_tac ctxt (v^"=UU") i THEN |
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asm_simp_tac (HOLCF_ss addsimps rews) i; |
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(* ----- general proofs ----------------------------------------------------- *) |
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val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp} |
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fun theorems |
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(((dname, _), cons) : eq, eqs : eq list) |
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(dbind : binding) |
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(spec : (binding * (bool * binding option * typ) list * mixfix) list) |
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(iso_info : Domain_Take_Proofs.iso_info) |
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(take_info : Domain_Take_Proofs.take_induct_info) |
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(thy : theory) = |
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let |
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val _ = message ("Proving isomorphism properties of domain "^dname^" ..."); |
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val map_tab = Domain_Take_Proofs.get_map_tab thy; |
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(* ----- getting the axioms and definitions --------------------------------- *) |
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val ax_abs_iso = #abs_inverse iso_info; |
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val ax_rep_iso = #rep_inverse iso_info; |
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val abs_const = #abs_const iso_info; |
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val rep_const = #rep_const iso_info; |
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local |
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fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s); |
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in |
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val ax_take_0 = ga "take_0" dname; |
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val ax_take_strict = ga "take_strict" dname; |
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end; (* local *) |
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val {take_Suc_thms, deflation_take_thms, ...} = take_info; |
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(* ----- define constructors ------------------------------------------------ *) |
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val (result, thy) = |
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Domain_Constructors.add_domain_constructors |
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(Long_Name.base_name dname) spec iso_info thy; |
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val con_appls = #con_betas result; |
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val {exhaust, casedist, ...} = result; |
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val {con_compacts, con_rews, inverts, injects, dist_les, dist_eqs, ...} = result; |
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val {sel_rews, ...} = result; |
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val when_rews = #cases result; |
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val when_strict = hd when_rews; |
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val dis_rews = #dis_rews result; |
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val mat_rews = #match_rews result; |
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val pat_rews = #pat_rews result; |
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(* ----- theorems concerning the isomorphism -------------------------------- *) |
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val pg = pg' thy; |
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val retraction_strict = @{thm retraction_strict}; |
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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 iso_rews = map Drule.export_without_context [ax_abs_iso, ax_rep_iso, abs_strict, rep_strict]; |
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(* ----- theorems concerning one induction step ----------------------------- *) |
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local |
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fun dc_take dn = %%:(dn^"_take"); |
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val dnames = map (fst o fst) eqs; |
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val deflation_thms = Domain_Take_Proofs.get_deflation_thms thy; |
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fun copy_of_dtyp tab r dt = |
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if Datatype_Aux.is_rec_type dt then copy tab r dt else ID |
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and copy tab r (Datatype_Aux.DtRec i) = r i |
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| copy tab r (Datatype_Aux.DtTFree a) = ID |
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| copy tab r (Datatype_Aux.DtType (c, ds)) = |
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case Symtab.lookup tab c of |
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SOME f => list_ccomb (%%:f, map (copy_of_dtyp tab r) ds) |
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| NONE => (warning ("copy_of_dtyp: unknown type constructor " ^ c); ID); |
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fun one_take_app (con, args) = |
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let |
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fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n"; |
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fun one_rhs arg = |
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if Datatype_Aux.is_rec_type (dtyp_of arg) |
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then copy_of_dtyp map_tab |
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mk_take (dtyp_of arg) ` (%# arg) |
