author | skalberg |
Fri, 04 Mar 2005 15:07:34 +0100 | |
changeset 15574 | b1d1b5bfc464 |
parent 15570 | 8d8c70b41bab |
child 16123 | 1381e90c2694 |
permissions | -rw-r--r-- |
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Module for defining realizers for induction and case analysis theorems
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(* Title: HOL/Tools/datatype_realizer.ML |
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ID: $Id$ |
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Author: Stefan Berghofer, TU Muenchen |
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Porgram extraction from proofs involving datatypes: |
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Module for defining realizers for induction and case analysis theorems
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Realizers for induction and case analysis |
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*) |
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Module for defining realizers for induction and case analysis theorems
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signature DATATYPE_REALIZER = |
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sig |
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val add_dt_realizers: (string * sort) list -> |
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DatatypeAux.datatype_info list -> theory -> theory |
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end; |
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structure DatatypeRealizer : DATATYPE_REALIZER = |
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struct |
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open DatatypeAux; |
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fun subsets i j = if i <= j then |
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let val is = subsets (i+1) j |
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in map (fn ks => i::ks) is @ is end |
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else [[]]; |
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fun forall_intr_prf (t, prf) = |
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let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p) |
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in Abst (a, SOME T, Proofterm.prf_abstract_over t prf) end; |
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fun prf_of thm = |
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let val {sign, prop, der = (_, prf), ...} = rep_thm thm |
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in Reconstruct.reconstruct_proof sign prop prf end; |
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fun prf_subst_vars inst = |
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Proofterm.map_proof_terms (subst_vars ([], inst)) I; |
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fun is_unit t = snd (strip_type (fastype_of t)) = HOLogic.unitT; |
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fun tname_of (Type (s, _)) = s |
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| tname_of _ = ""; |
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fun mk_realizes T = Const ("realizes", T --> HOLogic.boolT --> HOLogic.boolT); |
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fun make_ind sorts ({descr, rec_names, rec_rewrites, induction, ...} : datatype_info) (is, thy) = |
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let |
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val sg = sign_of thy; |
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val recTs = get_rec_types descr sorts; |
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val pnames = if length descr = 1 then ["P"] |
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else map (fn i => "P" ^ string_of_int i) (1 upto length descr); |
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val rec_result_Ts = map (fn ((i, _), P) => |
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if i mem is then TFree ("'" ^ P, HOLogic.typeS) else HOLogic.unitT) |
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(descr ~~ pnames); |
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fun make_pred i T U r x = |
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if i mem is then |
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Free (List.nth (pnames, i), T --> U --> HOLogic.boolT) $ r $ x |
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else Free (List.nth (pnames, i), U --> HOLogic.boolT) $ x; |
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fun mk_all i s T t = |
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if i mem is then list_all_free ([(s, T)], t) else t; |
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val (prems, rec_fns) = split_list (List.concat (snd (foldl_map |
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(fn (j, ((i, (_, _, constrs)), T)) => foldl_map (fn (j, (cname, cargs)) => |
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let |
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val Ts = map (typ_of_dtyp descr sorts) cargs; |
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val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames); |
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val recs = List.filter (is_rec_type o fst o fst) (cargs ~~ tnames ~~ Ts); |
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val frees = tnames ~~ Ts; |
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fun mk_prems vs [] = |
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let |
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val rT = List.nth (rec_result_Ts, i); |
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val vs' = filter_out is_unit vs; |
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val f = mk_Free "f" (map fastype_of vs' ---> rT) j; |
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val f' = Pattern.eta_contract (list_abs_free |
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(map dest_Free vs, if i mem is then list_comb (f, vs') |
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else HOLogic.unit)); |
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in (HOLogic.mk_Trueprop (make_pred i rT T (list_comb (f, vs')) |
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(list_comb (Const (cname, Ts ---> T), map Free frees))), f') |
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end |
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| mk_prems vs (((dt, s), T) :: ds) = |
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let |
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val k = body_index dt; |
