theory Fvimports "Nominal2_Atoms" "Abs" "Perm" "Rsp"begin(* Bindings are a list of lists of lists of triples. The first list represents the datatypes defined. The second list represents the constructors. The internal list is a list of all the bndings that concern the constructor. Every triple consists of a function, the binding and the body. Eg:nominal_datatype C1 | C2 x y z bind x in z | C3 x y z bind f x in z bind g y in zyields:[ [], [(NONE, 0, 2)], [(SOME (Const f), 0, 2), (Some (Const g), 1, 2)]]A SOME binding has to have a function which takes an appropriateargument and returns an atom set. A NONE binding has to be on anargument that is an atom or an atom set.How the procedure works: For each of the defined datatypes, For each of the constructors, It creates a union of free variables for each argument. For an argument the free variables are the variables minus bound variables. The variables are: For an atom, a singleton set with the atom itself. For an atom set, the atom set itself. For a recursive argument, the appropriate fv function applied to it. (* TODO: This one is not implemented *) For other arguments it should be an appropriate fv function stored in the database. The bound variables are a union of results of all bindings that involve the given argument. For a paricular binding the result is: For a function applied to an argument this function with the argument. For an atom, a singleton set with the atom itself. For an atom set, the atom set itself. For a recursive argument, the appropriate fv function applied to it. (* TODO: This one is not implemented *) For other arguments it should be an appropriate fv function stored in the database. For every argument that is a binding, we add a the same binding in its body.*)ML {*fun is_atom thy typ = Sign.of_sort thy (typ, @{sort at})fun is_atom_set thy (Type ("fun", [t, @{typ bool}])) = is_atom thy t | is_atom_set thy _ = false;*}(* Like map2, only if the second list is empty passes empty lists insted of error *)ML {*fun map2i _ [] [] = [] | map2i f (x :: xs) (y :: ys) = f x y :: map2i f xs ys | map2i f (x :: xs) [] = f x [] :: map2i f xs [] | map2i _ _ _ = raise UnequalLengths;*}(* Finds bindings with the same function and binding, and gathers all bodys for such pairs *)ML {*fun gather_binds binds =let fun gather_binds_cons binds = let val common = map (fn (f, bi, _) => (f, bi)) binds val nodups = distinct (op =) common fun find_bodys (sf, sbi) = filter (fn (f, bi, _) => f = sf andalso bi = sbi) binds val bodys = map ((map (fn (_, _, bo) => bo)) o find_bodys) nodups in nodups ~~ bodys endin map (map gather_binds_cons) bindsend*}ML {*fun un_gather_binds_cons binds = flat (map (fn (((f, bi), bos), pi) => map (fn bo => ((f, bi, bo), pi)) bos) binds)*}ML {* open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *); (* TODO: It is the same as one in 'nominal_atoms' *) fun mk_atom ty = Const (@{const_name atom}, ty --> @{typ atom}); val noatoms = @{term "{} :: atom set"}; fun mk_single_atom x = HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x]; fun mk_union sets = fold (fn a => fn b => if a = noatoms then b else if b = noatoms then a else if a = b then a else HOLogic.mk_binop @{const_name sup} (a, b)) (rev sets) noatoms; val mk_inter = foldr1 (HOLogic.mk_binop @{const_name inf}) fun mk_conjl props = fold (fn a => fn b => if a = @{term True} then b else if b = @{term True} then a else HOLogic.mk_conj (a, b)) props @{term True}; fun mk_diff a b = if b = noatoms then a else if b = a then noatoms else HOLogic.mk_binop @{const_name minus} (a, b); fun mk_atoms t = let val ty = fastype_of t; val atom_ty = HOLogic.dest_setT ty --> @{typ atom}; val img_ty = atom_ty --> ty --> @{typ "atom set"}; in (Const (@{const_name image}, img_ty) $ Const (@{const_name atom}, atom_ty) $ t) end; (* Similar to one in USyntax *) fun mk_pair (fst, snd) = let val ty1 = fastype_of fst val ty2 = fastype_of snd val c = HOLogic.pair_const ty1 ty2 in c $ fst $ snd end;*}ML {* fun add_perm (p1, p2) = Const(@{const_name plus}, @{typ "perm \<Rightarrow> perm \<Rightarrow> perm"}) $ p1 $ p2 *}(* TODO: Notice datatypes without bindings and replace alpha with equality *)ML {*fun define_fv_alpha (dt_info : Datatype_Aux.info) bindsall lthy =let val thy = ProofContext.theory_of lthy; val {descr, sorts, ...} = dt_info; fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i); val fv_names = Datatype_Prop.indexify_names (map (fn (i, _) => "fv_" ^ name_of_typ (nth_dtyp i)) descr); val fv_types = map (fn (i, _) => nth_dtyp i --> @{typ "atom set"}) descr; val fv_frees = map Free (fv_names ~~ fv_types); val alpha_names = Datatype_Prop.indexify_names (map (fn (i, _) => "alpha_" ^ name_of_typ (nth_dtyp i)) descr); val alpha_types = map (fn (i, _) => nth_dtyp i --> nth_dtyp i --> @{typ bool}) descr; val alpha_frees = map Free (alpha_names ~~ alpha_types); fun fv_alpha_constr ith_dtyp (cname, dts) bindcs = let val Ts = map (typ_of_dtyp descr sorts) dts; val bindslen = length bindcs val pi_strs_same = replicate bindslen "pi" val pi_strs = Name.variant_list [] pi_strs_same; val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs; val bind_pis_gath = bindcs ~~ pis; val bind_pis = un_gather_binds_cons bind_pis_gath; val bindcs = map fst bind_pis; val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts); val args = map Free (names ~~ Ts); val names2 = Name.variant_list (pi_strs @ names) (Datatype_Prop.make_tnames Ts); val args2 = map Free (names2 ~~ Ts); val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp)); val fv_c = nth fv_frees ith_dtyp; val alpha = nth alpha_frees ith_dtyp; val arg_nos = 0 upto (length dts - 1) fun fv_bind args (NONE, i, _) = if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atoms (nth args i) else (* TODO we do not know what to do with non-atomizable things *) @{term "{} :: atom set"} | fv_bind args (SOME f, i, _) = f $ (nth args i); fun fv_binds args relevant = mk_union (map (fv_bind args) relevant) fun fv_arg ((dt, x), arg_no) = let val arg = if is_rec_type dt then nth fv_frees (body_index dt) $ x else if ((is_atom thy) o fastype_of) x then mk_single_atom x else if ((is_atom_set thy) o fastype_of) x then mk_atoms x else (* TODO we do not know what to do with non-atomizable things *) @{term "{} :: atom set"} (* If i = j then we generate it only once *) val relevant = filter (fn (_, i, j) => ((i = arg_no) orelse (j = arg_no))) bindcs; val sub = fv_binds args relevant in mk_diff arg sub end; val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (fv_c $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ arg_nos)))) val alpha_rhs = HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2))); fun alpha_arg ((dt, arg_no), (arg, arg2)) = let