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else (%# arg); |
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val lhs = (dc_take dname $ (%%:"Suc" $ %:"n"))`(con_app con args); |
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val rhs = con_app2 con one_rhs args; |
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val goal = mk_trp (lhs === rhs); |
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val rules = |
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[ax_abs_iso] |
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@ @{thms take_con_rules ID1 cfcomp2 deflation_strict} |
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@ take_Suc_thms @ deflation_thms @ deflation_take_thms; |
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val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1]; |
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in pg con_appls goal (K tacs) end; |
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val take_apps = map one_take_app cons; |
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in |
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val take_rews = ax_take_0 :: ax_take_strict :: take_apps; |
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end; |
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val case_ns = |
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"bottom" :: map (fn (b,_,_) => Binding.name_of b) spec; |
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|
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fun qualified name = Binding.qualified true name dbind; |
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val simp = Simplifier.simp_add; |
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val fixrec_simp = Fixrec.fixrec_simp_add; |
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in |
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thy |
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|> PureThy.add_thmss [ |
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((qualified "iso_rews" , iso_rews ), [simp]), |
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((qualified "exhaust" , [exhaust] ), []), |
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((qualified "casedist" , [casedist] ), |
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[Rule_Cases.case_names case_ns, Induct.cases_type dname]), |
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((qualified "when_rews" , when_rews ), [simp]), |
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((qualified "compacts" , con_compacts), [simp]), |
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((qualified "con_rews" , con_rews ), [simp, fixrec_simp]), |
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((qualified "sel_rews" , sel_rews ), [simp]), |
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((qualified "dis_rews" , dis_rews ), [simp]), |
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((qualified "pat_rews" , pat_rews ), [simp]), |
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((qualified "dist_les" , dist_les ), [simp]), |
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((qualified "dist_eqs" , dist_eqs ), [simp]), |
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((qualified "inverts" , inverts ), [simp]), |
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((qualified "injects" , injects ), [simp]), |
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202 |
((qualified "take_rews" , take_rews ), [simp]), |
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((qualified "match_rews", mat_rews ), [simp, fixrec_simp])] |
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204 |
|> snd |
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|> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @ |
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pat_rews @ dist_les @ dist_eqs) |
23152 | 207 |
end; (* let *) |
208 |
||
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209 |
(******************************************************************************) |
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210 |
(****************************** induction rules *******************************) |
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211 |
(******************************************************************************) |
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212 |
|
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213 |
fun prove_induction |
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(comp_dbind : binding, eqs : eq list) |
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(take_rews : thm list) |
35659 | 216 |
(take_info : Domain_Take_Proofs.take_induct_info) |
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217 |
(thy : theory) = |
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let |
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val comp_dname = Sign.full_name thy comp_dbind; |
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val dnames = map (fst o fst) eqs; |
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val conss = map snd eqs; |
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fun dc_take dn = %%:(dn^"_take"); |
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val x_name = idx_name dnames "x"; |
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val P_name = idx_name dnames "P"; |
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225 |
val pg = pg' thy; |
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226 |
|
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227 |
local |
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228 |
fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s); |
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fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s); |
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230 |
in |
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val axs_rep_iso = map (ga "rep_iso") dnames; |
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val axs_abs_iso = map (ga "abs_iso") dnames; |
