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Quellcode-Bibliothek© Kompilation durch diese Firma[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]Datei: IO.vdmrt Sprache: VDMOriginal von: Isabelle© |
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(* Title: HOL/Tools/BNF/bnf_lift.ML
Author: Julian Biendarra, TU Muenchen Author: Basil Fürer, ETH Zurich Author: Joshua Schneider, ETH Zurich Author: Dmitriy Traytel, ETH Zurich Copyright 2015, 2019 Lifting of BNFs through typedefs. *) signature BNF_LIFT = sig datatype lift_bnf_option = Plugins_Option of Proof.context -> Plugin_Name.filter | No_Warn_Wits | No_Warn_Transfer val copy_bnf: (((lift_bnf_option list * (binding option * (string * sort option)) list) * string) * thm option) * (binding * binding * binding) -> local_theory -> local_theory val copy_bnf_cmd: (((lift_bnf_option list * (binding option * (string * string option)) list) * string) * (Facts.ref * Token.src list) option) * (binding * binding * binding) -> local_theory -> local_theory val lift_bnf: ((((lift_bnf_option list * (binding option * (string * sort option)) list) * string) * term list option) * thm list option) * (binding * binding * binding) -> ({context: Proof.context, prems: thm list} -> tactic) list -> local_theory -> local_theory val lift_bnf_cmd: ((((lift_bnf_option list * (binding option * (string * string option)) list) * string) * string list) * (Facts.ref * Token.src list) list option) * (binding * binding * binding) -> local_theory -> Proof.state end structure BNF_Lift : BNF_LIFT = struct open Ctr_Sugar_Tactics open BNF_Util open BNF_Comp open BNF_Def datatype lift_bnf_option = Plugins_Option of Proof.context -> Plugin_Name.filter | No_Warn_Wits | No_Warn_Transfer; datatype equiv_thm = Typedef | Quotient of thm (** Util **) fun last2 [x, y] = ([], (x, y)) | last2 (x :: xs) = last2 xs |>> (fn y => x :: y) | last2 [] = raise Match; fun strip3 thm = (case Term.strip_comb (HOLogic.dest_Trueprop (Thm.prop_of thm)) of (_, [x1, x2, x3]) => (x1, x2, x3) | _ => error "strip3: wrong number of arguments"); val mk_Free = yield_singleton o mk_Frees; fun TWICE t = t THEN' t; fun prove lthy fvars tm tac = Goal.prove_sorry lthy (map (fst o dest_Free) fvars) [] tm (fn {context, ...} => tac context); (** Term construction **) fun mk_relT aT bT = aT --> bT --> HOLogic.boolT; fun mk_relcompp r s = let val (rT, sT) = apply2 fastype_of (r, s); val ((xT, _), (_, zTs)) = apply2 dest_funT (rT, sT); val T = rT --> sT --> mk_relT xT (fst (dest_funT zTs)); in Const (@{const_name relcompp}, T) $ r $ s end; val mk_OO = fold_rev mk_relcompp; (* option from sum *) fun mk_MaybeT T = mk_sumT (HOLogic.unitT, T); fun mk_Nothing T = BNF_FP_Util.mk_Inl T HOLogic.unit; val Just_const = BNF_FP_Util.Inr_const HOLogic.unitT; fun mk_Just tm = Just_const (fastype_of tm) $ tm; fun Maybe_map_const T = let val (xT, yT) = dest_funT T in Const (@{const_name map_sum}, (HOLogic.unitT --> HOLogic.unitT) --> T --> mk_MaybeT xT --> mk_Ma HOLogic.id_const HOLogic.unitT end; fun mk_Maybe_map tm = Maybe_map_const (fastype_of tm) $ tm; fun fromJust_const T = Const (@{const_name sum.projr}, mk_MaybeT T --> T) fun rel_Maybe_const T U = Const (@{const_name rel_sum}, (HOLogic.unitT --> HOLogic.unitT --> HOLogic.boolT) --> (T --> U --> HOLogic.boolT) --> mk_MaybeT T --> mk_MaybeT U --> HOLogic.boolT) $ HOLogic.eq_const HOLogic.unitT fun set_Maybe_const T = Const (@{const_name Basic_BNFs.setr}, mk_MaybeT T --> HOLogic.mk_se fun Inf_const T = Const (@{const_name Inf}, HOLogic.mk_setT T --> T); fun Image_const T = let val relT = HOLogic.mk_setT (HOLogic.mk_prodT (T, T)); val setT = HOLogic.mk_setT T; in Const (@{const_name Image}, relT --> setT --> setT) end; fun bot_const T = Const (@{const_name bot}, T); fun mk_insert x S = Const (@{const_name Set.insert}, fastype_of x --> fastype_of S --> fastype_of S) $ x $ S; fun mk_vimage f s = let val (xT, yT) = dest_funT (fastype_of f) in Const (@{const_name vimage}, (xT --> yT) --> HOLogic.mk_setT yT --> HOLogic.mk_setT xT) $ f $ s end; fun mk_case_prod (x, y) tm = let val ((x, xT), (y, yT)) = apply2 dest_Free (x, y); val prodT = HOLogic.mk_prodT (xT, yT); in HOLogic.Collect_const prodT $ (Const (@{const_name case_prod}, (xT --> yT --> HOLogic.boolT) --> prodT --> HOLogic.boolT) $ absfree (x, xT) (absfree (y, yT) tm)) end; fun mk_Trueprop_implies (ps, c) = Logic.list_implies (map HOLogic.mk_Trueprop ps, HOLogic.mk_Trueprop c); fun mk_Collect (v, tm) = let val (n, T) = dest_Free v in HOLogic.mk_Collect (n, T, tm) end; (** witnesses **) fun prepare_wits is_quotient RepT wits opts alphas wits_F var_as var_as' sets lthy = let fun binder_types_until_eq V T = let fun strip (TU as Type ("fun", [T, U])) = if V = TU then [] else T :: strip U | strip T = if V = T then [] else error ("Bad type for witness: " ^ quote (Syntax.string_of_typ lthy T)); in strip T end; val Iwits = the_default wits_F (Option.map (map (`(map (fn T => find_index (fn U => T = U) alphas) o binder_types_until_eq RepT o fastype_of))) wits); val Iwits = if is_quotient then let val subsumed_Iwits = filter (fn (J, _) => exists (fn (I, _) => subset (op =) (I, J)) wits_F) Iwits; val _ = if null subsumed_Iwits orelse exists (fn No_Warn_Wits => true | _ => false) opts then () else let val (suffix1, suffix2, be) = (if length subsumed_Iwits = 1 then ("", "", "is") else ("s", "es", "are")) in subsumed_Iwits |> map (Syntax.pretty_typ lthy o fastype_of o snd) |> Pretty.big_list ("The following type" ^ suffix1 ^ " of nonemptiness witness" ^ suffix2 ^ " of the raw type's BNF " ^ be ^ " subsumed by the existing raw witnesses:") |> (fn pt => Pretty.chunks [pt, Pretty.para ("You do not need to lift these subsumed witnesses.")]) |> Pretty.string_of |> warning end; in filter_out (fn (J, _) => exists (fn (I, _) => subset (op =) (I, J)) wits_F) Iwits end else Iwits; val wit_goals = maps (BNF_Def.mk_wit_goals var_as var_as' sets) Iwits; val lost_wits = if is_quotient then [] else filter_out (fn (J, _) => exists (fn (I, _) => subset (op =) (I, J)) Iwits) wits_F; val _ = if null lost_wits orelse exists (fn No_Warn_Wits => true | _ => false) opts then () else let val what = (if is_quotient then "quotient type" else "typedef"); val (suffix1, suffix2, be) = (if length lost_wits = 1 then ("", "", "was") else ("s", "es", "were")) in lost_wits |> map (Syntax.pretty_typ lthy o fastype_of o snd) |> Pretty.big_list ("The following type" ^ suffix1 ^ " of nonemptiness witness" ^ suffix2 ^ " of the raw type's BNF " ^ be ^ " lost:") |> (fn pt => Pretty.chunks [pt, Pretty.para ("You can specify a liftable witness (e.g., a term of one of the abov \ that satisfies the " ^ what ^ "'s invariant)\ \ using the annotation [wits: <term>].")]) |> Pretty.string_of |> warning end; in (Iwits, wit_goals) end; (** transfer theorems **) fun