Use fs typeclass in showing finite support + some cheat cleaning.
theory Fv+ −
imports "Nominal2_Atoms" "Abs" "Perm" "Rsp" "Nominal2_FSet"+ −
begin+ −
+ −
(* The bindings data structure:+ −
+ −
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 z+ −
+ −
yields:+ −
[+ −
[],+ −
[(NONE, 0, 2)],+ −
[(SOME (Const f), 0, 2), (Some (Const g), 1, 2)]]+ −
+ −
A SOME binding has to have a function which takes an appropriate+ −
argument and returns an atom set. A NONE binding has to be on an+ −
argument that is an atom or an atom set.+ −
*)+ −
+ −
(*+ −
An overview of the generation of free variables:+ −
+ −
1) fv_bn functions are generated only for the non-recursive binds.+ −
+ −
An fv_bn for a constructor is a union of values for the arguments:+ −
+ −
For an argument x that is in the bn function+ −
- if it is a recursive argument bn' we return: fv_bn' x+ −
- otherwise empty+ −
+ −
For an argument x that is not in the bn function+ −
- for atom we return: {atom x}+ −
- for atom set we return: atom ` x+ −
- for a recursive call to type ty' we return: fv_ty' x+ −
with fv of the appropriate type+ −
- otherwise empty+ −
+ −
2) fv_ty functions generated for all types being defined:+ −
+ −
fv_ty for a constructor is a union of values for the arguments.+ −
+ −
For an argument that is bound in a shallow binding we return empty.+ −
+ −
For an argument x that bound in a non-recursive deep binding+ −
we return: fv_bn x.+ −
+ −
Otherwise we return the free variables of the argument minus the+ −
bound variables of the argument.+ −
+ −
The free variables for an argument x are:+ −
- for an atom: {atom x}+ −
- for atom set: atom ` x+ −
- for recursive call to type ty' return: fv_ty' x+ −
- for nominal datatype ty' return: fv_ty' x+ −
+ −
The bound variables are a union of results of all bindings that+ −
involve the given argument. For a paricular binding:+ −
+ −
- for a binding function bn: bn x+ −
- for a recursive argument of type ty': fv_fy' x+ −
- for nominal datatype ty' return: fv_ty' x+ −
*)+ −
+ −
(*+ −
An overview of the generation of alpha-equivalence:+ −
+ −
1) alpha_bn relations are generated for binding functions.+ −
+ −
An alpha_bn for a constructor is true if a conjunction of+ −
propositions for each argument holds.+ −
+ −
For an argument a proposition is build as follows from+ −
th:+ −
+ −
- for a recursive argument in the bn function, we return: alpha_bn argl argr+ −
- for a recursive argument for type ty not in bn, we return: alpha_ty argl argr+ −
- for other arguments in the bn function we return: True+ −
- for other arguments not in the bn function we return: argl = argr+ −
+ −
2) alpha_ty relations are generated for all the types being defined:+ −
+ −
For each constructor we gather all the arguments that are bound,+ −
and for each of those we add a permutation. We associate those+ −
permutations with the bindings. Note that two bindings can have+ −
the same permutation if the arguments being bound are the same.+ −
+ −
An alpha_ty for a constructor is true if there exist permutations+ −
as above such that a conjunction of propositions for all arguments holds.+ −
+ −
For an argument we allow bindings where only one of the following+ −
holds:+ −
+ −
- Argument is bound in some shallow bindings: We return true+ −
- Argument of type ty is bound recursively in some other+ −
arguments [i1, .. in] with one binding function bn.+ −
We return:+ −
+ −
(bn argl, (argl, argl_i1, ..., argl_in)) \<approx>gen+ −
\<lambda>(argl,argl1,..,argln) (argr,argr1,..,argrn). + −
(alpha_ty argl argr) \<and> (alpha_i1 argl1 argr1) \<and> .. \<and> (alpha_in argln argrn)+ −
\<lambda>(arg,arg1,..,argn). (fv_ty arg) \<union> (fv_i1 arg1) \<union> .. \<union> (fv_in argn)+ −
pi+ −