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233 |
val cases = map (ga "casedist" ) dnames; |
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val con_rews = maps (gts "con_rews" ) dnames; |
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235 |
end; |
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236 |
|
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237 |
val {take_consts, ...} = take_info; |
35659 | 238 |
val {take_0_thms, take_Suc_thms, chain_take_thms, ...} = take_info; |
35660 | 239 |
val {lub_take_thms, finite_defs, reach_thms, ...} = take_info; |
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val {take_induct_thms, ...} = take_info; |
35658 | 241 |
|
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242 |
fun one_con p (con, args) = |
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243 |
let |
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244 |
val P_names = map P_name (1 upto (length dnames)); |
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val vns = Name.variant_list P_names (map vname args); |
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246 |
val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns)); |
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247 |
fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg; |
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248 |
val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args); |
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249 |
val t2 = lift ind_hyp (filter is_rec args, t1); |
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250 |
val t3 = lift_defined (bound_arg vns) (nonlazy_vns, t2); |
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251 |
in Library.foldr mk_All (vns, t3) end; |
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252 |
|
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253 |
fun one_eq ((p, cons), concl) = |
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254 |
mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl); |
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255 |
|
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256 |
fun ind_term concf = Library.foldr one_eq |
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257 |
(mapn (fn n => fn x => (P_name n, x)) 1 conss, |
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258 |
mk_trp (foldr1 mk_conj (mapn concf 1 dnames))); |
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259 |
val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews); |
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260 |
fun quant_tac ctxt i = EVERY |
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261 |
(mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames); |
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262 |
|
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263 |
fun ind_prems_tac prems = EVERY |
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264 |
(maps (fn cons => |
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265 |
(resolve_tac prems 1 :: |
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266 |
maps (fn (_,args) => |
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267 |
resolve_tac prems 1 :: |
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268 |
map (K(atac 1)) (nonlazy args) @ |
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269 |
map (K(atac 1)) (filter is_rec args)) |
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270 |
cons)) |
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271 |
conss); |
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272 |
local |
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273 |
(* check whether every/exists constructor of the n-th part of the equation: |
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274 |
it has a possibly indirectly recursive argument that isn't/is possibly |
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275 |
indirectly lazy *) |
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276 |
fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => |
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277 |
is_rec arg andalso not(rec_of arg mem ns) andalso |
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278 |
((rec_of arg = n andalso nfn(lazy_rec orelse is_lazy arg)) orelse |
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279 |
rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) |
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280 |
(lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg)))) |
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281 |
) o snd) cons; |
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282 |
fun all_rec_to ns = rec_to forall not all_rec_to ns; |
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283 |
fun warn (n,cons) = |
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284 |
if all_rec_to [] false (n,cons) |
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285 |
then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true) |
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286 |
else false; |
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287 |
fun lazy_rec_to ns = rec_to exists I lazy_rec_to ns; |
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288 |
|
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289 |
in |
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290 |
val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs; |
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291 |
val is_emptys = map warn n__eqs; |
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292 |
val is_finite = #is_finite take_info; |
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293 |
val _ = if is_finite |
35774 | 294 |
then message ("Proving finiteness rule for domain "^comp_dname^" ...") |
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295 |
else (); |
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296 |
end; |
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297 |
val _ = trace " Proving finite_ind..."; |