mk_transfer_thms' bnf_F bnf_G name consts Tss crel_def pcrel_def defs lthy = val live = length (#alphas Tss); val (pcrel_tm, crel_tm) = apply2 (Thm.prop_of #> Logic.dest_equals #> fst #> head_of) (pcrel_def, crel_def); val (var_Qs, var_Rs) = fold Variable.declare_typ (#alphas Tss @ #deads Tss) lthy |> mk_Frees "Q" (map2 mk_relT (#alphas Tss) (#betas Tss)) ||>> mk_Frees "R" (map2 mk_relT (#gammas Tss) (#deltas Tss)) |> fst; (* get the "pcrel :: args_raw => rep => abs \<Rightarrow> bool" term and instantiate types *) val (args_raw, (rep, abs)) = pcrel_tm |> fastype_of |> binder_types |> last2; val thy = Proof_Context.theory_of lthy; val tyenv_match = Vartab.empty |> Sign.typ_match thy ((rep, #rep Tss)) |> Sign.typ_match thy ((abs, #abs Tss)); val args = map (Envir.subst_type tyenv_match) args_raw; val (pcrel_a, pcrel_b) = Envir.subst_term (tyenv_match, Vartab.empty) pcrel_tm |> `(subst_atomic_types (#alphas Tss @ #betas Tss ~~ #gammas Tss @ #deltas Tss)); (* match "crel :: {?a F} \<Rightarrow> a G" with "rep_G :: {a F}" *) val tyenv_match = Vartab.empty |> Sign.typ_match thy (crel_tm |> fastype_of |> binder_types |> hd, #rep Tss); val crel_b = Envir.subst_term (tyenv_match, Vartab.empty) crel_tm |> subst_atomic_types (#alphas Tss ~~ #betas Tss) val crel_d = subst_atomic_types (#betas Tss ~~ #deltas Tss) crel_b; (* instantiate pcrel with Qs and Rs*) val dead_args = map binder_types args |> map (fn [T,U] => if T = U then SOME T else NONE | _ => NONE); val parametrize = let fun go (SOME T :: dTs) tms = HOLogic.eq_const T :: go dTs tms | go (_ :: dTs) (tm :: tms) = tm :: go dTs tms | go (_ :: dTs) tms = go dTs tms | go _ _ = []; in go dead_args end; val pcrel_Qs = list_comb (pcrel_b, parametrize var_Qs); val pcrel_Rs = list_comb (pcrel_a, parametrize var_Rs); (* get the order of the dead variables right *) val Ds0 = maps the_list dead_args; val permute_Ds = (mk_T_of_bnf Ds0 (#betas Tss) bnf_G, nth (binder_types (type_of pcrel_Qs)) |> apply2 (fn Type (_, Ts) => Ts | _ => []) |> op~~ |> typ_subst_atomic; val Ds0 = map permute_Ds Ds0; (* terms for sets of the set_transfer thms *) val rel_sets_Q = @{map 3} (fn aT => fn bT => fn Q => mk_rel 1 [aT] [bT] @{term rel_set} $ Q) (#alphas Tss) (#betas Tss) var_Qs; (* rewrite rules for pcrel and BNF's relator: "pcrel Q = rel_F OO crel" *) fun mk_pcr_rel_F_eq Ts Us pcrel vs crel = let val thm = HOLogic.mk_Trueprop (HOLogic.mk_eq (pcrel, mk_relcompp (list_comb (mk_rel_of_bnf (#deads Tss) (Ts Tss) (Us Tss) bnf_F, vs)) crel)); fun tac ctxt = unfold_thms_tac ctxt (pcrel_def :: defs @ @{thm id_bnf_apply} :: Transfer.get_relator_eq ctxt) THEN (HEADGOAL (rtac ctxt refl)) in prove lthy vs thm tac |> mk_abs_def end; val pcr_rel_F_eqs = [mk_pcr_rel_F_eq #alphas #betas pcrel_Qs var_Qs crel_b, mk_pcr_rel_F_eq #gammas #deltas pcrel_Rs var_Rs crel_d]; (* create transfer-relations from term('s type) *) fun mk_transfer_rel' i tm = let fun go T' (n, q) = case T' of Type ("fun", [T as Type ("fun", _), U]) => go U (n+1, q) |>> mk_rel_fun (fst (go T (n, q))) | Type ("fun", [T, U]) => go T (n, q) |-> (fn x => fn st => go U st |>> mk_rel_fun x) | Type (@{type_name bool}, _) => (HOLogic.eq_const HOLogic.boolT, (n, q)) | Type (@{type_name set}, _) => (nth rel_sets_Q n, (n, q)) | Type _ => (if q then pcrel_Qs else pcrel_Rs, (n, not q)) | TFree _ => (nth (if q then var_Qs else var_Rs) n, (n, not q)) | _ => raise Match in go (fastype_of tm) (i, true) |> fst end; (* prove transfer rules *) fun prove_transfer_thm' i vars new_tm const = let val tm = HOLogic.mk_Trueprop (mk_transfer_rel' i new_tm $ #raw const $ new_tm); val tac = (fn ctxt => unfold_thms_tac ctxt (pcr_rel_F_eqs @ [crel_def]) THEN #tac const {Rs=var_Rs,Qs=var_Qs} ctxt); in prove lthy vars tm tac end; val prove_transfer_thm = prove_transfer_thm' 0; (* map transfer: "((Q ===> R) ===> pcr_G Q ===> pcr_G R) map_F' map_G" *) val map_G = mk_map_of_bnf Ds0 (#betas Tss) (#deltas Tss) bnf_G; val map_transfer = prove_transfer_thm (var_Qs @ var_Rs) map_G (#map consts); (* pred_transfer: "((Q ===> (=)) ===> pcr_G Q ===> (=)) pred_F' pred_G" *) val pred_G = mk_pred_of_bnf Ds0 (#betas Tss) bnf_G; val pred_transfer = #pred consts |> Option.map (fn p => ("pred", [prove_transfer_thm (var_Qs @ var_Rs) pred_G p])); (* rel_transfer: "((Q ===> R ===> (=)) ===> pcr_G Q ===> pcr_G R ===> (=)) rel_F' rel_G" *) val rel_G = mk_rel_of_bnf Ds0 (#betas Tss) (#deltas Tss) bnf_G; val rel_transfer = prove_transfer_thm (var_Qs @ var_Rs) rel_G (#rel consts); (* set_transfer: "(pcr_G Q ===> rel_set Q) set_F' set_G" *) val sets_G = mk_sets_of_bnf (replicate live Ds0) (replicate live (#betas Tss)) bnf_G; fun mk_set_transfer i set_G raw tac = prove_transfer_thm' i var_Qs set_G {raw=raw, tac= val sets_transfer = @{map 4} mk_set_transfer (0 upto (live-1)) sets_G (#raws (#sets consts)) (#tacs (#sets consts)); (* export transfer theorems *) val transform = Morphism.thm (Morphism.thm_morphism "BNF" (unfold_thms lthy defs)) |> map; val b = Binding.qualified_name name; val qualify = let val qs = Binding.path_of b; in fold_rev (fn (s, mand) => Binding.qualify mand s) qs end; fun mk_binding n = Binding.name (n ^ "_transfer_raw") |> Binding.qualify true (Binding.name_of b) |> qualify; val notes = [("map", [map_transfer]), ("rel", [rel_transfer])] @ the_list pred_transfer @ [("set", sets_transfer)] |> map (fn (n, thms) => ((mk_binding n, []), [(transform thms, @{attributes [transfer_rule]})])); in lthy |> Local_Theory.notes notes |> snd end; (* transfer theorems for map, pred (in case of a typedef), rel and sets *) fun mk_transfer_thms quiet bnf_F bnf_G name consts thm Tss defs lthy = let fun mk_crel_def quot_thm = (case thm of Quotient equiv => @{thm Quotient_crel_quotient} OF [quot_thm, equiv] | Typedef => hd ([quot_thm] RL @{thms Quotient_crel_typedef Quotient_crel_typecopy})); fun no_quotient _ = [Pretty.para ("No quotient theorem has been registered for " ^ Binding.name_of (name_of_bnf bnf_G) ^ "."), Pretty.para "Use setup_lifting to register a quotient or type definition theorem. fun wrong_quotient T lthy = [Pretty.para ("A wrong quotient theorem has been registere Binding.name_of (name_of_bnf bnf_G) ^ "."), Pretty.para ("Expected raw type " ^ Pretty.string_of (Syntax.pretty_typ lthy (T_of_bnf bnf_F)) ^ " but the quotient theorem has raw type " ^ Pretty.string_of (Syntax.pretty_typ lthy T) ^ "."), Pretty.para "Use setup_lifting to register a different quotient or type definitio fun pcr_why _ = [Pretty.para ("The pcr_" ^ Binding.name_of (name_of_bnf bnf_G) ^ " relator has not been defined.")]; fun warn_transfer why lthy = (Pretty.para "The transfer theorems can't be generated:" :: why lthy) |> Pretty.chunks |> Pretty.string_of |> warning |> K lthy; fun maybe_warn_transfer why = not quiet ? warn_transfer why; in case Lifting_Info.lookup_quotients lthy name of SOME {pcr_info, quot_thm} => (let val crel_def = mk_crel_def quot_thm; val rty = Lifting_Util.quot_thm_rty_qty quot_thm |> fst; val thy = Proof_Context.theory_of lthy; in if Sign.typ_instance thy (rty, T_of_bnf bnf_F) then (case pcr_info of SOME {pcrel_def, ...