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val (Us, U) = strip_type T; |
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val i = length Us; |
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val rT = List.nth (rec_result_Ts, k); |
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val r = Free ("r" ^ s, Us ---> rT); |
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val (p, f) = mk_prems (vs @ [r]) ds |
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in (mk_all k ("r" ^ s) (Us ---> rT) (Logic.mk_implies |
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(list_all (map (pair "x") Us, HOLogic.mk_Trueprop |
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(make_pred k rT U (app_bnds r i) |
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(app_bnds (Free (s, T)) i))), p)), f) |
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end |
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in (j + 1, |
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apfst (curry list_all_free frees) (mk_prems (map Free frees) recs)) |
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end) (j, constrs)) (1, descr ~~ recTs)))); |
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fun mk_proj j [] t = t |
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| mk_proj j (i :: is) t = if null is then t else |
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if j = i then HOLogic.mk_fst t |
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else mk_proj j is (HOLogic.mk_snd t); |
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val tnames = DatatypeProp.make_tnames recTs; |
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val fTs = map fastype_of rec_fns; |
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val ps = map (fn ((((i, _), T), U), s) => Abs ("x", T, make_pred i U T |
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(list_comb (Const (s, fTs ---> T --> U), rec_fns) $ Bound 0) (Bound 0))) |
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(descr ~~ recTs ~~ rec_result_Ts ~~ rec_names); |
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val r = if null is then Extraction.nullt else |
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foldr1 HOLogic.mk_prod (List.mapPartial (fn (((((i, _), T), U), s), tname) => |
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if i mem is then SOME |
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(list_comb (Const (s, fTs ---> T --> U), rec_fns) $ Free (tname, T)) |
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else NONE) (descr ~~ recTs ~~ rec_result_Ts ~~ rec_names ~~ tnames)); |
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val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &") |
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(map (fn ((((i, _), T), U), tname) => |
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make_pred i U T (mk_proj i is r) (Free (tname, T))) |
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(descr ~~ recTs ~~ rec_result_Ts ~~ tnames))); |
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val cert = cterm_of sg; |
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val inst = map (pairself cert) (map head_of (HOLogic.dest_conj |
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(HOLogic.dest_Trueprop (concl_of induction))) ~~ ps); |
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val thm = simple_prove_goal_cterm (cert (Logic.list_implies (prems, concl))) |
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(fn prems => |
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[rewrite_goals_tac (map mk_meta_eq [fst_conv, snd_conv]), |
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rtac (cterm_instantiate inst induction) 1, |
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ALLGOALS ObjectLogic.atomize_tac, |
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rewrite_goals_tac (o_def :: map mk_meta_eq rec_rewrites), |
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REPEAT ((resolve_tac prems THEN_ALL_NEW (fn i => |
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REPEAT (etac allE i) THEN atac i)) 1)]); |
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val {path, ...} = Sign.rep_sg sg; |
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val ind_name = Thm.name_of_thm induction; |
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val vs = map (fn i => List.nth (pnames, i)) is; |
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val (thy', thm') = thy |
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|> Theory.absolute_path |
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|> PureThy.store_thm |
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((space_implode "_" (ind_name :: vs @ ["correctness"]), thm), []) |
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|>> Theory.add_path (NameSpace.pack (getOpt (path,[]))); |
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val ivs = Drule.vars_of_terms |
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[Logic.varify (DatatypeProp.make_ind [descr] sorts)]; |
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val rvs = Drule.vars_of_terms [prop_of thm']; |
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val ivs1 = map Var (filter_out (fn (_, T) => |
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tname_of (body_type T) mem ["set", "bool"]) ivs); |
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val ivs2 = map (fn (ixn, _) => Var (ixn, valOf (assoc (rvs, ixn)))) ivs; |
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val prf = foldr forall_intr_prf |
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(foldr (fn ((f, p), prf) => |
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(case head_of (strip_abs_body f) of |
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Free (s, T) => |
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let val T' = Type.varifyT T |
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in Abst (s, SOME T', Proofterm.prf_abstract_over |
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(Var ((s, 0), T')) (AbsP ("H", SOME p, prf))) |
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end |
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| _ => AbsP ("H", SOME p, prf))) |
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(Proofterm.proof_combP |
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(prf_of thm', map PBound (length prems - 1 downto 0))) (rec_fns ~~ prems_of thm)) ivs2; |
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158 |
|