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298 |
val finite_ind = |
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299 |
let |
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300 |
fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n)); |
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301 |
val goal = ind_term concf; |
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302 |
|
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303 |
fun tacf {prems, context} = |
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304 |
let |
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305 |
val tacs1 = [ |
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306 |
quant_tac context 1, |
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307 |
simp_tac HOL_ss 1, |
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308 |
InductTacs.induct_tac context [[SOME "n"]] 1, |
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309 |
simp_tac (take_ss addsimps prems) 1, |
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310 |
TRY (safe_tac HOL_cs)]; |
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|
311 |
fun arg_tac arg = |
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312 |
(* FIXME! case_UU_tac *) |
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313 |
case_UU_tac context (prems @ con_rews) 1 |
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|
314 |
(List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg); |
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315 |
fun con_tacs (con, args) = |
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|
316 |
asm_simp_tac take_ss 1 :: |
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317 |
map arg_tac (filter is_nonlazy_rec args) @ |
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318 |
[resolve_tac prems 1] @ |
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|
319 |
map (K (atac 1)) (nonlazy args) @ |
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320 |
map (K (etac spec 1)) (filter is_rec args); |
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|
321 |
fun cases_tacs (cons, cases) = |
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322 |
res_inst_tac context [(("y", 0), "x")] cases 1 :: |
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323 |
asm_simp_tac (take_ss addsimps prems) 1 :: |
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|
324 |
maps con_tacs cons; |
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325 |
in |
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|
326 |
tacs1 @ maps cases_tacs (conss ~~ cases) |
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|
327 |
end; |
35663 | 328 |
in pg'' thy [] goal tacf end; |
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|
329 |
|
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|
330 |
(* ----- theorems concerning finiteness and induction ----------------------- *) |
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|
331 |
|
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|
332 |
val global_ctxt = ProofContext.init thy; |
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|
333 |
|
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|
334 |
val _ = trace " Proving ind..."; |
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|
335 |
val ind = |
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|
336 |
if is_finite |
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|
337 |
then (* finite case *) |
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|
338 |
let |
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|
339 |
fun concf n dn = %:(P_name n) $ %:(x_name n); |
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|
340 |
fun tacf {prems, context} = |
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|
341 |
let |
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|
342 |
fun finite_tacs (take_induct, fin_ind) = [ |
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|
343 |
rtac take_induct 1, |
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|
344 |
rtac fin_ind 1, |
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|
345 |
ind_prems_tac prems]; |
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|
346 |
in |
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|
347 |
TRY (safe_tac HOL_cs) :: |
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|
348 |
maps finite_tacs (take_induct_thms ~~ atomize global_ctxt finite_ind) |
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|
349 |
end; |
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|
350 |
in pg'' thy [] (ind_term concf) tacf end |
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|
351 |
|
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|
352 |
else (* infinite case *) |
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|
353 |
let |
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|
354 |
val goal = |
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|
355 |
let |
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|
356 |
fun one_adm n _ = mk_trp (mk_adm (%:(P_name n))); |
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|
357 |
fun concf n dn = %:(P_name n) $ %:(x_name n); |
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|
358 |
in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end; |
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|
359 |
val cont_rules = |
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|
360 |
@{thms cont_id cont_const cont2cont_Rep_CFun |
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|
361 |
cont2cont_fst cont2cont_snd}; |
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|
362 |
val subgoal = |
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|
363 |
let |
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|
364 |
val Ts = map (Type o fst) eqs; |
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|
365 |
val P_names = Datatype_Prop.indexify_names (map (K "P") dnames); |
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|
366 |
val x_names = Datatype_Prop.indexify_names (map (K "x") dnames); |
