(* Title: nominal_dt_alpha.ML Author: Cezary Kaliszyk Author: Christian Urban Definitions and proofs for the alpha-relations.*)signature NOMINAL_DT_ALPHA =sig val comb_binders: Proof.context -> bmode -> term list -> (term option * int) list -> term val define_raw_alpha: string list -> typ list -> cns_info list -> bn_info list -> bclause list list list -> term list -> Proof.context -> term list * term list * thm list * thm list * thm * local_theory val mk_alpha_distincts: Proof.context -> thm list -> thm list -> term list -> typ list -> thm list val mk_alpha_eq_iff: Proof.context -> thm list -> thm list -> thm list -> thm list -> thm list val induct_prove: typ list -> (typ * (term -> term)) list -> thm -> (Proof.context -> int -> tactic) -> Proof.context -> thm list val alpha_prove: term list -> (term * ((term * term) -> term)) list -> thm -> (Proof.context -> int -> tactic) -> Proof.context -> thm list val raw_prove_refl: term list -> term list -> thm list -> thm -> Proof.context -> thm list val raw_prove_sym: term list -> thm list -> thm -> Proof.context -> thm list val raw_prove_trans: term list -> thm list -> thm list -> thm -> thm list -> Proof.context -> thm list val raw_prove_equivp: term list -> term list -> thm list -> thm list -> thm list -> Proof.context -> thm list * thm list val raw_prove_bn_imp: term list -> term list -> thm list -> thm -> Proof.context -> thm list val raw_fv_bn_rsp_aux: term list -> term list -> term list -> term list -> term list -> thm -> thm list -> Proof.context -> thm list val raw_size_rsp_aux: term list -> thm -> thm list -> Proof.context -> thm list val raw_constrs_rsp: term list -> term list -> thm list -> thm list -> Proof.context -> thm list val raw_alpha_bn_rsp: term list -> thm list -> thm list -> thm list val raw_perm_bn_rsp: term list -> term list -> thm -> thm list -> thm list -> Proof.context -> thm list val mk_funs_rsp: thm -> thm val mk_alpha_permute_rsp: thm -> thm endstructure Nominal_Dt_Alpha: NOMINAL_DT_ALPHA =structopen Nominal_Permeqfun lookup xs x = the (AList.lookup (op=) xs x)fun group xs = AList.group (op=) xs(** definition of the inductive rules for alpha and alpha_bn **)(* construct the compound terms for prod_fv and prod_alpha *)fun mk_prod_fv (t1, t2) = let val ty1 = fastype_of t1 val ty2 = fastype_of t2 val resT = HOLogic.mk_prodT (domain_type ty1, domain_type ty2) --> @{typ "atom set"} in Const (@{const_name "prod_fv"}, [ty1, ty2] ---> resT) $ t1 $ t2 endfun mk_prod_alpha (t1, t2) = let val ty1 = fastype_of t1 val ty2 = fastype_of t2 val prodT = HOLogic.mk_prodT (domain_type ty1, domain_type ty2) val resT = [prodT, prodT] ---> @{typ "bool"} in Const (@{const_name "prod_alpha"}, [ty1, ty2] ---> resT) $ t1 $ t2 end(* generates the compound binder terms *)fun comb_binders lthy bmode args binders = let fun bind_set lthy args (NONE, i) = setify lthy (nth args i) | bind_set _ args (SOME bn, i) = bn $ (nth args i) fun bind_lst lthy args (NONE, i) = listify lthy (nth args i) | bind_lst _ args (SOME bn, i) = bn $ (nth args i) val (combine_fn, bind_fn) = case bmode of Lst => (mk_append, bind_lst) | Set => (mk_union, bind_set) | Res => (mk_union, bind_set) in binders |> map (bind_fn lthy