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val r' = if null is then r else Logic.varify (foldr (uncurry lambda) |
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r (map Logic.unvarify ivs1 @ filter_out is_unit |
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(map (head_of o strip_abs_body) rec_fns))); |
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|
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in Extraction.add_realizers_i [(ind_name, (vs, r', prf))] thy' end; |
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|
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fun make_casedists sorts ({index, descr, case_name, case_rewrites, exhaustion, ...} : datatype_info, thy) = |
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let |
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val sg = sign_of thy; |
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val cert = cterm_of sg; |
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val rT = TFree ("'P", HOLogic.typeS); |
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val rT' = TVar (("'P", 0), HOLogic.typeS); |
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fun make_casedist_prem T (cname, cargs) = |
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let |
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val Ts = map (typ_of_dtyp descr sorts) cargs; |
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val frees = variantlist |
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(DatatypeProp.make_tnames Ts, ["P", "y"]) ~~ Ts; |
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val free_ts = map Free frees; |
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val r = Free ("r" ^ NameSpace.base cname, Ts ---> rT) |
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in (r, list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop |
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(HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))), |
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HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $ |
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list_comb (r, free_ts))))) |
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end; |
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|
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val SOME (_, _, constrs) = assoc (descr, index); |
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val T = List.nth (get_rec_types descr sorts, index); |
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val (rs, prems) = split_list (map (make_casedist_prem T) constrs); |
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val r = Const (case_name, map fastype_of rs ---> T --> rT); |
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|
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val y = Var (("y", 0), Type.varifyT T); |
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val y' = Free ("y", T); |
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193 |
|
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val thm = prove_goalw_cterm [] (cert (Logic.list_implies (prems, |
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HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $ |
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list_comb (r, rs @ [y']))))) |
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(fn prems => |
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[rtac (cterm_instantiate [(cert y, cert y')] exhaustion) 1, |
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ALLGOALS (EVERY' |
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[asm_simp_tac (HOL_basic_ss addsimps case_rewrites), |
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resolve_tac prems, asm_simp_tac HOL_basic_ss])]); |
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202 |
|
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val {path, ...} = Sign.rep_sg sg; |
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val exh_name = Thm.name_of_thm exhaustion; |
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val (thy', thm') = thy |
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|> Theory.absolute_path |
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|> PureThy.store_thm ((exh_name ^ "_P_correctness", thm), []) |
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|>> Theory.add_path (NameSpace.pack (getOpt (path,[]))); |
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209 |
|
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val P = Var (("P", 0), rT' --> HOLogic.boolT); |
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val prf = forall_intr_prf (y, forall_intr_prf (P, |
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foldr (fn ((p, r), prf) => |
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forall_intr_prf (Logic.varify r, AbsP ("H", SOME (Logic.varify p), |
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prf))) (Proofterm.proof_combP (prf_of thm', |
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map PBound (length prems - 1 downto 0))) (prems ~~ rs))); |
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val r' = Logic.varify (Abs ("y", Type.varifyT T, |
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list_abs (map dest_Free rs, list_comb (r, |
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map Bound ((length rs - 1 downto 0) @ [length rs]))))); |
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|
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in Extraction.add_realizers_i |
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[(exh_name, (["P"], r', prf)), |
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(exh_name, ([], Extraction.nullt, prf_of exhaustion))] thy' |
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223 |
end; |
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|
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225 |
|
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fun add_dt_realizers sorts infos thy = if !proofs < 2 then thy else |
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(message "Adding realizers for induction and case analysis ..."; thy |
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|> curry (Library.foldr (make_ind sorts (hd infos))) |
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(subsets 0 (length (#descr (hd infos)) - 1)) |
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|> curry (Library.foldr (make_casedists sorts)) infos); |
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|
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232 |
end; |