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|
367 |
val P_types = map (fn T => T --> HOLogic.boolT) Ts; |
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|
368 |
val Ps = map Free (P_names ~~ P_types); |
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|
369 |
val xs = map Free (x_names ~~ Ts); |
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|
370 |
val n = Free ("n", HOLogic.natT); |
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|
371 |
val goals = |
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|
372 |
map (fn ((P,t),x) => P $ HOLCF_Library.mk_capply (t $ n, x)) |
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|
373 |
(Ps ~~ take_consts ~~ xs); |
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|
374 |
in |
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|
375 |
HOLogic.mk_Trueprop |
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|
376 |
(HOLogic.mk_all ("n", HOLogic.natT, foldr1 HOLogic.mk_conj goals)) |
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|
377 |
end; |
35585
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|
378 |
fun tacf {prems, context} = |
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|
379 |
let |
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|
380 |
val subtac = |
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|
381 |
EVERY [rtac allI 1, rtac finite_ind 1, ind_prems_tac prems]; |
35662
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|
382 |
val subthm = Goal.prove context [] [] subgoal (K subtac); |
35585
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changeset
|
383 |
in |
35660 | 384 |
map (fn ax_reach => rtac (ax_reach RS subst) 1) reach_thms @ [ |
35585
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|
385 |
cut_facts_tac (subthm :: take (length dnames) prems) 1, |
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changeset
|
386 |
REPEAT (rtac @{thm conjI} 1 ORELSE |
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|
387 |
EVERY [etac @{thm admD [OF _ ch2ch_Rep_CFunL]} 1, |
35659 | 388 |
resolve_tac chain_take_thms 1, |
35585
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|
389 |
asm_simp_tac HOL_basic_ss 1]) |
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|
390 |
] |
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|
391 |
end; |
35663 | 392 |
in pg'' thy [] goal tacf end; |
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changeset
|
393 |
|
35630
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changeset
|
394 |
val case_ns = |
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changeset
|
395 |
let |
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changeset
|
396 |
val bottoms = |
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changeset
|
397 |
if length dnames = 1 then ["bottom"] else |
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changeset
|
398 |
map (fn s => "bottom_" ^ Long_Name.base_name s) dnames; |
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changeset
|
399 |
fun one_eq bot (_,cons) = |
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changeset
|
400 |
bot :: map (fn (c,_) => Long_Name.base_name c) cons; |
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changeset
|
401 |
in flat (map2 one_eq bottoms eqs) end; |
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changeset
|
402 |
|
35585
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changeset
|
403 |
val inducts = Project_Rule.projections (ProofContext.init thy) ind; |
35630
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changeset
|
404 |
fun ind_rule (dname, rule) = |
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diff
changeset
|
405 |
((Binding.empty, [rule]), |
8e562d56d404
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changeset
|
406 |
[Rule_Cases.case_names case_ns, Induct.induct_type dname]); |
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changeset
|
407 |
|
35774 | 408 |
in |
409 |
thy |
|
410 |
|> snd o PureThy.add_thmss [ |
|
411 |
((Binding.qualified true "finite_ind" comp_dbind, [finite_ind]), []), |
|
412 |
((Binding.qualified true "ind" comp_dbind, [ind] ), [])] |
|
413 |
|> (snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts))) |
|
35585
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|
414 |
end; (* prove_induction *) |
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changeset
|
415 |
|
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|
416 |
(******************************************************************************) |
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|
417 |
(************************ bisimulation and coinduction ************************) |
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|
418 |
(******************************************************************************) |
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changeset
|
419 |
|
35574
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changeset
|
420 |
fun prove_coinduction |
35774 | 421 |
(comp_dbind : binding, eqs : eq list) |
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changeset
|
422 |
(take_lemmas : thm list) |
35599
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changeset
|
423 |
(thy : theory) : theory = |
23152 | 424 |
let |
27232 | 425 |
|
23152 | 426 |
val dnames = map (fst o fst) eqs; |
35774 | 427 |
val comp_dname = Sign.full_name thy comp_dbind; |
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changeset
|
428 |
fun dc_take dn = %%:(dn^"_take"); |
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diff
changeset
|
429 |
val x_name = idx_name dnames "x"; |
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changeset
|
430 |
val n_eqs = length eqs; |
23152 | 431 |
|
35574
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changeset
|
432 |
val take_rews = |
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changeset
|
433 |
maps (fn dn => PureThy.get_thms thy (dn ^ ".take_rews")) dnames; |
35497 | 434 |
|
435 |
(* ----- define bisimulation predicate -------------------------------------- *) |
|
436 |
||
437 |
local |
|
438 |
open HOLCF_Library |
|
439 |