args) |> foldl1 combine_fn end(* produces the term for an alpha with abstraction *)fun mk_alpha_term bmode fv alpha args args' binders binders' = let val (alpha_name, binder_ty) = case bmode of Lst => (@{const_name "alpha_lst"}, @{typ "atom list"}) | Set => (@{const_name "alpha_set"}, @{typ "atom set"}) | Res => (@{const_name "alpha_res"}, @{typ "atom set"}) val ty = fastype_of args val pair_ty = HOLogic.mk_prodT (binder_ty, ty) val alpha_ty = [ty, ty] ---> @{typ "bool"} val fv_ty = ty --> @{typ "atom set"} val pair_lhs = HOLogic.mk_prod (binders, args) val pair_rhs = HOLogic.mk_prod (binders', args') in HOLogic.exists_const @{typ perm} $ Abs ("p", @{typ perm}, Const (alpha_name, [pair_ty, alpha_ty, fv_ty, @{typ "perm"}, pair_ty] ---> @{typ bool}) $ pair_lhs $ alpha $ fv $ (Bound 0) $ pair_rhs) end(* for non-recursive binders we have to produce alpha_bn premises *)fun mk_alpha_bn_prem alpha_bn_map args args' bodies binder = case binder of (NONE, _) => [] | (SOME bn, i) => if member (op=) bodies i then [] else [lookup alpha_bn_map bn $ nth args i $ nth args' i](* generate the premises for an alpha rule; mk_frees is used if no binders are present *)fun mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args') bclause = let fun mk_frees i = let val arg = nth args i val arg' = nth args' i val ty = fastype_of arg in if nth is_rec i then fst (lookup alpha_map ty) $ arg $ arg' else HOLogic.mk_eq (arg, arg') end fun mk_alpha_fv i = let val ty = fastype_of (nth args i) in case AList.lookup (op=) alpha_map ty of NONE => (HOLogic.eq_const ty, supp_const ty) | SOME (alpha, fv) => (alpha, fv) end in case bclause of BC (_, [], bodies) => map (HOLogic.mk_Trueprop o mk_frees) bodies | BC (bmode, binders, bodies) => let val (alphas, fvs) = split_list (map mk_alpha_fv bodies) val comp_fv = foldl1 mk_prod_fv fvs val comp_alpha = foldl1 mk_prod_alpha alphas val comp_args = foldl1 HOLogic.mk_prod (map (nth args) bodies) val comp_args' = foldl1 HOLogic.mk_prod (map (nth args') bodies) val comp_binders = comb_binders lthy bmode args binders val comp_binders' = comb_binders lthy bmode args' binders val alpha_prem = mk_alpha_term bmode comp_fv comp_alpha comp_args comp_args' comp_binders comp_binders' val alpha_bn_prems = flat (map (mk_alpha_bn_prem alpha_bn_map args args' bodies) binders) in map HOLogic.mk_Trueprop (alpha_prem::alpha_bn_prems) end end(* produces the introduction rule for an alpha rule *)fun mk_alpha_intros lthy alpha_map alpha_bn_map (constr, ty, arg_tys, is_rec) bclauses = let val arg_names = Datatype_Prop.make_tnames arg_tys val arg_names' = Name.variant_list arg_names arg_names val args = map Free (arg_names ~~ arg_tys) val args' = map Free (arg_names' ~~ arg_tys) val alpha = fst (lookup alpha_map ty) val concl = HOLogic.mk_Trueprop (alpha $ list_comb (constr, args) $ list_comb (constr, args')) val prems = map (mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args')) bclauses in Library.foldr Logic.mk_implies (flat prems, concl) end(* produces the premise of an alpha-bn rule; we only need to treat the case special where the binding clause is empty; - if the body is not included in the bn_info, then we either produce an equation or an alpha-premise - if the body is included in the bn_info, then we create either a recursive call to alpha-bn, or no premise *)fun mk_alpha_bn lthy alpha_map alpha_bn_map bn_args is_rec (args, args') bclause = let fun mk_alpha_bn_prem i = let val arg = nth args i val arg' = nth args' i val ty = fastype_of arg in case AList.lookup (op=) bn_args i of NONE => (case (AList.lookup (op=) alpha_map ty) of NONE => [HOLogic.mk_eq (arg, arg')] | SOME (alpha, _) => [alpha $ arg $ arg']) | SOME (NONE) => [] | SOME (SOME bn) => [lookup alpha_bn_map bn $ arg $ arg'] end in case bclause of BC (_, [], bodies) => map HOLogic.mk_Trueprop (flat (map mk_alpha_bn_prem bodies)) | _ => mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args') bclause endfun mk_alpha_bn_intro lthy bn_trm alpha_map alpha_bn_map (bn_args, (constr, _, arg_tys, is_rec)) bclauses = let val arg_names = Datatype_Prop.make_tnames arg_tys val arg_names' = Name.variant_list arg_names arg_names val args = map Free (arg_names ~~ arg_tys) val args' = map Free (arg_names' ~~ arg_tys) val alpha_bn = lookup alpha_bn_map bn_trm val concl = HOLogic.mk_Trueprop (alpha_bn $ list_comb (constr, args) $ list_comb (constr, args')) val prems = map (mk_alpha_bn lthy alpha_map alpha_bn_map bn_args is_rec (args, args')) bclauses in Library.foldr Logic.mk_implies (flat prems, concl) endfun mk_alpha_bn_intros lthy alpha_map alpha_bn_map constrs_info bclausesss (bn_trm, bn_n, bn_argss) = let val nth_constrs_info = nth constrs_info bn_n val nth_bclausess = nth bclausesss bn_n in map2 (mk_alpha_bn_intro lthy bn_trm alpha_map alpha_bn_map) (bn_argss ~~ nth_constrs_info) nth_bclausess endfun define_raw_alpha raw_full_ty_names raw_tys cns_info bn_info bclausesss fvs lthy = let val alpha_names = map (prefix "alpha_" o Long_Name.base_name) raw_full_ty_names val alpha_tys = map (fn ty => [ty, ty] ---> @{typ bool}) raw_tys val alpha_frees = map Free (alpha_names ~~ alpha_tys) val alpha_map = raw_tys ~~ (alpha_frees ~~ fvs) val (bns, bn_tys) = split_list (map (fn (bn, i, _) => (bn, i)) bn_info) val bn_names = map (fn bn => Long_Name.base_name (fst (dest_Const bn))) bns val alpha_bn_names = map (prefix "alpha_") bn_names val alpha_bn_arg_tys = map (nth raw_tys) bn_tys val alpha_bn_tys = map (fn ty => [ty, ty] ---> @{typ "bool"}) alpha_bn_arg_tys val alpha_bn_frees = map Free (alpha_bn_names ~~ alpha_bn_tys) val alpha_bn_map = bns ~~ alpha_bn_frees val alpha_intros = map2 (map2 (mk_alpha_intros lthy alpha_map alpha_bn_map)) cns_info bclausesss val alpha_bn_intros = map (mk_alpha_bn_intros lthy alpha_map alpha_bn_map cns_info bclausesss) bn_info val all_alpha_names = map (fn (a, ty) => ((Binding.name a, ty), NoSyn)) (alpha_names @ alpha_bn_names ~~ alpha_tys @ alpha_bn_tys) val all_alpha_intros = map (pair Attrib.empty_binding) (flat alpha_intros @ flat alpha_bn_intros) val (alphas, lthy') = Inductive.add_inductive_i {quiet_mode = true, verbose = false, alt_name = Binding.empty, coind = false, no_elim = false, no_ind = false, skip_mono = false, fork_mono = false} all_alpha_names [] all_alpha_intros [] lthy val all_alpha_trms_loc = #preds alphas; val alpha_induct_loc = #raw_induct alphas; val alpha_intros_loc = #intrs alphas; val alpha_cases_loc = #elims alphas; val phi = ProofContext.export_morphism lthy' lthy; val all_alpha_trms = map (Morphism.term phi) all_alpha_trms_loc val (all_alpha_trms', _) = Variable.importT_terms all_alpha_trms lthy val alpha_induct = Morphism.thm phi alpha_induct_loc; val alpha_intros = map (Morphism.thm phi) alpha_intros_loc val alpha_cases = map (Morphism.thm phi) alpha_cases_loc val (alpha_trms, alpha_bn_trms) = chop (length fvs) all_alpha_trms' in (alpha_trms, alpha_bn_trms, alpha_intros, alpha_cases, alpha_induct, lthy') end(** produces the distinctness theorems **)(* transforms the distinctness theorems of the constructors to "not-alphas" of the constructors *)fun mk_distinct_goal ty_trm_assoc neq = let val (lhs, rhs) = neq |> HOLogic.dest_Trueprop |> HOLogic.dest_not |> HOLogic.dest_eq val ty = fastype_of lhs in (lookup ty_trm_assoc ty) $ lhs $ rhs |> HOLogic.mk_not |> HOLogic.mk_Trueprop endfun distinct_tac cases_thms distinct_thms = rtac notI THEN' eresolve_tac cases_thms THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps distinct_thms)fun mk_alpha_distincts ctxt cases_thms distinct_thms alpha_trms alpha_tys = let (* proper import of type-variables does not work, since then they are replaced by new variables, messing up the ty_trm assoc list *) val distinct_thms' = map Thm.legacy_freezeT distinct_thms val ty_trm_assoc = alpha_tys ~~ alpha_trms fun mk_alpha_distinct distinct_trm = let val ([trm'], ctxt') = Variable.import_terms true [distinct_trm] ctxt val goal = mk_distinct_goal ty_trm_assoc distinct_trm in Goal.prove ctxt' [] [] goal (K (distinct_tac cases_thms distinct_thms 1)) |> singleton (Variable.export ctxt' ctxt) end in map (mk_alpha_distinct o prop_of) distinct_thms' |> map Thm.varifyT_global end(** produces the alpha_eq_iff simplification rules **)(* in case a theorem is of the form (Rel Const Const), it will be rewritten to ((Rel Const = Const) = True) *)fun mk_simp_rule thm = case (prop_of thm) of @{term "Trueprop"} $ (_ $ Const _ $ Const _) => thm RS @{thm eqTrueI} | _ => thmfun alpha_eq_iff_tac dist_inj intros elims = SOLVED' (asm_full_simp_tac (HOL_ss addsimps intros)) ORELSE' (rtac @{thm iffI} THEN' RANGE [eresolve_tac elims THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps dist_inj), asm_full_simp_tac (HOL_ss addsimps intros)])fun mk_alpha_eq_iff_goal thm = let val prop = prop_of thm; val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl prop); val hyps = map HOLogic.dest_Trueprop (Logic.strip_imp_prems prop); fun list_conj l = foldr1 HOLogic.mk_conj l; in if hyps = [] then HOLogic.mk_Trueprop concl else HOLogic.mk_Trueprop (HOLogic.mk_eq (concl, list_conj hyps)) end;fun mk_alpha_eq_iff ctxt alpha_intros distinct_thms inject_thms alpha_elims = let val ((_, thms_imp), ctxt') = Variable.import false alpha_intros ctxt; val goals = map mk_alpha_eq_iff_goal thms_imp; val tac = alpha_eq_iff_tac (distinct_thms @ inject_thms) alpha_intros alpha_elims 1; val thms = map (fn goal => Goal.prove ctxt' [] [] goal (K tac)) goals; in Variable.export ctxt' ctxt thms |> map mk_simp_rule end(** proof by induction over types **)fun induct_prove tys props induct_thm cases_tac ctxt = let val (arg_names, ctxt') = Variable.variant_fixes (replicate (length tys) "x") ctxt val args = map2 (curry Free) arg_names tys val true_trms = replicate (length tys) (K @{term True}) fun apply_all x fs = map (fn f => f x) fs val goals = group (props @ (tys ~~ true_trms)) |> map snd |> map2 apply_all args |> map fold_conj |> foldr1 HOLogic.mk_conj |> HOLogic.mk_Trueprop fun tac ctxt = HEADGOAL (DETERM o (rtac induct_thm) THEN_ALL_NEW (REPEAT_ALL_NEW (FIRST' [resolve_tac @{thms TrueI conjI}, cases_tac ctxt]))) in Goal.prove ctxt' [] [] goals (fn {context, ...} => tac context) |> singleton (ProofContext.export ctxt' ctxt) |> Datatype_Aux.split_conj_thm |> map Datatype_Aux.split_conj_thm |> flat |> filter_out (is_true o concl_of) |> map zero_var_indexes end(** proof by induction over the alpha-definitions **)fun alpha_prove alphas props alpha_induct_thm cases_tac ctxt = let val arg_tys = map (domain_type o fastype_of) alphas val ((arg_names1, arg_names2), ctxt') = ctxt |> Variable.variant_fixes (replicate (length alphas) "x") ||>> Variable.variant_fixes (replicate (length alphas) "y") val args1 = map2 (curry Free) arg_names1 arg_tys val args2 = map2 (curry Free) arg_names2 arg_tys val true_trms = replicate (length alphas) (K @{term True}) fun apply_all x fs = map (fn f => f x) fs val goals_rhs = group (props @ (alphas ~~ true_trms)) |> map snd |> map2 apply_all (args1 ~~ args2) |> map fold_conj fun apply_trm_pair t (ar1, ar2) = t $ ar1 $ ar2 val goals_lhs = map2 apply_trm_pair alphas (args1 ~~ args2) val goals = (map2 (curry HOLogic.mk_imp) goals_lhs goals_rhs) |> foldr1 HOLogic.mk_conj |> HOLogic.mk_Trueprop fun tac ctxt = HEADGOAL (DETERM o (rtac alpha_induct_thm) THEN_ALL_NEW FIRST' [rtac @{thm TrueI}, cases_tac ctxt]) in Goal.prove ctxt' [] [] goals (fn {context, ...} => tac context) |> singleton (ProofContext.export ctxt' ctxt) |> Datatype_Aux.split_conj_thm |> map (fn th => th RS mp) |> map Datatype_Aux.split_conj_thm |> flat |> filter_out (is_true o concl_of) |> map zero_var_indexes end(** reflexivity proof for the alphas **)val exi_zero = @{lemma "P (0::perm) ==> (? x. P x)" by auto}fun cases_tac intros ctxt = let val prod_simps = @{thms split_conv prod_alpha_def prod_rel_def} val unbound_tac = REPEAT o (etac @{thm conjE}) THEN' atac val bound_tac = EVERY' [ rtac exi_zero, resolve_tac @{thms alpha_refl}, asm_full_simp_tac (HOL_ss addsimps prod_simps) ] in resolve_tac intros THEN_ALL_NEW FIRST' [rtac @{thm refl}, unbound_tac, bound_tac] endfun raw_prove_refl alpha_trms alpha_bns alpha_intros raw_dt_induct ctxt = let val arg_tys = alpha_trms |> map fastype_of |> map domain_type val arg_bn_tys = alpha_bns |> map fastype_of |> map domain_type val props = map (fn (ty, c) => (ty, fn x => c $ x $ x)) ((arg_tys ~~ alpha_trms) @ (arg_bn_tys ~~ alpha_bns)) in induct_prove arg_tys props raw_dt_induct (cases_tac alpha_intros) ctxt end(** symmetry proof for the alphas **)val exi_neg = @{lemma "(EX (p::perm). P p) ==> (!!q. P q ==> Q (- q)) ==> EX p. Q p" by (erule exE, rule_tac x="-p" in exI, auto)}(* for premises that contain binders *)fun prem_bound_tac pred_names ctxt = let fun trans_prem_tac pred_names ctxt = SUBPROOF (fn {prems, context, ...