} => mk_transfer_thms' bnf_F bnf_G name consts Tss crel_def pcrel_def defs lthy | _ => maybe_warn_transfer pcr_why lthy) else maybe_warn_transfer (wrong_quotient rty) lthy end) | _ => maybe_warn_transfer no_quotient lthy end; (** typedef_bnf **) fun mk_typedef_transfer_tacs bnf_F bnf_G thms old_defs map_raw rel_raw pred_raw sets_raw = let val live = live_of_bnf bnf_G; val Abs_G_inverse = @{thm type_definition.Abs_inverse} OF [#typedef thms]; val Rep_G = @{thm type_definition.Rep} OF [#typedef thms]; fun common_tac addefs tac = (fn _ => fn ctxt => HEADGOAL (EVERY' [SELECT_GOAL (unfold_thms_tac ctxt addefs), REPEAT_DETERM o rtac ctxt rel_funI, SELECT_GOAL (unfold_thms_tac ctxt @{thms o_apply}), REPEAT_DETERM o eresolve_tac ctxt @{thms relcomppE exE conjE}, hyp_subst_tac ctxt]) THEN tac ctxt) fun map_tac ctxt = (HEADGOAL o EVERY') [rtac ctxt @{thm relcomppI}, etac ctxt (mk_rel_funDN (live+1) (map_transfer_of_bnf bnf_F)), REPEAT_DETERM_N live o assume_tac ctxt, SELECT_GOAL (unfold_thms_tac ctxt [Abs_G_inverse OF [#map_closed thms] OF [Rep_G]]), REPEAT_DETERM o rtac ctxt refl]; val map_tac = common_tac [#map old_defs] map_tac; fun rel_tac ctxt = HEADGOAL (etac ctxt (mk_rel_funDN (live+2) (rel_transfer_of_bnf bnf_F)) THEN' REPEAT_DETERM_N (live+1) o assume_tac ctxt); val rel_tac = common_tac (#rel old_defs :: @{thms vimage2p_def}) rel_tac; fun pred_tac ctxt = HEADGOAL (etac ctxt (mk_rel_funDN (live+1) (pred_transfer_of_bnf bnf_F)) THEN' REPEAT_DETERM_N live o (assume_tac ctxt)); val pred_tac = common_tac [#pred old_defs] pred_tac; fun set_tac set_transfer_thm ctxt = HEADGOAL (etac ctxt (rel_funD OF [set_transfer_thm])); fun mk_set_tac set_def set_transfer = common_tac [set_def] (set_tac set_transfer); val set_tacs = map2 mk_set_tac (#sets old_defs) (set_transfer_of_bnf bnf_F); in {map={raw=map_raw,tac=map_tac},rel={raw=rel_raw,tac=rel_tac}, sets={raws=sets_raw,tacs=set_tacs},pred=SOME{raw=pred_raw,tac=pred_tac}} end; fun typedef_bnf thm wits specs map_b rel_b pred_b opts lthy = let val plugins = get_first (fn Plugins_Option f => SOME (f lthy) | _ => NONE) (rev opts) |> the_default Plugin_Name.default_filter; (* extract Rep Abs F RepT AbsT *) val (Rep_G, Abs_G, F) = strip3 thm; val typ_Abs_G = dest_funT (fastype_of Abs_G); val RepT = fst typ_Abs_G; (* F *) val AbsT = snd typ_Abs_G; (* G *) val AbsT_name = fst (dest_Type AbsT); val tvs = AbsT |> dest_Type |> snd |> map (fst o dest_TVar); val alpha0s = map (TFree o snd) specs; val _ = length tvs = length alpha0s orelse error ("Expected " ^ string_of_int (length tvs) ^ " type argument(s) to " ^ quote AbsT_nam (* instantiate the new type variables newtvs to oldtvs *) val subst = subst_TVars (tvs ~~ alpha0s); val typ_subst = typ_subst_TVars (tvs ~~ alpha0s); val Rep_G = subst Rep_G; val Abs_G = subst Abs_G; val F = subst F; val RepT = typ_subst RepT; val AbsT = typ_subst AbsT; fun flatten_tyargs Ass = map dest_TFree alpha0s |> filter (fn T => exists (fn Ts => member op= Ts T) Ass); val Ds0 = filter (is_none o fst) specs |> map snd; (* get the bnf for RepT *) val ((bnf_F, (deads, alphas)), ((_, unfolds), lthy)) = bnf_of_typ true Dont_Inline (Binding.qualify true AbsT_name) flatten_tyargs [] Ds0 RepT ((empty_comp_cache, empty_unfolds), lthy); val set_bs = map (fn T => find_index (fn U => T = U) alpha0s) alphas |> map (the_default Binding.empty o fst o nth specs); val _ = (case alphas of [] => error "No live variables" | _ => ()); val defs = #map_unfolds unfolds @ flat (#set_unfoldss unfolds) @ #rel_unfolds unfolds @ #pred (* number of live variables *) val live = length alphas; (* state the three required properties *) val sorts = map Type.sort_of_atyp alphas; val names_lthy = fold Variable.declare_typ (alphas @ deads) lthy; val (((alphas', betas), betas'), names_lthy) = names_lthy |> mk_TFrees' sorts ||>> mk_TFrees' sorts ||>> mk_TFrees' sorts; val map_F = mk_map_of_bnf deads alphas betas bnf_F; val (typ_fs, typ_aF) = fastype_of map_F |> strip_typeN live ||> domain_type; val typ_pairs = map HOLogic.mk_prodT (alphas ~~ alphas'); val typ_subst_pair = typ_subst_atomic (alphas ~~ typ_pairs); val typ_pair = typ_subst_pair RepT; val subst_b = subst_atomic_types (alphas ~~ betas); val subst_a' = subst_atomic_types (alphas ~~ alphas'); val subst_pair = subst_atomic_types (alphas ~~ typ_pairs); val aF_set = F; val aF_set' = subst_a' F; val pairF_set = subst_pair F; val bF_set = subst_b F; val map_F_fst = mk_map_of_bnf deads typ_pairs alphas bnf_F; val map_F_snd = mk_map_of_bnf deads typ_pairs alphas' bnf_F val sets_F_pairs = mk_sets_of_bnf (replicate live deads) (replicate live typ_pairs) bnf_F val wits_F = mk_wits_of_bnf (replicate (nwits_of_bnf bnf_F) deads) (replicate (nwits_of_bnf bnf_F) alphas) bnf_F; (* val map_closed_F = @{term "\<And>f x. x \<in> F \<Longrightarrow> map_F f x \<in> F"}; *) val (var_fs, names_lthy) = mk_Frees "f" typ_fs names_lthy; val (var_x, names_lthy) = mk_Frees "x" [typ_aF] names_lthy |>> the_single; val mem_x = HOLogic.mk_Trueprop (HOLogic.mk_mem (var_x, aF_set)); val map_f = list_comb (map_F, var_fs); val mem_map = HOLogic.mk_Trueprop (HOLogic.mk_mem (map_f $ var_x, bF_set)); val imp_map = Logic.mk_implies (mem_x, mem_map); val map_closed_F = fold_rev Logic.all var_fs (Logic.all var_x imp_map); (* val zip_closed_F = @{term "\<And>z. \<lbrakk>map_F fst z \<in> F; map_F snd z \<in> F\<rbrakk> \<Longrightarrow> \<exists>z' \<in> F. set_F z' \<subseteq> set_F z \<and> map_F fst z' = map_F fst z \<and> map_F snd z' = map_F val (var_z, names_lthy) = mk_Free "z" typ_pair names_lthy; val (var_z', names_lthy) = mk_Free "z'" typ_pair names_lthy; val (pairs, names_lthy) = mk_Frees "tmp" typ_pairs names_lthy; fun mk_map mfs f z = Term.list_comb (mfs, map (fst o Term.strip_comb o f) pairs) $ z; fun mk_set var = map (fn t => t $ var) sets_F_pairs; val (map_fst', map_fst) = apply2 (mk_map map_F_fst HOLogic.mk_fst) (var_z', var_z) val (map_snd', map_snd) = apply2 (mk_map map_F_snd HOLogic.mk_snd) (var_z', var_z) val mem_map_fst = HOLogic.mk_Trueprop (HOLogic.mk_mem (map_fst, aF_set)); val mem_map_snd = HOLogic.mk_Trueprop (HOLogic.mk_mem (map_snd, aF_set')); val ex_conj = foldr1 HOLogic.mk_conj (map2 mk_leq (mk_set var_z') (mk_set var_z) @ [HOLogic.mk_eq (map_fst', map_fst), HOLogic.mk_eq (map_snd', map_snd)]); val zip_concl = HOLogic.mk_Trueprop (mk_Bex pairF_set (absfree (dest_Free var_z') ex_con val zip_closed_F = Logic.all var_z (Logic.list_implies ([mem_map_fst, mem_map_snd], zip_ (* val wit_closed_F = @{term "wit_F a \<in> F"}; *) val (var_as, names_lthy) = mk_Frees "a" alphas names_lthy; val (var_bs, _) = mk_Frees "a" alphas names_lthy; val sets = mk_sets_of_bnf (replicate live deads) (replicate live alphas) bnf_F; val (Iwits, wit_goals) = prepare_wits false RepT wits opts alphas wits_F var_as var_bs sets lthy; val wit_closed_Fs = Iwits |> map (fn (I, wit_F) => let val vars = map (nth var_as) I; val wit_a = list_comb (wit_F, vars); in fold_rev Logic.all vars (HOLogic.mk_Trueprop (HOLogic.mk_mem (wit_a, aF_set))) end); val goals = [map_closed_F, zip_closed_F] @ wit_closed_Fs @ (case wits of NONE => [] | _ => wit_goals); fun after_qed ([map_closed_thm] :: [zip_closed_thm] :: wit_thmss) lthy = let val (wit_closed_thms, wit_thms) = (case wits of NONE => (map the_single wit_thmss, wit_thms_of_bnf bnf_F) | _ => chop (length wit_closed_Fs) (map the_single wit_thmss)) (* construct map set bd rel wit *) (* val map_G = @{term "\<lambda>f. Abs_G o map_F f o Rep_G"}; *) val Abs_Gb = subst_b Abs_G; val map_G = fold_rev lambda var_fs (HOLogic.mk_comp (HOLogic.mk_comp (Abs_Gb, map_f), Rep_G)); val map_raw = fold_rev lambda var_fs map_f; (* val sets_G = [@{term "set_F o Rep_G"}]; *) val sets_F = mk_sets_of_bnf (replicate live deads) (replicate live alphas) bnf_F; val sets_G = map (fn set_F => HOLogic.mk_comp (set_F, Rep_G)) sets_F; (* val bd_G = @{term "bd_F"}; *) val bd_F = mk_bd_of_bnf deads alphas bnf_F; val bd_G = bd_F; (* val rel_G = @{term "\<lambda>R. BNF_Def.vimage2p Rep_G Rep_G (rel_F R)"}; *) val rel_F = mk_rel_of_bnf deads alphas betas bnf_F; val (typ_Rs, _) = strip_typeN live (fastype_of rel_F); val (var_Rs, names_lthy) = mk_Frees "R" typ_Rs lthy; val Rep_Gb = subst_b Rep_G; val rel_G = fold_rev absfree (map dest_Free var_Rs) (mk_vimage2p Rep_G Rep_Gb $ list_comb (rel_F, var_Rs)); val rel_raw = fold_rev absfree (map dest_Free var_Rs) (list_comb (rel_F, var_Rs)); (* val pred_G = @{term "\<lambda>P. pred_F P o Rep_G"}; *) val pred_F = mk_pred_of_bnf deads alphas bnf_F; val (typ_Ps, _) = strip_typeN live (fastype_of pred_F); val (var_Ps, names_lthy) = mk_Frees "P" typ_Ps names_lthy; val pred_G = fold_rev absfree (map dest_Free var_Ps) (HOLogic.mk_comp (list_comb (pred_F, var_Ps), Rep_G)); val pred_raw = fold_rev absfree (map dest_Free var_Ps) (list_comb (pred_F, var_Ps)); (* val wits_G = [@{term "Abs_G o wit_F"}]; *) val (var_as, _) = mk_Frees "a" alphas names_lthy; val wits_G = map (fn (I, wit_F) => let val vs = map (nth var_as) I; in fold_rev absfree (map dest_Free vs) (Abs_G $ (list_comb (wit_F, vs))) end) Iwits; (* tactics *) val Rep_thm = thm RS @{thm type_definition.Rep}; val Abs_inverse_thm = thm RS @{thm type_definition.Abs_inverse}; val Abs_inject_thm = thm RS @{thm type_definition.Abs_inject}; val Rep_cases_thm = thm RS @{thm type_definition.Rep_cases}; val Rep_inverse_thm = thm RS @{thm type_definition.Rep_inverse}; fun map_id0_tac ctxt = HEADGOAL (EVERY' [rtac ctxt ext, SELECT_GOAL (unfold_thms_tac ctxt [map_id0_of_bnf bnf_F, id_apply, o_apply, Rep_inverse_thm]), rtac ctxt refl]); fun map_comp0_tac ctxt = HEADGOAL (EVERY' [rtac ctxt ext, SELECT_GOAL (unfold_thms_tac ctxt [map_comp0_of_bnf bnf_F, o_apply, Rep_thm RS (map_closed_thm RS Abs_inverse_thm)]), rtac ctxt refl]); fun map_cong0_tac ctxt = HEADGOAL (EVERY' ([SELECT_GOAL (unfold_thms_tac ctxt [o_apply]), rtac ctxt (([Rep_thm RS map_closed_thm, Rep_thm RS map_closed_thm] MRS Abs_inject_thm) RS iffD2), rtac ctxt (map_cong0_of_bnf bnf_F)] @ replicate live (Goal.assume_rule_tac ctxt))); val set_map0s_tac = map (fn set_map => fn ctxt => HEADGOAL (EVERY' [rtac ctxt ext, SELECT_GOAL (unfold_thms_tac ctxt [set_map, o_apply, Rep_thm RS (map_closed_thm RS Abs_inverse_thm)]), rtac ctxt refl])) (set_map_of_bnf bnf_F); fun card_order_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_card_order_of_bnf bnf_F)); fun cinfinite_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_cinfinite_of_bnf bnf_F)); val set_bds_tac = map (fn set_bd => fn ctxt => HEADGOAL (EVERY' [SELECT_GOAL (unfold_thms_tac ctxt [o_apply]), rtac ctxt set_bd] (set_bd_of_bnf bnf_F); fun le_rel_OO_tac ctxt = HEADGOAL (EVERY' [rtac ctxt @{thm vimage2p_relcompp_mono}, rtac ctxt ((rel_OO_of_bnf bnf_F RS sym) RS @{thm ord_eq_le_trans}), rtac ctxt @{thm order_refl}]); fun rel_OO_Grp_tac ctxt = HEADGOAL (EVERY' ([SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt ext))), SELECT_GOAL (unfold_thms_tac ctxt [@{thm OO_Grp_alt}, mem_Collect_eq, o_apply, @{thm vimage2p_def}, in_rel_of_bnf bnf_F, Bex_def, mem_Collect_eq]), rtac ctxt iffI, SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve_tac ctxt [exE, conjE]))), forward_tac ctxt [zip_closed_thm OF (replicate 2 (Rep_thm RSN (2, @{thm ssubst_mem})))], assume_tac ctxt, SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve_tac ctxt [bexE, conjE]))), etac ctxt Rep_cases_thm, hyp_subst_tac ctxt, SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt exI))), rtac ctxt conjI] @ replicate live (EVERY' [TRY o rtac ctxt conjI, etac ctxt @{thm order_trans}, assume_tac ctxt]) [SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt conjI))), REPEAT_DETERM_N 2 o EVERY' [rtac ctxt (trans OF [iffD2 OF [Abs_inject_thm OF [map_closed_thm OF [Rep_thm], Rep_thm]], Rep_inverse_thm]), etac ctxt trans, assume_tac ctxt], SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve_tac ctxt [exE, conjE]))), rtac ctxt exI, rtac ctxt conjI] @ replicate (live-1) (rtac ctxt conjI THEN' assume_tac ctxt) @ [assume_tac ctxt, rtac ctxt conjI, REPEAT_DETERM_N 2 o EVERY' [rtac ctxt (iffD1 OF [Abs_inject_thm OF [map_closed_thm OF [Rep_thm], Rep_thm]]), etac ctxt (Rep_inverse_thm RS sym RSN (2, trans))]])); fun pred_set_tac ctxt = HEADGOAL (EVERY' [rtac ctxt (pred_set_of_bnf bnf_F RS @{thm arg_cong[of _ _ "\ SELECT_GOAL (unfold_thms_tac ctxt (@{thms Ball_comp_iff conj_comp_iff})), rtac ctxt refl]); fun wit_tac ctxt = HEADGOAL (EVERY' (map (fn thm => (EVERY' [SELECT_GOAL (unfold_thms_tac ctxt (o_apply :: (wit_closed_thms RL [Abs_inverse_thm]))), dtac ctxt thm, assume_tac ctxt])) wit_thms)); val tactics = [map_id0_tac, map_comp0_tac, map_cong0_tac] @ set_map0s_tac @ [card_order_bd_tac, cinfinite_bd_tac] @ set_bds_tac @ [le_rel_OO_tac, rel_OO_Grp_tac, pred_set_tac]; val (bnf_G, lthy) = bnf_def Dont_Inline (user_policy Note_Some) true I tactics wit_tac NONE map_b rel_b pred_b set_bs (((((((Binding.empty, AbsT), map_G), sets_G), bd_G), wits_G), SOME rel_G), SOME pred_G) lthy; val old_defs = {sets = set_defs_of_bnf bnf_G, map = map_def_of_bnf bnf_G, rel = rel_def_of_bnf bnf_G, pred = pred_def_of_bnf bnf_G}; val unfold_morphism = Morphism.thm_morphism "BNF" (unfold_thms lthy defs); val (bnf_G, lthy) = morph_bnf_defs unfold_morphism bnf_G |> (fn bnf => note_bnf_defs bnf lthy); val quiet = exists (fn No_Warn_Transfer => true | _ => false) opts; val transfer_consts = mk_typedef_transfer_tacs