val dtypes = map (Type o fst) eqs; |
|
440 |
val relprod = mk_tupleT (map (fn tp => tp --> tp --> boolT) dtypes); |
|
35774 | 441 |
val bisim_bind = Binding.suffix_name "_bisim" comp_dbind; |
35497 | 442 |
val bisim_type = relprod --> boolT; |
443 |
in |
|
444 |
val (bisim_const, thy) = |
|
445 |
Sign.declare_const ((bisim_bind, bisim_type), NoSyn) thy; |
|
446 |
end; |
|
447 |
||
448 |
local |
|
449 |
||
450 |
fun legacy_infer_term thy t = |
|
451 |
singleton (Syntax.check_terms (ProofContext.init thy)) (Sign.intern_term thy t); |
|
452 |
fun legacy_infer_prop thy t = legacy_infer_term thy (TypeInfer.constrain propT t); |
|
453 |
fun infer_props thy = map (apsnd (legacy_infer_prop thy)); |
|
454 |
fun add_defs_i x = PureThy.add_defs false (map Thm.no_attributes x); |
|
455 |
fun add_defs_infer defs thy = add_defs_i (infer_props thy defs) thy; |
|
456 |
||
35521 | 457 |
fun one_con (con, args) = |
35497 | 458 |
let |
459 |
val nonrec_args = filter_out is_rec args; |
|
460 |
val rec_args = filter is_rec args; |
|
461 |
val recs_cnt = length rec_args; |
|
462 |
val allargs = nonrec_args @ rec_args |
|
463 |
@ map (upd_vname (fn s=> s^"'")) rec_args; |
|
464 |
val allvns = map vname allargs; |
|
465 |
fun vname_arg s arg = if is_rec arg then vname arg^s else vname arg; |
|
466 |
val vns1 = map (vname_arg "" ) args; |
|
467 |
val vns2 = map (vname_arg "'") args; |
|
468 |
val allargs_cnt = length nonrec_args + 2*recs_cnt; |
|
469 |
val rec_idxs = (recs_cnt-1) downto 0; |
|
470 |
val nonlazy_idxs = map snd (filter_out (fn (arg,_) => is_lazy arg) |
|
471 |
(allargs~~((allargs_cnt-1) downto 0))); |
|
472 |
fun rel_app i ra = proj (Bound(allargs_cnt+2)) eqs (rec_of ra) $ |
|
473 |
Bound (2*recs_cnt-i) $ Bound (recs_cnt-i); |
|
474 |
val capps = |
|
475 |
List.foldr |
|
476 |
mk_conj |
|
477 |
(mk_conj( |
|
478 |
Bound(allargs_cnt+1)===list_ccomb(%%:con,map (bound_arg allvns) vns1), |
|
479 |
Bound(allargs_cnt+0)===list_ccomb(%%:con,map (bound_arg allvns) vns2))) |
|
480 |
(mapn rel_app 1 rec_args); |
|
481 |
in |
|
482 |
List.foldr |
|
483 |
mk_ex |
|
484 |
(Library.foldr mk_conj |
|
485 |
(map (defined o Bound) nonlazy_idxs,capps)) allvns |
|
486 |
end; |
|
487 |
fun one_comp n (_,cons) = |
|
488 |
mk_all (x_name(n+1), |
|
489 |
mk_all (x_name(n+1)^"'", |
|
490 |
mk_imp (proj (Bound 2) eqs n $ Bound 1 $ Bound 0, |
|
491 |
foldr1 mk_disj (mk_conj(Bound 1 === UU,Bound 0 === UU) |
|
492 |
::map one_con cons)))); |
|
493 |
val bisim_eqn = |
|
494 |
%%:(comp_dname^"_bisim") == |
|
495 |
mk_lam("R", foldr1 mk_conj (mapn one_comp 0 eqs)); |
|
496 |
||
497 |
in |
|
35774 | 498 |
val (ax_bisim_def, thy) = |
499 |
yield_singleton add_defs_infer |
|
500 |
(Binding.qualified true "bisim_def" comp_dbind, bisim_eqn) thy; |
|
35497 | 501 |
end; (* local *) |
502 |
||
35574
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huffman
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35560
diff
changeset
|
503 |
(* ----- theorem concerning coinduction ------------------------------------- *) |
ee5df989b7c4
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huffman
parents:
35560
diff
changeset
|
504 |
|
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
505 |
local |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
506 |
val pg = pg' thy; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
507 |
val xs = mapn (fn n => K (x_name n)) 1 dnames; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
508 |
fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
509 |
val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
510 |
val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")")); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
511 |
val _ = trace " Proving coind_lemma..."; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
512 |
val coind_lemma = |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
513 |
let |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
514 |
fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
515 |
fun mk_eqn n dn = |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
516 |
(dc_take dn $ %:"n" ` bnd_arg n 0) === |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
517 |
(dc_take dn $ %:"n" ` bnd_arg n 1); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
518 |
fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t)); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
519 |
val goal = |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
520 |
mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R", |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
521 |
Library.foldr mk_all2 (xs, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
522 |
Library.foldr mk_imp (mapn mk_prj 0 dnames, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
523 |
foldr1 mk_conj (mapn mk_eqn 0 dnames))))); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
524 |
fun x_tacs ctxt n x = [ |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
525 |
rotate_tac (n+1) 1, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
526 |
etac all2E 1, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
527 |
eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
528 |
TRY (safe_tac HOL_cs), |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
529 |
REPEAT (CHANGED (asm_simp_tac take_ss 1))]; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
530 |
fun tacs ctxt = [ |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
531 |
rtac impI 1, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
532 |
InductTacs.induct_tac ctxt [[SOME "n"]] 1, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
533 |
simp_tac take_ss 1, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
534 |
safe_tac HOL_cs] @ |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
535 |
flat (mapn (x_tacs ctxt) 0 xs); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
536 |
in pg [ax_bisim_def] goal tacs end; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
537 |
in |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
538 |
val _ = trace " Proving coind..."; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
539 |