} => let val prems' = map (transform_prem1 context pred_names) prems in resolve_tac prems' 1 end) ctxt val prod_simps = @{thms split_conv permute_prod.simps prod_alpha_def prod_rel_def alphas} in EVERY' [ etac exi_neg, resolve_tac @{thms alpha_sym_eqvt}, asm_full_simp_tac (HOL_ss addsimps prod_simps), eqvt_tac ctxt eqvt_relaxed_config THEN' rtac @{thm refl}, trans_prem_tac pred_names ctxt ] endfun raw_prove_sym alpha_trms alpha_intros alpha_induct ctxt = let val props = map (fn t => fn (x, y) => t $ y $ x) alpha_trms fun tac ctxt = let val alpha_names = map (fst o dest_Const) alpha_trms in resolve_tac alpha_intros THEN_ALL_NEW FIRST' [atac, rtac @{thm sym} THEN' atac, prem_bound_tac alpha_names ctxt] end in alpha_prove alpha_trms (alpha_trms ~~ props) alpha_induct tac ctxt end(** transitivity proof for alphas **)(* applies cases rules and resolves them with the last premise *)fun ecases_tac cases = Subgoal.FOCUS (fn {prems, ...} => HEADGOAL (resolve_tac cases THEN' rtac (List.last prems)))fun aatac pred_names = SUBPROOF (fn {prems, context, ...} => HEADGOAL (resolve_tac (map (transform_prem1 context pred_names) prems)))(* instantiates exI with the permutation p + q *)val perm_inst_tac = Subgoal.FOCUS (fn {params, ...} => let val (p, q) = pairself snd (last2 params) val pq_inst = foldl1 (uncurry Thm.capply) [@{cterm "plus::perm => perm => perm"}, p, q] val exi_inst = Drule.instantiate' [SOME (@{ctyp "perm"})] [NONE, SOME pq_inst] @{thm exI} in HEADGOAL (rtac exi_inst) end)fun non_trivial_cases_tac pred_names intros ctxt = let val prod_simps = @{thms split_conv alphas permute_prod.simps prod_alpha_def prod_rel_def} in resolve_tac intros THEN_ALL_NEW (asm_simp_tac HOL_basic_ss THEN' TRY o EVERY' (* if binders are present *) [ etac @{thm exE}, etac @{thm exE}, perm_inst_tac ctxt, resolve_tac @{thms alpha_trans_eqvt}, atac, aatac pred_names ctxt, eqvt_tac ctxt eqvt_relaxed_config THEN' rtac @{thm refl}, asm_full_simp_tac (HOL_ss addsimps prod_simps) ]) endfun prove_trans_tac pred_names raw_dt_thms intros cases ctxt = let fun all_cases ctxt = asm_full_simp_tac (HOL_basic_ss addsimps raw_dt_thms) THEN' TRY o non_trivial_cases_tac pred_names intros ctxt in EVERY' [ rtac @{thm allI}, rtac @{thm impI}, ecases_tac cases ctxt THEN_ALL_NEW all_cases ctxt ] endfun prep_trans_goal alpha_trm (arg1, arg2) = let val arg_ty = fastype_of arg1 val mid = alpha_trm $ arg2 $ (Bound 0) val rhs = alpha_trm $ arg1 $ (Bound 0) in HOLogic.all_const arg_ty $ Abs ("z", arg_ty, HOLogic.mk_imp (mid, rhs)) endfun raw_prove_trans alpha_trms raw_dt_thms alpha_intros alpha_induct alpha_cases ctxt = let val alpha_names = map (fst o dest_Const) alpha_trms val props = map prep_trans_goal alpha_trms in alpha_prove alpha_trms (alpha_trms ~~ props) alpha_induct (prove_trans_tac alpha_names raw_dt_thms alpha_intros alpha_cases) ctxt end(** proves the equivp predicate for all alphas **)val reflp_def' = @{lemma "reflp R == !x. R x x" by (simp add: reflp_def refl_on_def)}val symp_def' = @{lemma "symp R == !x y . R x y --> R y x" by (simp add: symp_def sym_def)}val transp_def' = @{lemma "transp R == !x y. R x y --> (!z. R y z --> R x z)" by (rule eq_reflection, auto simp add: trans_def transp_def)}fun raw_prove_equivp alphas alpha_bns refl symm trans ctxt = let val refl' = map (fold_rule [reflp_def'] o atomize) refl val symm' = map (fold_rule [symp_def'] o atomize) symm val trans' = map (fold_rule [transp_def'] o atomize) trans fun prep_goal t = HOLogic.mk_Trueprop (Const (@{const_name "equivp"}, fastype_of t --> @{typ bool}) $ t) in Goal.prove_multi ctxt [] [] (map prep_goal (alphas @ alpha_bns)) (K (HEADGOAL (Goal.conjunction_tac THEN_ALL_NEW (rtac @{thm equivpI} THEN' RANGE [resolve_tac refl', resolve_tac symm', resolve_tac trans'])))) |> chop (length alphas) end(* proves that alpha_raw implies alpha_bn *)fun raw_prove_bn_imp_tac pred_names alpha_intros ctxt = SUBPROOF (fn {prems, context, ...} => let val prems' = flat (map Datatype_Aux.split_conj_thm prems) val prems'' = map (transform_prem1 context pred_names) prems' in HEADGOAL (REPEAT_ALL_NEW (FIRST' [ rtac @{thm TrueI}, rtac @{thm conjI}, resolve_tac prems', resolve_tac prems'', resolve_tac alpha_intros ])) end) ctxtfun raw_prove_bn_imp alpha_trms alpha_bn_trms alpha_intros alpha_induct ctxt = let val arg_ty = domain_type o fastype_of val alpha_names = map (fst o dest_Const) alpha_trms val ty_assoc = map (fn t => (arg_ty t, t)) alpha_trms val props = map (fn t => (lookup ty_assoc (arg_ty t), fn (x, y) => t $ x $ y)) alpha_bn_trms in alpha_prove (alpha_trms @ alpha_bn_trms) props alpha_induct (raw_prove_bn_imp_tac alpha_names alpha_intros) ctxt end(* respectfulness for fv_raw / bn_raw *)fun raw_fv_bn_rsp_aux alpha_trms alpha_bn_trms fvs bns fv_bns alpha_induct simps ctxt = let val arg_ty = domain_type o fastype_of val ty_assoc = map (fn t => (arg_ty t, t)) alpha_trms fun mk_eq' t x y = HOLogic.mk_eq (t $ x, t $ y) val prop1 = map (fn t => (lookup ty_assoc (arg_ty t), fn (x, y) => mk_eq' t x y)) fvs val prop2 = map (fn t => (lookup ty_assoc (arg_ty t), fn (x, y) => mk_eq' t x y)) (bns @ fv_bns) val prop3 = map2 (fn t1 => fn t2 => (t1, fn (x, y) => mk_eq' t2 x y)) alpha_bn_trms fv_bns val ss = HOL_ss addsimps (simps @ @{thms alphas prod_fv.simps set.simps append.simps} @ @{thms Un_assoc Un_insert_left Un_empty_right Un_empty_left}) in alpha_prove (alpha_trms @ alpha_bn_trms) (prop1 @ prop2 @ prop3) alpha_induct (K (asm_full_simp_tac ss)) ctxt end(* respectfulness for size *)fun raw_size_rsp_aux all_alpha_trms alpha_induct simps ctxt = let val arg_tys = map (domain_type o fastype_of) all_alpha_trms fun mk_prop ty (x, y) = HOLogic.mk_eq (HOLogic.size_const ty $ x, HOLogic.size_const ty $ y) val props = map2 (fn trm => fn ty => (trm, mk_prop ty)) all_alpha_trms arg_tys val ss = HOL_ss addsimps (simps @ @{thms alphas prod_alpha_def prod_rel_def permute_prod_def prod.cases prod.recs}) val tac = (TRY o REPEAT o etac @{thm exE}) THEN' asm_full_simp_tac ss in alpha_prove all_alpha_trms props alpha_induct (K tac) ctxt end(* respectfulness for constructors *)fun raw_constr_rsp_tac alpha_intros simps = let val pre_ss = HOL_ss addsimps @{thms fun_rel_def} val post_ss = HOL_ss addsimps @{thms alphas prod_alpha_def prod_rel_def prod_fv.simps fresh_star_zero permute_zero prod.cases} @ simps in asm_full_simp_tac pre_ss THEN' REPEAT o (resolve_tac @{thms allI impI}) THEN' resolve_tac alpha_intros THEN_ALL_NEW (TRY o (rtac exi_zero) THEN' asm_full_simp_tac