(bn argr, (argr, argr_i1, ..., argr_in))+ −
+ −
- Argument is bound in some deep non-recursive bindings.+ −
We return: alpha_bn argl argr+ −
- Argument of type ty has some shallow bindings [b1..bn] and/or+ −
non-recursive bindings [f1 a1, .., fm am], where the bindings+ −
have the permutations p1..pl. We return:+ −
+ −
(b1l \<union>..\<union> bnl \<union> f1 a1l \<union>..\<union> fn anl, argl) \<approx>gen+ −
alpha_ty fv_ty (p1 +..+ pl)+ −
(b1r \<union>..\<union> bnr \<union> f1 a1r \<union>..\<union> fn anr, argr)+ −
+ −
- Argument has some recursive bindings. The bindings were+ −
already treated in 2nd case so we return: True+ −
- Argument has no bindings and is not bound.+ −
If it is recursive for type ty, we return: alpha_ty argl argr+ −
Otherwise we return: argl = argr+ −
+ −
*)+ −
+ −
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;+ −
+ −
fun is_atom_fset thy (Type ("FSet.fset", [t])) = is_atom thy t+ −
| is_atom_fset thy _ = false;+ −
+ −
val fset_to_set = @{term "fset_to_set :: atom fset \<Rightarrow> atom set"}+ −
*}+ −
+ −
+ −
+ −
+ −
(* 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+ −
end+ −
in+ −
map (map gather_binds_cons) binds+ −
end+ −
*}+ −
+ −
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)) (rev 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_atom_set 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;+ −
fun mk_atom_fset t =+ −
let+ −
val ty = fastype_of t;+ −
val atom_ty = dest_fsetT ty --> @{typ atom};+ −
val fmap_ty = atom_ty --> ty --> @{typ "atom fset"};+ −
in+ −
fset_to_set $ ((Const (@{const_name fmap}, fmap_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;+ −
*}+ −
+ −
(* Given [fv1, fv2, fv3] creates %(x, y, z). fv1 x u fv2 y u fv3 z *)+ −
ML {*+ −
fun mk_compound_fv fvs =+ −
let+ −
val nos = (length fvs - 1) downto 0;+ −
val fvs_applied = map (fn (fv, no) => fv $ Bound no) (fvs ~~ nos);+ −
val fvs_union = mk_union fvs_applied;+ −
val (tyh :: tys) = rev (map (domain_type o fastype_of) fvs);+ −
fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))+ −
in+ −
fold fold_fun tys (Abs ("", tyh, fvs_union))+ −
end;+ −
*}+ −
+ −
ML {* @{term "\<lambda>(x, y, z). \<lambda>(x', y', z'). R x x' \<and> R2 y y' \<and> R3 z z'"} *}+ −
+ −
(* Given [R1, R2, R3] creates %(x,x'). %(y,y'). %(z,z'). R x x' \<and> R y y' \<and> R z z' *)+ −
ML {*+ −
fun mk_compound_alpha Rs =+ −
let+ −
val nos = (length Rs - 1) downto 0;+ −
val nos2 = (2 * length Rs - 1) downto length Rs;+ −
val Rs_applied = map (fn (R, (no2, no)) => R $ Bound no2 $ Bound no) (Rs ~~ (nos2 ~~ nos));+ −
val Rs_conj = mk_conjl Rs_applied;+ −
val (tyh :: tys) = rev (map (domain_type o fastype_of) Rs);+ −
fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))+ −
val abs_rhs = fold fold_fun tys (Abs ("", tyh, Rs_conj))+ −
in+ −
fold fold_fun tys (Abs ("", tyh, abs_rhs))+ −
end;+ −
*}+ −
+ −
ML {* cterm_of @{theory} (mk_compound_alpha [@{term "R :: 'a \<Rightarrow> 'a \<Rightarrow> bool"}, @{term "R2 :: 'b \<Rightarrow> 'b \<Rightarrow> bool"}, @{term "R3 :: 'b \<Rightarrow> 'b \<Rightarrow> bool"}]) *}+ −
+ −
ML {* fun add_perm (p1, p2) = Const(@{const_name plus}, @{typ "perm \<Rightarrow> perm \<Rightarrow> perm"}) $ p1 $ p2 *}+ −
+ −
ML {*+ −
fun non_rec_binds l =+ −
let+ −
fun is_non_rec (SOME (f, false), _, _) = SOME f+ −
| is_non_rec _ = NONE+ −
in+ −
distinct (op =) (map_filter is_non_rec (flat (flat l)))+ −
end+ −
*}+ −
+ −
(* We assume no bindings in the type on which bn is defined *)+ −
(* TODO: currently works only with current fv_bn function *)+ −
ML {*+ −
fun fv_bn thy (dt_info : Datatype_Aux.info) fv_frees (bn, ith_dtyp, args_in_bns) =+ −
let+ −
val {descr, sorts, ...} = dt_info;+ −
fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+ −