val coind = |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
540 |
let |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
541 |
fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'")); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
542 |
fun mk_eqn x = %:x === %:(x^"'"); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
543 |
val goal = |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
544 |
mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===> |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
545 |
Logic.list_implies (mapn mk_prj 0 xs, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
546 |
mk_trp (foldr1 mk_conj (map mk_eqn xs))); |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
547 |
val tacs = |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
548 |
TRY (safe_tac HOL_cs) :: |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
549 |
maps (fn take_lemma => [ |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
550 |
rtac take_lemma 1, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
551 |
cut_facts_tac [coind_lemma] 1, |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
552 |
fast_tac HOL_cs 1]) |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
553 |
take_lemmas; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
554 |
in pg [] goal (K tacs) end; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
555 |
end; (* local *) |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
556 |
|
35774 | 557 |
in thy |> snd o PureThy.add_thmss |
558 |
[((Binding.qualified true "coind" comp_dbind, [coind]), [])] |
|
35599
20670f5564e9
skip coinduction proofs for indirect-recursive domain definitions
huffman
parents:
35597
diff
changeset
|
559 |
end; (* let *) |
35574
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
560 |
|
35657
0537c34c6067
pass take_induct_info as an argument to comp_theorems
huffman
parents:
35654
diff
changeset
|
561 |
fun comp_theorems |
35774 | 562 |
(comp_dbind : binding, eqs : eq list) |
35659 | 563 |
(take_info : Domain_Take_Proofs.take_induct_info) |
35657
0537c34c6067
pass take_induct_info as an argument to comp_theorems
huffman
parents:
35654
diff
changeset
|
564 |
(thy : theory) = |
35574
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
565 |
let |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
566 |
val map_tab = Domain_Take_Proofs.get_map_tab thy; |
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
567 |
|
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
568 |
val dnames = map (fst o fst) eqs; |
35774 | 569 |
val comp_dname = Sign.full_name thy comp_dbind; |
35574
ee5df989b7c4
move coinduction-related stuff into function prove_coindunction
huffman
parents:
35560
diff
changeset
|
570 |
|
35585
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
571 |
(* ----- getting the composite axiom and definitions ------------------------ *) |
23152 | 572 |
|
35585
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
573 |
(* Test for indirect recursion *) |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
574 |
local |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
575 |
fun indirect_arg arg = |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
576 |
rec_of arg = ~1 andalso Datatype_Aux.is_rec_type (dtyp_of arg); |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
577 |
fun indirect_con (_, args) = exists indirect_arg args; |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
578 |
fun indirect_eq (_, cons) = exists indirect_con cons; |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
579 |
in |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
580 |
val is_indirect = exists indirect_eq eqs; |
35599
20670f5564e9
skip coinduction proofs for indirect-recursive domain definitions
huffman
parents:
35597
diff
changeset
|
581 |
val _ = |
20670f5564e9
skip coinduction proofs for indirect-recursive domain definitions
huffman
parents:
35597
diff
changeset
|
582 |
if is_indirect |
20670f5564e9
skip coinduction proofs for indirect-recursive domain definitions
huffman
parents:
35597
diff
changeset
|
583 |
then message "Indirect recursion detected, skipping proofs of (co)induction rules" |
20670f5564e9
skip coinduction proofs for indirect-recursive domain definitions
huffman
parents:
35597
diff
changeset
|
584 |
else message ("Proving induction properties of domain "^comp_dname^" ..."); |
35585
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
585 |
end; |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
586 |
|
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
587 |
(* theorems about take *) |
23152 | 588 |
|
35659 | 589 |
val take_lemmas = #take_lemma_thms take_info; |
23152 | 590 |
|
35585
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
591 |
val take_rews = |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
592 |
maps (fn dn => PureThy.get_thms thy (dn ^ ".take_rews")) dnames; |
23152 | 593 |
|
35585
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
594 |
(* prove induction rules, unless definition is indirect recursive *) |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
595 |
val thy = |
555f26f00e47
skip proof of induction rule for indirect-recursive domain definitions
huffman
parents:
35574
diff
changeset
|
596 |
if is_indirect then thy else |
35774 | 597 |
prove_induction (comp_dbind, eqs) take_rews take_info thy; |
23152 | 598 |
|
35599
20670f5564e9
skip coinduction proofs for indirect-recursive domain definitions
huffman
parents:
35597
diff
changeset
|
599 |
val thy = |
20670f5564e9
skip coinduction proofs for indirect-recursive domain definitions
huffman
parents:
35597
diff
changeset
|
600 |
if is_indirect then thy else |
35774 | 601 |
prove_coinduction (comp_dbind, eqs) take_lemmas thy; |
23152 | 602 |
|
35642
f478d5a9d238
generate separate qualified theorem name for each type's reach and take_lemma
huffman
parents:
35630
diff
changeset
|
603 |
in |
f478d5a9d238
generate separate qualified theorem name for each type's reach and take_lemma
huffman
parents:
35630
diff
changeset
|
604 |
(take_rews, thy) |
23152 | 605 |
end; (* let *) |
606 |
end; (* struct *) |