post_ss) endfun raw_constrs_rsp constrs alpha_trms alpha_intros simps ctxt = let val alpha_arg_tys = map (domain_type o fastype_of) alpha_trms fun lookup ty = case AList.lookup (op=) (alpha_arg_tys ~~ alpha_trms) ty of NONE => HOLogic.eq_const ty | SOME alpha => alpha fun fun_rel_app t1 t2 = Const (@{const_name "fun_rel"}, dummyT) $ t1 $ t2 fun prep_goal trm = trm |> strip_type o fastype_of |>> map lookup ||> lookup |> uncurry (fold_rev fun_rel_app) |> (fn t => t $ trm $ trm) |> Syntax.check_term ctxt |> HOLogic.mk_Trueprop in Goal.prove_multi ctxt [] [] (map prep_goal constrs) (K (HEADGOAL (Goal.conjunction_tac THEN_ALL_NEW raw_constr_rsp_tac alpha_intros simps))) end(* rsp lemmas for alpha_bn relations *)val rsp_equivp = @{lemma "[|equivp Q; !!x y. R x y ==> Q x y|] ==> (R ===> R ===> op =) Q Q" by (simp only: fun_rel_def equivp_def, metis)}(* we have to reorder the alpha_bn_imps theorems in order to be in order with alpha_bn_trms *)fun raw_alpha_bn_rsp alpha_bn_trms alpha_bn_equivp alpha_bn_imps = let fun mk_map thm = thm |> `prop_of |>> List.last o snd o strip_comb |>> HOLogic.dest_Trueprop |>> head_of |>> fst o dest_Const val alpha_bn_imps' = map (lookup (map mk_map alpha_bn_imps) o fst o dest_Const) alpha_bn_trms fun mk_thm thm1 thm2 = (forall_intr_vars thm2) COMP (thm1 RS rsp_equivp) in map2 mk_thm alpha_bn_equivp alpha_bn_imps' end(* rsp for permute_bn functions *)val perm_bn_rsp = @{lemma "(!x y p. R x y --> R (f p x) (f p y)) ==> (op= ===> R ===> R) f f" by (simp add: fun_rel_def)}fun raw_prove_perm_bn_tac pred_names alpha_intros simps ctxt = SUBPROOF (fn {prems, context, ...} => let val prems' = flat (map Datatype_Aux.split_conj_thm prems) val prems'' = map (transform_prem1 context pred_names) prems' in HEADGOAL (simp_tac (HOL_basic_ss addsimps (simps @ prems')) THEN' TRY o REPEAT_ALL_NEW (FIRST' [ rtac @{thm TrueI}, rtac @{thm conjI}, rtac @{thm refl}, resolve_tac prems', resolve_tac prems'', resolve_tac alpha_intros ])) end) ctxtfun raw_perm_bn_rsp alpha_trms perm_bns alpha_induct alpha_intros simps ctxt = let val arg_ty = domain_type o fastype_of val perm_bn_ty = range_type o range_type o fastype_of val ty_assoc = map (fn t => (arg_ty t, t)) alpha_trms val ([p], ctxt') = Variable.variant_fixes ["p"] ctxt val p = Free (p, @{typ perm}) fun mk_prop t = let val alpha_trm = lookup ty_assoc (perm_bn_ty t) in (alpha_trm, fn (x, y) => alpha_trm $ (t $ p $ x) $ (t $ p $ y)) end val goals = map mk_prop perm_bns val alpha_names = map (fst o dest_Const) alpha_trms in alpha_prove alpha_trms goals alpha_induct (raw_prove_perm_bn_tac alpha_names alpha_intros simps) ctxt |> ProofContext.export ctxt' ctxt |> map atomize |> map single |> map (curry (op OF) perm_bn_rsp) end(* transformation of the natural rsp-lemmas into standard form *)val fun_rsp = @{lemma "(!x y. R x y --> f x = f y) ==> (R ===> (op =)) f f" by (simp add: fun_rel_def)}fun mk_funs_rsp thm = thm |> atomize |> single |> curry (op OF) fun_rspval permute_rsp = @{lemma "(!x y p. R x y --> R (permute p x) (permute p y)) ==> ((op =) ===> R ===> R) permute permute" by (simp add: fun_rel_def)}fun mk_alpha_permute_rsp thm = thm |> atomize |> single |> curry (op OF) permute_rspend (* structure *)