val fvbn_name = "fv_" ^ (Long_Name.base_name (fst (dest_Const bn)));+ −
val fvbn = Free (fvbn_name, fastype_of (nth fv_frees ith_dtyp));+ −
fun fv_bn_constr (cname, dts) args_in_bn =+ −
let+ −
val Ts = map (typ_of_dtyp descr sorts) dts;+ −
val names = Datatype_Prop.make_tnames Ts;+ −
val args = map Free (names ~~ Ts);+ −
val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+ −
fun fv_arg ((dt, x), arg_no) =+ −
let+ −
val ty = fastype_of x+ −
in+ −
if arg_no mem args_in_bn then + −
(if is_rec_type dt then+ −
(if body_index dt = ith_dtyp then fvbn $ x else error "fv_bn: recursive argument, but wrong datatype.")+ −
else @{term "{} :: atom set"}) else+ −
if is_atom thy ty then mk_single_atom x else+ −
if is_atom_set thy ty then mk_atom_set x else+ −
if is_atom_fset thy ty then mk_atom_fset x else+ −
if is_rec_type dt then nth fv_frees (body_index dt) $ x else+ −
@{term "{} :: atom set"}+ −
end;+ −
val arg_nos = 0 upto (length dts - 1)+ −
in+ −
HOLogic.mk_Trueprop (HOLogic.mk_eq+ −
(fvbn $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ arg_nos))))+ −
end;+ −
val (_, (_, _, constrs)) = nth descr ith_dtyp;+ −
val eqs = map2i fv_bn_constr constrs args_in_bns+ −
in+ −
((bn, fvbn), (fvbn_name, eqs))+ −
end+ −
*}+ −
+ −
ML {*+ −
fun alpha_bn thy (dt_info : Datatype_Aux.info) alpha_frees ((bn, ith_dtyp, args_in_bns), is_rec) =+ −
let+ −
val {descr, sorts, ...} = dt_info;+ −
fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);+ −
val alpha_bn_name = "alpha_" ^ (Long_Name.base_name (fst (dest_Const bn)));+ −
val alpha_bn_type = + −
(*if is_rec then @{typ perm} --> nth_dtyp ith_dtyp --> nth_dtyp ith_dtyp --> @{typ bool} else*)+ −
nth_dtyp ith_dtyp --> nth_dtyp ith_dtyp --> @{typ bool};+ −
val alpha_bn_free = Free(alpha_bn_name, alpha_bn_type);+ −
val pi = Free("pi", @{typ perm})+ −
fun alpha_bn_constr (cname, dts) args_in_bn =+ −
let+ −
val Ts = map (typ_of_dtyp descr sorts) dts;+ −
val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);+ −
val names2 = Name.variant_list ("pi" :: names) (Datatype_Prop.make_tnames Ts);+ −
val args = map Free (names ~~ Ts);+ −
val args2 = map Free (names2 ~~ Ts);+ −
val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));+ −
val rhs = HOLogic.mk_Trueprop+ −
(alpha_bn_free $ (list_comb (c, args)) $ (list_comb (c, args2)));+ −
fun lhs_arg ((dt, arg_no), (arg, arg2)) =+ −
let+ −
val argty = fastype_of arg;+ −
val permute = Const (@{const_name permute}, @{typ perm} --> argty --> argty);+ −
in+ −
if is_rec_type dt then+ −
if arg_no mem args_in_bn then alpha_bn_free $ arg $ arg2+ −
else (nth alpha_frees (body_index dt)) $ arg $ arg2+ −
else+ −
if arg_no mem args_in_bn then @{term True}+ −
else HOLogic.mk_eq (arg, arg2)+ −
end+ −
val arg_nos = 0 upto (length dts - 1)+ −
val lhss = mk_conjl (map lhs_arg (dts ~~ arg_nos ~~ (args ~~ args2)))+ −
val eq = Logic.mk_implies (HOLogic.mk_Trueprop lhss, rhs)+ −
in+ −
eq+ −
end+ −
val (_, (_, _, constrs)) = nth descr ith_dtyp;+ −
val eqs = map2i alpha_bn_constr constrs args_in_bns+ −
in+ −
((bn, alpha_bn_free), (alpha_bn_name, eqs))+ −
end+ −
*}+ −
+ −
(* Checks that a list of bindings contains only compatible ones *)+ −
ML {*+ −
fun bns_same l =+ −
length (distinct (op =) (map (fn ((b, _, _), _) => b) l)) = 1+ −
*}+ −
+ −
(* TODO: Notice datatypes without bindings and replace alpha with equality *)+ −
ML {*+ −
fun define_fv_alpha (dt_info : Datatype_Aux.info) bindsall bns 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 nr_bns = non_rec_binds bindsall;+ −
val rel_bns = filter (fn (bn, _, _) => bn mem nr_bns) bns;+ −
val (bn_fv_bns, fv_bn_names_eqs) = split_list (map (fv_bn thy dt_info fv_frees) rel_bns);+ −
val (fv_bn_names, fv_bn_eqs) = split_list fv_bn_names_eqs;+ −
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);+ −