bnf_F bnf_G {map_closed=map_closed_thm,typedef=thm} old_defs map_raw rel_raw pred_raw sets_F; in lthy |> BNF_Def.register_bnf plugins AbsT_name bnf_G |> mk_transfer_thms quiet bnf_F bnf_G AbsT_name transfer_consts Typedef {abs=typ_subst_atomic (alphas ~~ alphas') AbsT, rep=RepT, Ds0=map TFree Ds0, deads = deads, alphas=alphas, betas=alphas', gammas=betas, deltas=betas'} defs end | after_qed _ _ = raise Match; in (goals, after_qed, defs, lthy) end; (** quotient_bnf **) fun mk_quotient_transfer_tacs bnf_F Tss live qthms thms set_F'_thmss old_defs inst_REL_pos_distrI map_raw rel_raw sets_raw = let fun common_tac ctxt addefs = unfold_thms_tac ctxt (#REL qthms :: addefs) THEN (REPEAT_DETERM o HEADGOAL) (rtac ctxt rel_funI); (* quotient.map_transfer tactic *) val map_F_transfer = map_transfer_of_bnf bnf_F |> mk_rel_funDN (live+1); fun map_transfer_q _ ctxt = common_tac ctxt (#map old_defs :: @{thms o_def}) THEN (HEADGOAL o EVERY') [REPEAT_DETERM o eresolve_tac ctxt @{thms relcomppE}, rtac ctxt @{thm relcomppI[rotated]}, hyp_subst_tac ctxt THEN' EVERY' (map (rtac ctxt) [#rel_abs qthms, #map_F_rsp thms, (#rep_abs_rsp qthms), (#reflp qthms)]), hyp_subst_tac ctxt, rtac ctxt map_F_transfer, REPEAT_DETERM_N (live+1) o assume_tac ctxt] (* quotient.rel_transfer tactic *) val rel_F_maps = rel_map_of_bnf bnf_F; val rel_F_map_iffD2s = map (fn thm => thm RS @{thm iffD2}) rel_F_maps; fun inst_REL_pos_distrI_order_refls vs aTs bTs ctxt = inst_REL_pos_distrI live vs aTs bTs ct OF (replicate (live+1) asm_rl @ replicate live @{thm order_refl}); fun rel_transfer_q {Qs, Rs} ctxt = EVERY [common_tac ctxt [#rel old_defs, @{thm vimage2p_def}], HEADGOAL (rtac ctxt iffI), (REPEAT_DETERM o ALLGOALS) (eresolve_tac ctxt @{thms exE conjE relcomppE} ORELSE' hyp_subst_tac ctxt), (HEADGOAL o EVERY') [REPEAT_DETERM o dtac ctxt @{thm rel_fun_rel_OO1}, rtac ctxt (inst_REL_pos_distrI 0 (map mk_conversep Qs) (#betas Tss) (#alphas Tss) ctxt), rtac ctxt @{thm relcomppI}, rtac ctxt (#symp qthms), rtac ctxt (#map_F_rsp thms), rtac ctxt (#rep_abs_rsp qthms), rtac ctxt (#reflp qthms), rtac ctxt @{thm relcomppI}, rtac ctxt (rel_flip_of_bnf bnf_F RS @{thm iffD1}), rtac ctxt (nth rel_F_map_iffD2s 0), rtac ctxt (nth rel_F_map_iffD2s 1), etac ctxt (#rel_monoD_rotated thms)], (REPEAT_DETERM_N live o HEADGOAL o EVERY') [rtac ctxt @{thm predicate2I}, rtac ctxt @{thm conversepI}, rtac ctxt @{thm Basic_BNFs.rel_sum_simps(4)[THEN iffD2]}, etac ctxt @{thm conversepI}], (HEADGOAL o EVERY') [rtac ctxt (inst_REL_pos_distrI_order_refls Rs (#gammas Tss) (#deltas Tss) ctxt), (SELECT_GOAL o REPEAT_DETERM o HEADGOAL) (etac ctxt @{thm relcomppI}), rtac ctxt @{thm relcomppI[rotated]}, rtac ctxt (#map_F_rsp thms), rtac ctxt (#rep_abs_rsp qthms OF [#reflp qthms]), SELECT_GOAL (unfold_thms_tac ctxt (@{thms rel_sum_simps} @ rel_F_maps)), assume_tac ctxt], (REPEAT_DETERM_N (2*live) o HEADGOAL) (rtac ctxt @{thm rel_sum_eq2_nonempty} ORELSE' rtac ctxt @{thm rel_sum_eq3_nonempty} (REPEAT_DETERM_N live) (unfold_thms_tac ctxt @{thms sum.rel_compp[symmetric] eq_OO} THEN HEADGOAL (etac ctxt @{thm sum.rel_mono[OF order_refl]})), (HEADGOAL o EVERY') [(SELECT_GOAL o REPEAT_DETERM o HEADGOAL) (dtac ctxt @{thm rel_fun_rel_OO2}), rtac ctxt (inst_REL_pos_distrI 0 Qs (#alphas Tss) (#betas Tss) ctxt), rtac ctxt @{thm relcomppI}, rtac ctxt (#reflp qthms), rtac ctxt @{thm relcomppI}, rtac ctxt (nth rel_F_map_iffD2s 0), rtac ctxt (nth rel_F_map_iffD2s 1), etac ctxt (#rel_monoD_rotated thms)], (REPEAT_DETERM_N live o HEADGOAL o EVERY') [rtac ctxt @{thm predicate2I}, rtac ctxt @{thm rel_sum.intros(2)}, assume_tac ctxt], (HEADGOAL o EVERY') [rtac ctxt (inst_REL_pos_distrI_order_refls (map mk_conversep Rs) (#deltas Tss) (#gammas Tss) ctxt) rtac ctxt @{thm relcomppI}, etac ctxt (rotate_prems 1 (#transp qthms)), rtac ctxt (#map_F_rsp thms), rtac ctxt (#rep_abs_rsp qthms OF [#reflp qthms]), etac ctxt @{thm relcomppI}, rtac ctxt @{thm relcomppI}, etac ctxt (#transp qthms), rtac ctxt (#symp qthms), rtac ctxt (#map_F_rsp thms), rtac ctxt (#rep_abs_rsp qthms), rtac ctxt (#reflp qthms), rtac ctxt @{thm relcomppI[rotated]}, rtac ctxt (#reflp qthms), rtac ctxt (rel_flip_of_bnf bnf_F RS @{thm iffD1}), rtac ctxt (nth rel_F_map_iffD2s 0), rtac ctxt (nth rel_F_map_iffD2s 1), etac ctxt (#rel_monoD_rotated thms)], (REPEAT_DETERM_N live o HEADGOAL o EVERY') [rtac ctxt @{thm predicate2I}, rtac ctxt @{thm conversepI}, rtac ctxt @{thm rel_sum.intros(2)}, etac ctxt @{thm conversepI}], (REPEAT_DETERM_N (2*live) o HEADGOAL) (rtac ctxt @{thm rel_sum_eq2_nonempty} ORELSE' rtac ctxt @{thm rel_sum_eq3_nonempty} (REPEAT_DETERM_N live o EVERY) [unfold_thms_tac ctxt @{thms sum.rel_compp[symmetric] eq_OO}, HEADGOAL (etac ctxt @{thm sum.rel_mono[OF order_refl]})]]; (* quotient.set_transfer tactics *) fun set_transfer_q set_G_def set_F'_thms _ ctxt = let val set_F'_rsp = mk_rel_funDN 1 (#set_F'_respect set_F'_thms) in common_tac ctxt (set_G_def :: @{thms o_def}) THEN (HEADGOAL o EVERY') [etac ctxt @{thm relcomppE}, hyp_subst_tac ctxt, SELECT_GOAL (unfold_thms_tac ctxt [set_F'_rsp OF [#rep_abs qthms] OF [#reflp qthms], @{thm rel_set_def}]), dtac ctxt (#rel_F_rel_F' thms), rtac ctxt conjI] THEN (REPEAT_DETERM_N 2 o HEADGOAL o EVERY') [SELECT_GOAL (unfold_thms_tac ctxt [#rel_F'_set thms]), REPEAT_DETERM o eresolve_tac ctxt [exE, conjE], REPEAT_DETERM o dtac ctxt (mk_sym set_F'_rsp), SELECT_GOAL (unfold_thms_tac ctxt [#set_map_F' set_F'_thms]), rtac ctxt ballI, dtac ctxt @{thm equalityD1[THEN subsetD]}, assume_tac ctxt, SELECT_GOAL (unfold_thms_tac ctxt @{thms image_iff}), etac ctxt bexE, dtac ctxt set_mp, assume_tac ctxt, SELECT_GOAL (unfold_thms_tac ctxt @{thms mem_Collect_eq case_prod_beta}), rtac ctxt bexI, hyp_subst_tac ctxt, assume_tac ctxt, etac ctxt @{thm hypsubst}, etac ctxt @{thm imageI}, assume_tac ctxt] end; in {map={raw=map_raw, tac=map_transfer_q}, rel={raw=rel_raw, tac=rel_transfer_q}, sets={raws=sets_raw,tacs=map2 set_transfer_q (#sets old_defs) set_F'_thmss}, pred=NONE} end; fun quotient_bnf (equiv_thm, quot_thm) wits specs map_b rel_b pred_b opts lthy = let (* extract rep_G and abs_G *) val (equiv_rel, abs_G, rep_G) = strip3 quot_thm; val (repT, absT) = dest_funT (fastype_of abs_G); (* ("?'a F", "?'a G") *) val absT_name = fst (dest_Type absT); val tvs = map (fst o dest_TVar) (snd (dest_Type absT)); val _ = length tvs = length specs orelse error ("Expected " ^ string_of_int (length tvs) ^ " type argument" ^ (if (length tvs) = 1 then "" else "s") ^ " to " ^ quote absT_name); (* instantiate TVars *) val alpha0s = map (TFree o snd) specs; val typ_subst = typ_subst_TVars (tvs ~~ alpha0s); val (repT, absT) = apply2 typ_subst (repT, absT); (* get the bnf for RepT *) val Ds0 = filter (is_none o fst) specs |> map snd; fun flatten_tyargs Ass = map dest_TFree alpha0s |> filter (fn T => exists (fn Ts => member op= Ts T) Ass); val ((bnf_F, (deads, alphas)), ((_, unfolds), lthy)) = bnf_of_typ true Dont_Inline (Binding.qualify true absT_name) flatten_tyargs [] Ds0 repT ((empty_comp_cache, empty_unfolds), lthy); val live = length alphas; val _ = (if live = 0 then error "No live variables" else ()); val defs = #map_unfolds unfolds @ flat (#set_unfoldss unfolds) @ #rel_unfolds unfolds; val set_bs = map (fn T => find_index (fn U => T = U) alpha0s) alphas |> map (the_default Binding.empty o fst o nth specs); (* create and instantiate all the needed type variables *) val subst = subst_TVars (tvs ~~ alpha0s); val (abs_G, rep_G) = apply2 subst (abs_G, rep_G); val sorts = map Type.sort_of_atyp alphas; val (((betas, gammas), deltas), names_lthy) = fold Variable.declare_typ (alphas @ deads) lt |> mk_TFrees' sorts ||>> mk_TFrees' sorts ||>> mk_TFrees' sorts; fun subst_Ts tm Ts = subst_atomic_types (alphas ~~ Ts) tm; val subst_b = subst_atomic_types (alphas ~~ betas); val subst_Maybe = subst_atomic_types o map (swap o `mk_MaybeT); val equiv_rel_a = subst equiv_rel; val map_F = mk_map_of_bnf deads alphas betas bnf_F; val rel_F_ab = mk_rel_of_bnf deads alphas betas bnf_F; val rel_F_bc = mk_rel_of_bnf deads betas gammas bnf_F; val rel_F_ac = mk_rel_of_bnf deads alphas gammas bnf_F; val rel_F_option = mk_rel_of_bnf deads (map mk_MaybeT alphas) (map mk_MaybeT betas) bnf_F; val sets_F = mk_sets_of_bnf (replicate live deads) (replicate live alphas) bnf_F; val wits_F = mk_wits_of_bnf (replicate (nwits_of_bnf bnf_F) deads) (replicate (nwits_of_bnf bnf_F) alphas) bnf_F; val (typ_fs, (typ_aF, typ_bF)) = strip_typeN live (fastype_of map_F) ||> dest_funT; val typ_MaybeF = typ_subst_atomic (alphas ~~ map mk_MaybeT alphas) typ_aF; val typ_a_sets = map HOLogic.mk_setT alphas; val typ_pairs = map HOLogic.mk_prodT (alphas ~~ betas); val typ_fs' = map (typ_subst_atomic (map (swap o `mk_MaybeT) betas)) typ_fs; (* create all the needed variables *) val ((((((((((((((((((((((var_Ps, var_Qs), var_Rs), var_x), var_x'), var_y), var_y'), var_As), var_As'), var_Ss), var_Bs), var_as), var_as'), var_bs), var_bs'), var_R), var_fs'), var_gs), var_gs'), var_z), var_ts) = names_lthy |> mk_Frees "Ps" (map2 mk_relT alphas betas) ||>> mk_Frees "Qs" (map2 mk_relT betas gammas) ||>> mk_Frees "Rs" (map2 mk_relT alphas gammas) ||>> mk_Free "x" typ_aF ||>> mk_Free "x'" typ_aF ||>> mk_Free "y" typ_bF ||>> mk_Free "y'" (typ_subst_atomic (alphas ~~ gammas) typ_aF) ||>> mk_Free "mx" typ_MaybeF ||>> mk_Frees "As" typ_a_sets ||>> mk_Frees "As'" typ_a_sets ||>> mk_Frees "Ss" (map HOLogic.mk_setT typ_a_sets) ||>> mk_Frees "Bs" (map HOLogic.mk_setT betas) ||>> mk_Frees "as" alphas ||>> mk_Frees "as'" alphas ||>> mk_Frees "bs" betas ||>> mk_Frees "bs'" betas ||>> mk_Free "R" (typ_aF --> typ_bF --> HOLogic.boolT) ||>> mk_Frees "fs" typ_fs ||>> mk_Frees "fs'" typ_fs' ||>> mk_Frees "gs" typ_fs ||>> mk_Frees "gs'" typ_fs' ||>> mk_Free "z" (typ_subst_atomic (alphas ~~ typ_pairs) typ_aF) ||>> mk_Frees "ts" typ_pairs |> fst; (* create local definitions `b = tm` with n arguments *) fun suffix tm s = (dest_Const tm |> fst |> Long_Name.base_name) ^ s; fun define lthy b n tm = let val b = Binding.qualify true absT_name (Binding.qualified_name b); val ((tm, (_, def)), (lthy, lthy_old)) = lthy |> (snd o Local_Theory.begin_nested) |> Local_Theory.define_internal (((Binding.concealed b, NoSyn), (Binding.empty_atts, tm ||> `Local_Theory.end_nested; val phi = Proof_Context.export_morphism lthy_old lthy; val tm = Term.subst_atomic_types (map (`(Morphism.typ phi)) (alphas @ betas @ gammas @ map TFree Ds0)) (Morphism.term phi tm); val def = mk_unabs_def n (HOLogic.mk_obj_eq (Morphism.thm phi def)); in ({def=def, tm=tm}, lthy) end; (* internally use REL, not the user-provided definition *) val (REL, lthy) = define lthy "REL" 0 equiv_rel_a; val REL_def = sym RS eq_reflection OF [#def REL]; fun REL_rewr_all ctxt thm = Conv.fconv_rule (Conv.top_conv (fn _ => Conv.try_conv (Conv.rewr_conv REL_def)) ctxt) thm; val equiv_rel_a' = equiv_rel_a; val equiv_rel_a = #tm REL; val (equiv_rel_b, equiv_rel_c) = apply2 (subst_Ts equiv_rel_a) (betas, gammas); (* rel_pos_distr: @{term "\<And>A B. A \<circ>\<circ> B \<noteq> bot \<Longrightarrow> REL \<circ>\<circ> rel_F A \<circ>\<circ> REL \<circ>\<circ> re fun compp_not_bot comp aT cT = let val T = mk_relT aT cT; val mk_eq = HOLogic.eq_const T; in HOLogic.mk_not (mk_eq $ comp $ bot_const T) end; val ab_comps = map2 mk_relcompp var_Ps var_Qs; val ne_comps = (@{map 3} compp_not_bot ab_comps alphas gammas); val ab_prem = foldr1 HOLogic.mk_conj ne_comps; val REL_pos_distrI_tm = let val le_relcomps = map2 mk_leq ab_comps var_Rs; val assm = mk_OO [equiv_rel_a, list_comb (rel_F_ab, var_Ps), equiv_rel_b, list_comb (rel_F_bc, var_Qs)] equiv_rel_c; val concl = mk_OO [equiv_rel_a, list_comb (rel_F_ac, var_Rs)] equiv_rel_c; in mk_Trueprop_implies ([assm $ var_x $ var_y'] @ ne_comps @ le_relcomps, concl $ var_x $ var_y') end; val ab_concl = mk_leq (mk_OO [list_comb (rel_F_ab, var_Ps), equiv_rel_b] (list_comb (rel_F_bc, var_Qs))) (mk_OO [equiv_rel_a, list_comb (rel_F_ac, ab_comps)] (equiv_rel_c)); val ab_imp = Logic.mk_implies (apply2 HOLogic.mk_Trueprop (ab_prem, ab_concl)); val rel_pos_distr = fold_rev Logic.all (var_Ps @ var_Qs) ab_imp; (* {(x, y) . REL x y} *) fun mk_rel_pairs rel = mk_case_prod (var_x, var_x') (rel $ var_x $ var_x') val rel_pairs = mk_rel_pairs equiv_rel_a; (* rel_Inter: \<And>S. \<lbrakk> S \<noteq> {}; \<Inter>S \<noteq> {} \<rbrakk> \<Longrightarrow> (\<Inter>A\<in>S. {(x, y). REL x y} `` {x. set_F x \<subseteq> A}) \<subseteq> {(x, y). REL x y} `` {x. set_ fun rel_Inter_from_set_F (var_A, var_S) set_F = let val typ_aset = fastype_of var_A; (* \<Inter>S *) val inter_S = Inf_const typ_aset $ var_S; (* [S \<noteq> {}, \<Inter>S \<noteq> {}] *) fun not_empty x = let val ty = fastype_of x in HOLogic.mk_not (HOLogic.mk_eq (x, bot_const ty)) end; val prems = map (HOLogic.mk_Trueprop o not_empty) [var_S, inter_S]; (* {x. set_F x \<subseteq> A} *) val setF_sub_A = mk_in [var_A] [set_F] typ_aF; (* {x. set_F x \<subseteq> \<Inter>S} *) val setF_sub_S = mk_in [inter_S] [set_F] typ_aF; val lhs = Inf_const (HOLogic.mk_setT typ_aF) $ (mk_image (absfree (dest_Free var_A) (Image_const typ_aF $ rel_pairs $ setF_sub_A)) $ var_S); val rhs = Image_const typ_aF $ rel_pairs $ setF_sub_S; val concl = HOLogic.mk_Trueprop (mk_leq lhs rhs); in Logic.all var_S (Logic.list_implies (prems, concl)) end; val rel_Inters = map2 rel_Inter_from_set_F (var_As ~~ var_Ss) sets_F; (* map_F_Just = map_F Just *) val map_F_Just = let val option_tys = map mk_MaybeT alphas; val somes = map Just_const alphas; in list_comb (subst_atomic_types (betas ~~ option_tys) map_F, somes) end; fun mk_set_F'_tm typ_a set_F = let val typ_aset = HOLogic.mk_setT