val relevant = filter (fn ((_, i, j), _) => i = arg_no orelse j = arg_no) bind_pis; in if relevant = [] then ( if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2) else (HOLogic.mk_eq (arg, arg2))) else if is_rec_type dt then let (* THE HARD CASE *) val (rbinds, rpis) = split_list relevant val lhs_binds = fv_binds args rbinds val lhs = mk_pair (lhs_binds, arg); val rhs_binds = fv_binds args2 rbinds; val rhs = mk_pair (rhs_binds, arg2); val alpha = nth alpha_frees (body_index dt); val fv = nth fv_frees (body_index dt); val pi = foldr1 add_perm (distinct (op =) rpis); val alpha_gen_pre = Const (@{const_name alpha_gen}, dummyT) $ lhs $ alpha $ fv $ pi $ rhs; val alpha_gen = Syntax.check_term lthy alpha_gen_pre(* val pi_supps = map ((curry op $) @{term "supp :: perm \<Rightarrow> atom set"}) rpis; val pi_supps_eq = HOLogic.mk_eq (mk_inter pi_supps, @{term "{} :: atom set"}) *) in (*if length pi_supps > 1 then HOLogic.mk_conj (alpha_gen, pi_supps_eq) else*) alpha_gen (* TODO Add some test that is makes sense *) end else @{term "True"} end val alphas = map alpha_arg (dts ~~ arg_nos ~~ (args ~~ args2)) val alpha_lhss = mk_conjl alphas val alpha_lhss_ex = fold (fn pi_str => fn t => HOLogic.mk_exists (pi_str, @{typ perm}, t)) pi_strs alpha_lhss val alpha_eq = Logic.mk_implies (HOLogic.mk_Trueprop alpha_lhss_ex, alpha_rhs) in (fv_eq, alpha_eq) end; fun fv_alpha_eq (i, (_, _, constrs)) binds = map2i (fv_alpha_constr i) constrs binds; val (fv_eqs, alpha_eqs) = split_list (flat (map2i fv_alpha_eq descr (gather_binds bindsall))) val add_binds = map (fn x => (Attrib.empty_binding, x)) val (fvs, lthy') = (Primrec.add_primrec (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names) (add_binds fv_eqs) lthy) 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 = true, fork_mono = false} (map2 (fn x => fn y => ((Binding.name x, y), NoSyn)) alpha_names alpha_types) [] (add_binds alpha_eqs) [] lthy')in ((fvs, alphas), lthy'')end*}(* testsatom_decl namedatatype ty = Var "name set"ML {* Syntax.check_term @{context} (mk_atoms @{term "a :: name set"}) *}local_setup {* define_fv_alpha "Fv.ty" [[[[]]]] *}print_theoremsdatatype rtrm1 = rVr1 "name"| rAp1 "rtrm1" "rtrm1"| rLm1 "name" "rtrm1" --"name is bound in trm1"| rLt1 "bp" "rtrm1" "rtrm1" --"all variables in bp are bound in the 2nd trm1"and bp = BUnit| BVr "name"| BPr "bp" "bp"(* to be given by the user *)primrec bv1where "bv1 (BUnit) = {}"| "bv1 (BVr x) = {atom x}"| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp1)"setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Fv.rtrm1", "Fv.bp"] *}local_setup {* define_fv_alpha "Fv.rtrm1" [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv1}, 0)], [], [(SOME @{term bv1}, 0)]]], [[], [[]], [[], []]]] *}print_theorems*)ML {*fun alpha_inj_tac dist_inj intrs elims = SOLVED' (asm_full_simp_tac (HOL_ss addsimps intrs)) 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 intrs)])*}ML {*fun build_alpha_inj_gl 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 concl else HOLogic.mk_eq (concl, list_conj hyps) end;*}ML {*fun build_alpha_inj intrs dist_inj elims ctxt =let val ((_, thms_imp), ctxt') = Variable.import false intrs ctxt; val gls = map (HOLogic.mk_Trueprop o build_alpha_inj_gl) thms_imp; fun tac _ = alpha_inj_tac dist_inj intrs elims 1; val thms = map (fn gl => Goal.prove ctxt' [] [] gl tac) gls;in Variable.export ctxt' ctxt thmsend*}ML {*fun build_alpha_refl_gl alphas (x, y, z) =let fun build_alpha alpha = let val ty = domain_type (fastype_of alpha); val var = Free(x, ty); val var2 = Free(y, ty); val var3 = Free(z, ty); val symp = HOLogic.mk_imp (alpha $ var $ var2, alpha $ var2 $ var); val transp = HOLogic.mk_imp (alpha $ var $ var2, HOLogic.mk_all (z, ty, HOLogic.mk_imp (alpha $ var2 $ var3, alpha $ var $ var3))) in ((alpha $ var $ var), (symp, transp)) end; val (refl_eqs, eqs) = split_list (map build_alpha alphas) val (sym_eqs, trans_eqs) = split_list eqs fun conj l = @{term Trueprop} $ foldr1 HOLogic.mk_conj lin (conj refl_eqs, (conj sym_eqs, conj trans_eqs))end*}ML {*fun reflp_tac induct inj ctxt = rtac induct THEN_ALL_NEW simp_tac ((mk_minimal_ss ctxt) addsimps inj) THEN_ALL_NEW split_conjs THEN_ALL_NEW REPEAT o rtac @{thm exI[of _ "0 :: perm"]} THEN_ALL_NEW split_conjs THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps @{thms alpha_gen fresh_star_def fresh_zero_perm permute_zero ball_triv add_0_left supp_zero_perm Int_empty_left})*}lemma exi_neg: "\<exists>(pi :: perm). P pi \<Longrightarrow> (\<And>(p :: perm). P p \<Longrightarrow> Q (- p)) \<Longrightarrow> \<exists>pi. Q pi"apply (erule exE)apply (rule_tac x="-pi" in exI)by autoML {*fun symp_tac induct inj eqvt ctxt = ind_tac induct THEN_ALL_NEW simp_tac ((mk_minimal_ss ctxt) addsimps inj) THEN_ALL_NEW split_conjs THEN_ALL_NEW REPEAT o etac @{thm exi_neg} THEN_ALL_NEW split_conjs THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps @{thms supp_minus_perm minus_add[symmetric]}) THEN_ALL_NEW (rtac @{thm alpha_gen_compose_sym} THEN' RANGE [ (asm_full_simp_tac (HOL_ss addsimps @{thms plus_perm_eq})), (asm_full_simp_tac (HOL_ss addsimps (eqvt @ all_eqvts ctxt))) ])*}ML {*fun imp_elim_tac case_rules = Subgoal.FOCUS (fn {concl, context, ...} => case term_of concl of _ $ (_ $ asm $ _) => let fun filter_fn case_rule = ( case Logic.strip_assums_hyp (prop_of case_rule) of ((_ $ asmc) :: _) => let val thy = ProofContext.theory_of context in Pattern.matches thy (asmc, asm) end | _ => false) val matching_rules = filter filter_fn case_rules in (rtac impI THEN' rotate_tac (~1) THEN' eresolve_tac matching_rules) 1 end | _ => no_tac )*}lemma exi_sum: "\<exists>(pi :: perm). P pi \<Longrightarrow> \<exists>(pi :: perm). Q pi \<Longrightarrow> (\<And>(p :: perm) (pi :: perm). P p \<Longrightarrow> Q pi \<Longrightarrow> R (pi + p)) \<Longrightarrow> \<exists>pi. R pi"apply (erule exE)+apply (rule_tac x="pia + pi" in exI)by autoML {*fun is_ex (Const ("Ex", _) $ Abs _) = true | is_ex _ = false;*}ML {*fun eetac rule = Subgoal.FOCUS_PARAMS (fn (focus) => let val concl = #concl focus val prems = Logic.strip_imp_prems (term_of concl) val exs = filter (fn x => is_ex (HOLogic.dest_Trueprop x)) prems val cexs = map (SOME o (cterm_of (ProofContext.theory_of (#context focus)))) exs val thins = map (fn cex => Drule.instantiate' [] [cex] Drule.thin_rl) cexs in (etac rule THEN' RANGE[ atac, eresolve_tac thins ]) 1 end )*}ML {*fun transp_tac ctxt induct alpha_inj term_inj distinct cases eqvt = ind_tac induct THEN_ALL_NEW (TRY o rtac allI THEN' imp_elim_tac cases ctxt) THEN_ALL_NEW asm_full_simp_tac ((mk_minimal_ss ctxt) addsimps alpha_inj) THEN_ALL_NEW split_conjs THEN_ALL_NEW REPEAT o (eetac @{thm exi_sum} ctxt) THEN_ALL_NEW