typ_a; (* set_F' x = (\<Inter>y\<in>{y. REL (map_F Just x) y}. UNION (set_F y) set_Maybe) *) val sbind = mk_UNION (subst_Maybe alphas set_F $ var_mx) (set_Maybe_const typ_a); val collection = HOLogic.Collect_const typ_MaybeF $ absfree (dest_Free var_mx) (subst_Maybe alphas equiv_rel_a $ (map_F_Just $ var_x) $ var_mx); val set_F'_tm = lambda var_x (Inf_const typ_aset $ (mk_image (absfree (dest_Free var_mx) sbind) $ collection)); in set_F'_tm end val sets = mk_sets_of_bnf (replicate live deads) (replicate live alphas) bnf_F; val sets' = map2 mk_set_F'_tm alphas sets; val (Iwits, wit_goals) = prepare_wits true repT wits opts alphas wits_F var_as var_as' sets' lthy; val goals = rel_pos_distr :: rel_Inters @ (case wits of NONE => [] | _ => wit_goals); val plugins = get_first (fn Plugins_Option f => SOME (f lthy) | _ => NONE) (rev opts) |> the_default Plugin_Name.default_filter; fun after_qed thmss lthy = (case thmss of [rel_pos_distr_thm0] :: thmss => let val equiv_thm' = REL_rewr_all lthy equiv_thm; val rel_pos_distr_thm = @{thm equivp_add_relconj} OF [equiv_thm', equiv_thm', rel_pos_distr_thm0]; val (rel_Inter_thms, wit_thmss) = apply2 (fn f => flat (f live thmss)) (take, drop); (* construct map_G, sets_G, bd_G, rel_G and wits_G *) (* map_G f = abs_G o map_F f o rep_G *) val map_G = fold_rev lambda var_fs (HOLogic.mk_comp (HOLogic.mk_comp (subst_Ts abs_G betas, list_comb (map_F, var_fs)), rep_G)); val map_raw = fold_rev lambda var_fs (list_comb (map_F, var_fs)) |> subst_atomic_types (betas ~~ gammas); (* Define set_G and the three auxiliary definitions (set_F', F_in, F_in') *) fun mk_set_G var_A set_F lthy = let val typ_a = HOLogic.dest_setT (fastype_of var_A); val set_F'_tm = mk_set_F'_tm typ_a set_F val (set_F', lthy) = define lthy (suffix set_F "'") 1 set_F'_tm; (* set_G = set_F' o rep_G *) val set_G = HOLogic.mk_comp (#tm set_F', rep_G); (* F_in A = {x. set_F x \<subseteq> A} *) val F_in_tm = lambda var_A (mk_in [var_A] [set_F] typ_aF); val (F_in, lthy) = define lthy (suffix set_F "_in") 1 F_in_tm; (* F_in' A = map_F Inr -` ({(x, y). REL x y} `` F_in (insert (Inl ()) (Inr ` A))) *) val F_in' = lambda var_A (mk_vimage map_F_Just (Image_const typ_MaybeF $ subst_Maybe alphas rel_pairs $ (subst_Maybe alphas (#tm F_in) $ mk_insert (mk_Nothing typ_a) (mk_image (Just_const typ_a) $ var_A)))); val (F_in', lthy) = define lthy (suffix set_F "_in'") 1 F_in'; in ((set_G, {set_F'=set_F', F_in=F_in, F_in'=F_in'}), lthy) end; val ((sets_G, set_F'_aux_defs), lthy) = @{fold_map 2} mk_set_G var_As sets_F lthy |>> split_list; (* bd_G = bd_F *) val bd_G = mk_bd_of_bnf deads alphas bnf_F; (* rel_F' A = BNF_Def.vimage2p (map_F Just) (map_F Just) ((\<cong>) OO rel_F (rel_Maybe A) OO (\<cong>)) *) val rel_Maybes = @{map 3} (fn v => fn aT => fn bT => rel_Maybe_const aT bT $ v); val rel_F'_tm = let val equiv_equiv_rel_option = subst_Ts equiv_rel_a' o map mk_Mayb mk_vimage2p map_F_Just (subst_atomic_types (alphas ~~ betas) map_F_Just) $ mk_OO [equiv_equiv_rel_option alphas, list_comb (rel_F_option, rel_Maybes var_Ps alphas (equiv_equiv_rel_option betas) end; val (rel_F', lthy) = define lthy (suffix rel_F_ab "'") (live+2) (fold_rev lambda var_Ps rel_F'_tm); (* rel_G A = vimage2p rep_G rep_G (rel_F' A) *) val rel_G = fold_rev lambda var_Ps (mk_vimage2p rep_G (subst_Ts rep_G betas) $ rel_F'_tm); val rel_raw = fold_rev lambda var_Ps rel_F'_tm |> subst_atomic_types (betas ~~ gammas); (* auxiliary lemmas *) val bd_card_order = bd_card_order_of_bnf bnf_F; val bd_cinfinite = bd_cinfinite_of_bnf bnf_F; val in_rel = in_rel_of_bnf bnf_F; val map_F_comp = map_comp_of_bnf bnf_F; val map_F_comp0 = map_comp0_of_bnf bnf_F; val map_F_cong = map_cong_of_bnf bnf_F; val map_F_id0 = map_id0_of_bnf bnf_F; val map_F_id = map_id_of_bnf bnf_F; val rel_conversep = rel_conversep_of_bnf bnf_F; val rel_flip = rel_flip_of_bnf bnf_F; val rel_eq_onp = rel_eq_onp_of_bnf bnf_F; val rel_Grp = rel_Grp_of_bnf bnf_F; val rel_OO = rel_OO_of_bnf bnf_F; val rel_map = rel_map_of_bnf bnf_F; val rel_refl_strong = rel_refl_strong_of_bnf bnf_F; val set_bd_thms = set_bd_of_bnf bnf_F; val set_map_thms = set_map_of_bnf bnf_F; (* map_F_respect: @{term "((=) ===> REL ===> REL) map_F map_F"} *) val map_F_respect = HOLogic.mk_Trueprop (fold_rev mk_rel_fun (map2 (fn xT => fn yT => HOLogic.eq_const (xT --> yT)) alphas betas @ [equiv_rel_a]) (equiv_rel_b) $ map_F $ map_F); fun map_F_respect_tac ctxt = HEADGOAL (EVERY' [REPEAT_DETERM_N (live + 1) o rtac ctxt @{thm rel_funI}, hyp_subst_tac ctxt, rtac ctxt (BNF_FP_Util.split_conj_prems live rel_pos_distr_thm0 OF replicate live @{thm Grp_conversep_nonempty} RS rev_mp), rtac ctxt impI, dtac ctxt @{thm predicate2D}, etac ctxt @{thm relcomppI2[rotated]}, rtac ctxt (rel_Grp RS @{thm predicate2_eqD} RS iffD2), rtac ctxt @{thm GrpI[OF refl]}, REPEAT_DETERM o resolve_tac ctxt [CollectI, conjI, subset_UNIV], rtac ctxt (rel_flip RS iffD2), rtac ctxt (rel_Grp RS @{thm predicate2_eqD} RS iffD2), rtac ctxt @{thm GrpI[OF refl]}, REPEAT_DETERM o resolve_tac ctxt [CollectI, conjI, subset_UNIV], SELECT_GOAL (unfold_thms_tac ctxt (rel_eq_onp :: @{thms Grp_conversep_eq_onp})), etac ctxt @{thm predicate2D[OF rel_conj_eq_onp, rotated]}, rtac ctxt equiv_thm']); val map_F_respect_thm = prove lthy [] map_F_respect map_F_respect_tac; val rel_funD = mk_rel_funDN (live+1); val map_F_rsp = (rel_funD map_F_respect_thm) OF (replicate live refl); fun map_F_rsp_of tms ctxt = (infer_instantiate' ctxt (NONE :: NONE :: map (SOME o Thm.cterm_of ctxt) tms) map_F_rsp) val qthms = let fun equivp_THEN thm = REL_rewr_all lthy equiv_thm RS thm; fun Quotient3_THEN thm = REL_rewr_all lthy quot_thm RS thm; in {abs_rep = Quotient3_THEN @{thm Quotient3_abs_rep}, rel_abs = Quotient3_THEN @{thm Quotient3_rel_abs}, rep_abs = Quotient3_THEN @{thm Quotient3_rep_abs}, rep_reflp = Quotient3_THEN @{thm Quotient3_rep_reflp}, rep_abs_rsp = Quotient3_THEN @{thm rep_abs_rsp}, reflp = equivp_THEN @{thm equivp_reflp}, symp = equivp_THEN @{thm equivp_symp}, transp = equivp_THEN @{thm equivp_transp}, REL = REL_def} end; (* lemma REL_OO_REL_left: REL OO REL OO R = REL OO R *) val REL_OO_REL_left_thm = let val tm = mk_Trueprop_eq (mk_OO [equiv_rel_a, equiv_rel_a] var_R, mk_relcompp equiv_rel_a var_R) fun tac ctxt = HEADGOAL (EVERY' [rtac ctxt ext, rtac ctxt ext, rtac ctxt iffI, TWICE (etac ctxt @{thm relcomppE}), rtac ctxt @{thm relcomppI}, etac ctxt (#transp qthms), TWICE (assume_tac ctxt), etac ctxt @{thm relcomppE}, etac ctxt @{thm relcomppI}, rtac ctxt @{thm relcomppI}, rtac ctxt (#reflp qthms), assume_tac ctxt]); in prove lthy [var_R] tm tac end; (* Generate theorems related to the setters *) val map_F_fs = list_comb (map_F, var_fs); (* aset aset asetset bset typ_b typ_b *) fun mk_set_F'_thmss (((((var_A, var_A'), var_S), var_B), var_b), var_b') set_F {set_F', F_in, F_in'} rel_Inter_thm set_map_F_thm (idx, vf) = let val (var_f, var_fs') = case vf of (f :: fs) => (f, fs) | _ => error "won't happen"; val typ_a = fastype_of var_f |> dest_funT |> fst; val typ_b = fastype_of var_b; val (typ_asetset, typ_aset) = `HOLogic.mk_setT (fastype_of var_A); val map_F_fs_x = map_F_fs $ var_x; (* F_in'_mono: A \<subseteq> B \<Longrightarrow> F_in' A \<subseteq> F_in' B *) val F_in'_mono_tm = mk_Trueprop_implies ([mk_leq var_A var_A'], mk_leq (#tm F_in' $ var_A) (#tm F_in' $ var_A')); fun F_in'_mono_tac ctxt = unfold_thms_tac ctxt [#def F_in', #def F_in] THEN HEADGOAL (EVERY' [rtac ctxt subsetI, etac ctxt vimageE, etac ctxt @{thm ImageE}, etac ctxt CollectE, etac ctxt CollectE, dtac ctxt @{thm case_prodD}, hyp_subst_tac ctxt, rtac ctxt (vimageI OF [refl]), rtac ctxt @{thm ImageI}, rtac ctxt CollectI, rtac ctxt @{thm case_prodI}, assume_tac ctxt ORELSE' rtac ctxt refl, rtac ctxt CollectI, etac ctxt subset_trans, etac ctxt (@{thm insert_mono} OF [@{thm image_mono}])]); val F_in'_mono_thm = prove lthy [var_A, var_A'] F_in'_mono_tm F_in'_mono_tac; (* F_in'_Inter: F_in' (\<Inter>S) = (\<Inter>A\<in>S. F_in' A) *) val F_in'_Inter_tm = mk_Trueprop_eq ((#tm F_in' $ (Inf_const typ_aset $ var_S)), (Inf_const (HOLogic.mk_setT typ_aF) $ (mk_image (#tm F_in') $ var_S))); fun F_in'_Inter_tac ctxt = Local_Defs.unfold_tac ctxt [#def F_in', #def F_in] THEN HEADGOAL (rtac ctxt (infer_instantiate' ctxt [SOME (Thm.cterm_of ctxt (HOLogic.mk_eq (var_S, bot_const typ_asetset)))] @{thm case_spl THEN' EVERY' [ hyp_subst_tac ctxt, SELECT_GOAL (unfold_thms_tac ctxt @{thms Inter_empty INT_empty UNIV_sum_unit_conv}), rtac ctxt @{thm set_eqI}, rtac ctxt iffI, rtac ctxt UNIV_I, rtac ctxt (vimageI OF [refl]), rtac ctxt @{thm ImageI}, rtac ctxt CollectI, rtac ctxt @{thm case_prodI}, rtac ctxt (#reflp qthms), rtac ctxt CollectI, rtac ctxt subset_UNIV, etac ctxt @{thm exE[OF ex_in_conv[THEN iffD2]]}, EqSubst.eqsubst_tac ctxt [0] @{thms image_INT[of _ UNIV _ id, simplified]}, rtac ctxt @{thm inj_Inr}, assume_tac ctxt, SELECT_GOAL (unfold_thms_tac ctxt @{thms INT_extend_simps vimage_INT[symmetric]}), rtac ctxt @{thm arg_cong2[where f=vimage, OF refl]}, rtac ctxt equalityI, rtac ctxt subsetI, rtac ctxt @{thm InterI}, etac ctxt imageE, etac ctxt @{thm ImageE}, etac ctxt CollectE, etac ctxt CollectE, dtac ctxt @{thm case_prodD}, hyp_subst_tac ctxt, rtac ctxt @{thm ImageI[OF CollectI]}, etac ctxt @{thm case_prodI} ORELSE' (SELECT_GOAL (unfold_thms_tac ctxt @{thms prod.case}) THEN' rtac ctxt @{thm refl}), rtac ctxt CollectI, etac ctxt subset_trans, etac ctxt @{thm INT_lower[OF imageI]}, rtac ctxt (@{thm subset_trans} OF [asm_rl, rel_Inter_thm]), K (unfold_thms_tac ctxt @{thms image_image}), rtac ctxt subset_refl, K (unfold_thms_tac ctxt @{thms INT_extend_simps ex_in_conv[symmetric]}), rtac ctxt exI, rtac ctxt imageI, assume_tac ctxt, rtac ctxt exI, rtac ctxt @{thm InterI}, etac ctxt imageE, hyp_subst_tac ctxt, rtac ctxt @{thm insertI1}]); val F_in'_Inter_thm = prove lthy [var_S] F_in'_Inter_tm F_in'_Inter_tac; (* set_F'_respect: (REL ===> (=)) set_F' set_F' *) val set_F'_respect_tm = HOLogic.mk_Trueprop (mk_rel_fun equiv_rel_a (HOLogic.eq_const typ_aset) $ #tm set_F' $ #tm set_F'); fun set_F'_respect_tac ctxt = unfold_thms_tac ctxt (#def set_F' :: @{thms rel_fun_de THEN HEADGOAL (EVERY' [TWICE (rtac ctxt allI), rtac ctxt impI, dtac ctxt (map_F_rsp_of (map Just_const alphas) ctxt), rtac ctxt @{thm INF_cong}, rtac ctxt @{thm Collect_eqI}, rtac ctxt iffI, etac ctxt (#transp qthms OF [#symp qthms]), assume_tac ctxt, etac ctxt (#transp qthms), assume_tac ctxt, rtac ctxt refl]); (* F_in'_alt2: F_in' A = {x. set_F' x \<subseteq> A} *) val F_in'_alt2_tm = mk_Trueprop_eq (#tm F_in' $ var_A, mk_in [var_A] [#tm set_F'] typ_aF); fun F_in'_alt2_tac ctxt = HEADGOAL (rtac ctxt equalityI THEN' (Subgoal.FOCUS o K) (unfold_thms_tac ctxt (map #def [set_F', F_in', F_in])) ctxt THEN' EVERY' [rtac ctxt subsetI, rtac ctxt CollectI, rtac ctxt subsetI, dtac ctxt vimageD, etac ctxt @{thm ImageE}, etac ctxt CollectE, etac ctxt CollectE, dtac ctxt @{thm case_prodD}, dtac ctxt @{thm InterD}, rtac ctxt @{thm imageI[OF CollectI]}, etac ctxt (#symp qthms), etac ctxt @{thm UnionE}, etac ctxt imageE, hyp_subst_tac ctxt, etac ctxt @{thm subset_lift_sum_unitD}, etac ctxt @{thm setr.cases}, hyp_subst_tac ctxt, assume_tac ctxt]) THEN unfold_thms_tac ctxt [#def set_F'] THEN (HEADGOAL o EVERY') [rtac ctxt subsetI, etac ctxt CollectE, etac ctxt (subsetD OF [F_in'_mono_thm]), EqSubst.eqsubst_tac ctxt [0] [F_in'_Inter_thm], rtac ctxt @{thm InterI}] THEN REPEAT_DETERM (HEADGOAL (etac ctxt @{thm imageE} THEN' hyp_subst_tac ctxt)) THEN (HEADGOAL o EVERY') [etac ctxt CollectE, SELECT_GOAL (unfold_thms_tac ctxt (map #def [F_in', F_in])), rtac ctxt @{thm vimageI[OF refl]}, rtac ctxt @{thm ImageI}, rtac ctxt CollectI, rtac ctxt @{thm case_prodI}, etac ctxt (#symp qthms), rtac ctxt CollectI, rtac ctxt subsetI, rtac ctxt @{thm sum_insert_Inl_unit}, assume_tac ctxt, hyp_subst_tac ctxt, rtac ctxt imageI, rtac ctxt @{thm UnionI}, rtac ctxt imageI, assume_tac ctxt, rtac ctxt @{thm setr.intros[OF refl]}]; val F_in'_alt2_thm = prove lthy [var_A] F_in'_alt2_tm F_in'_alt2_tac; (* set_F'_alt: set_F' x = \<Inter>{A. x \<in> F_in' A} *) val set_F'_alt_tm = mk_Trueprop_eq (#tm set_F' $ var_x, Inf_const typ_aset $ mk_Collect (var_A, HOLogic.mk_mem (var_x, #tm F_in' $ var_A))); fun set_F'_alt_tac ctxt = unfold_thms_tac ctxt [F_in'_alt2_thm] THEN HEADGOAL (EVERY' [rtac ctxt @{thm set_eqI}, rtac ctxt iffI, rtac ctxt @{thm InterI}, etac ctxt CollectE, etac ctxt CollectE, dtac ctxt subsetD, assume_tac ctxt, assume_tac ctxt, etac ctxt @{thm InterD}, rtac ctxt CollectI, rtac ctxt CollectI, rtac ctxt subset_refl]); val set_F'_alt_thm = prove lthy [var_x] set_F'_alt_tm set_F'_alt_tac; (* map_F_in_F_in'I: x \<in> F_in' B \<Longrightarrow> map_F f x \<in> F_in' (f ` B) *) val map_F_in_F_in'I_tm = mk_Trueprop_implies ([HOLogic.mk_mem (var_x, #tm F_in' $ v HOLogic.mk_mem (map_F_fs_x, subst_b (#tm F_in') $ (mk_image var_f $ var_A'))); fun map_F_in_F_in'I_tac ctxt = Local_Defs.unfold_tac ctxt ([#def F_in', #def F_in, Bex_def] @ @{thms vimage_def Im HEADGOAL (EVERY' [etac ctxt @{thm CollectE}, etac ctxt exE, etac ctxt conjE, etac ctxt @{thm CollectE}, etac ctxt @{thm CollectE}, dtac ctxt @{thm case_prodD}, rtac ctxt @{thm CollectI}, rtac ctxt exI, rtac ctxt @{thm conjI[rotated]}, rtac ctxt @{thm CollectI}, rtac ctxt @{thm case_prodI}, dtac ctxt (map_F_rsp_of (map mk_Maybe_map var_fs) ctxt), SELECT_GOAL (unfold_thms_tac ctxt (map_F_comp :: @{thms o_def map_sum.simps})), --> -------------------- --> maximum size reached --> -------------------- ¤ Dauer der Verarbeitung: 0.73 Sekunden (vorverarbeitet) ¤ |
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