split_conjs THEN_ALL_NEW (asm_full_simp_tac (HOL_ss addsimps (term_inj @ distinct))) THEN_ALL_NEW split_conjs THEN_ALL_NEW TRY o (etac @{thm alpha_gen_compose_trans} THEN' RANGE[atac]) THEN_ALL_NEW (asm_full_simp_tac (HOL_ss addsimps (all_eqvts ctxt @ eqvt @ term_inj @ distinct)))*}lemma transp_aux: "(\<And>xa ya. R xa ya \<longrightarrow> (\<forall>z. R ya z \<longrightarrow> R xa z)) \<Longrightarrow> transp R" unfolding transp_def by blastML {*fun equivp_tac reflps symps transps = simp_tac (HOL_ss addsimps @{thms equivp_reflp_symp_transp reflp_def symp_def}) THEN' rtac conjI THEN' rtac allI THEN' resolve_tac reflps THEN' rtac conjI THEN' rtac allI THEN' rtac allI THEN' resolve_tac symps THEN' rtac @{thm transp_aux} THEN' resolve_tac transps*}ML {*fun build_equivps alphas term_induct alpha_induct term_inj alpha_inj distinct cases eqvt ctxt =let val ([x, y, z], ctxt') = Variable.variant_fixes ["x","y","z"] ctxt; val (reflg, (symg, transg)) = build_alpha_refl_gl alphas (x, y, z) fun reflp_tac' _ = reflp_tac term_induct alpha_inj ctxt 1; fun symp_tac' _ = symp_tac alpha_induct alpha_inj eqvt ctxt 1; fun transp_tac' _ = transp_tac ctxt alpha_induct alpha_inj term_inj distinct cases eqvt 1; val reflt = Goal.prove ctxt' [] [] reflg reflp_tac'; val symt = Goal.prove ctxt' [] [] symg symp_tac'; val transt = Goal.prove ctxt' [] [] transg transp_tac'; val [refltg, symtg, transtg] = Variable.export ctxt' ctxt [reflt, symt, transt] val reflts = HOLogic.conj_elims refltg val symts = HOLogic.conj_elims symtg val transts = HOLogic.conj_elims transtg fun equivp alpha = let val equivp = Const (@{const_name equivp}, fastype_of alpha --> @{typ bool}) val goal = @{term Trueprop} $ (equivp $ alpha) fun tac _ = equivp_tac reflts symts transts 1 in Goal.prove ctxt [] [] goal tac endin map equivp alphasend*}(*Tests:prove alpha1_reflp_aux: {* fst (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}by (tactic {* reflp_tac @{thm rtrm1_bp.induct} @{thms alpha1_inj} 1 *})prove alpha1_symp_aux: {* (fst o snd) (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}by (tactic {* symp_tac @{thm alpha_rtrm1_alpha_bp.induct} @{thms alpha1_inj} @{thms alpha1_eqvt} 1 *})prove alpha1_transp_aux: {* (snd o snd) (build_alpha_refl_gl [@{term alpha_rtrm1}, @{term alpha_bp}] ("x","y","z")) *}by (tactic {* transp_tac @{context} @{thm alpha_rtrm1_alpha_bp.induct} @{thms alpha1_inj} @{thms rtrm1.inject bp.inject} @{thms rtrm1.distinct bp.distinct} @{thms alpha_rtrm1.cases alpha_bp.cases} @{thms alpha1_eqvt} 1 *})lemma alpha1_equivp: "equivp alpha_rtrm1" "equivp alpha_bp"apply (tactic {* (simp_tac (HOL_ss addsimps @{thms equivp_reflp_symp_transp reflp_def symp_def}) THEN' rtac @{thm conjI} THEN' rtac @{thm allI} THEN' resolve_tac (HOLogic.conj_elims @{thm alpha1_reflp_aux}) THEN' rtac @{thm conjI} THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' resolve_tac (HOLogic.conj_elims @{thm alpha1_symp_aux}) THEN' rtac @{thm transp_aux} THEN' resolve_tac (HOLogic.conj_elims @{thm alpha1_transp_aux}))1 *})done*)ML {*fun dtyp_no_of_typ _ (TFree (n, _)) = error "dtyp_no_of_typ: Illegal free" | dtyp_no_of_typ _ (TVar _) = error "dtyp_no_of_typ: Illegal schematic" | dtyp_no_of_typ dts (Type (tname, Ts)) = case try (find_index (curry op = tname o fst)) dts of NONE => error "dtyp_no_of_typ: Illegal recursion" | SOME i => i*}end