(* We assume that a bn is either recursive or not *)+ −
val bns_rec = map (fn (bn, _, _) => not (bn mem nr_bns)) bns;+ −
val (bn_alpha_bns, alpha_bn_names_eqs) = split_list (map (alpha_bn thy dt_info alpha_frees) (bns ~~ bns_rec))+ −
val (alpha_bn_names, alpha_bn_eqs) = split_list alpha_bn_names_eqs;+ −
val alpha_bn_frees = map snd bn_alpha_bns;+ −
val alpha_bn_types = map fastype_of alpha_bn_frees;+ −
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_atom_set (nth args i) else+ −
if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (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 find_nonrec_binder j (SOME (f, false), i, _) = if i = j then SOME f else NONE+ −
| find_nonrec_binder _ _ = NONE+ −
fun fv_arg ((dt, x), arg_no) =+ −
case get_first (find_nonrec_binder arg_no) bindcs of+ −
SOME f =>+ −
(case get_first (fn (x, y) => if x = f then SOME y else NONE) bn_fv_bns of+ −
SOME fv_bn => fv_bn $ x+ −
| NONE => error "bn specified in a non-rec binding but not in bn list")+ −
| NONE =>+ −
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_atom_set x else+ −
if ((is_atom_fset thy) o fastype_of) x then mk_atom_fset 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 rel_in_simp_binds = filter (fn ((NONE, i, _), _) => i = arg_no | _ => false) bind_pis;+ −
val rel_in_comp_binds = filter (fn ((SOME _, i, _), _) => i = arg_no | _ => false) bind_pis;+ −
val rel_has_binds = filter (fn ((NONE, _, j), _) => j = arg_no+ −
| ((SOME (_, false), _, j), _) => j = arg_no+ −
| _ => false) bind_pis;+ −
val rel_has_rec_binds = filter + −
(fn ((SOME (_, true), _, j), _) => j = arg_no | _ => false) bind_pis;+ −
in+ −
case (rel_in_simp_binds, rel_in_comp_binds, rel_has_binds, rel_has_rec_binds) of+ −
([], [], [], []) =>+ −
if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2)+ −
else (HOLogic.mk_eq (arg, arg2))+ −
| (_, [], [], []) => @{term True}+ −
| ([], [], [], _) => @{term True}+ −
| ([], ((((SOME (bn, is_rec)), _, _), pi) :: _), [], []) =>+ −
if not (bns_same rel_in_comp_binds) then error "incompatible bindings for an argument" else+ −
if is_rec then+ −
let+ −
val (rbinds, rpis) = split_list rel_in_comp_binds+ −
val bound_in_nos = map (fn (_, _, i) => i) rbinds+ −
val bound_in_ty_nos = map (fn i => body_index (nth dts i)) bound_in_nos;+ −
val bound_args = arg :: map (nth args) bound_in_nos;+ −
val bound_args2 = arg2 :: map (nth args2) bound_in_nos;+ −
fun bound_in args (_, _, i) = nth args i;+ −
val lhs_binds = fv_binds args rbinds+ −
val lhs_arg = foldr1 HOLogic.mk_prod bound_args+ −
val lhs = mk_pair (lhs_binds, lhs_arg);+ −
val rhs_binds = fv_binds args2 rbinds;+ −
val rhs_arg = foldr1 HOLogic.mk_prod bound_args2;+ −
val rhs = mk_pair (rhs_binds, rhs_arg);+ −
val fvs = map (nth fv_frees) ((body_index dt) :: bound_in_ty_nos);+ −
val fv = mk_compound_fv fvs;+ −
val alphas = map (nth alpha_frees) ((body_index dt) :: bound_in_ty_nos);+ −
val alpha = mk_compound_alpha alphas;+ −
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+ −
in+ −
alpha_gen+ −
end+ −
else+ −
let+ −
val alpha_bn_const =+ −
nth alpha_bn_frees (find_index (fn (b, _, _) => b = bn) bns)+ −
in+ −
alpha_bn_const $ arg $ arg2+ −
end+ −
| ([], [], relevant, []) =>+ −
let+ −
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+ −
in+ −
alpha_gen+ −
end+ −
| _ => error "Fv.alpha: not supported binding structure"+ −
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 fveqs_alphaeqs = map2i fv_alpha_eq descr (gather_binds bindsall)+ −
val (fv_eqs_perfv, alpha_eqs) = apsnd flat (split_list (map split_list fveqs_alphaeqs))+ −
val rel_bns_nos = map (fn (_, i, _) => i) rel_bns;+ −
fun filter_fun (_, b) = b mem rel_bns_nos;+ −
val all_fvs = (fv_names ~~ fv_eqs_perfv) ~~ (0 upto (length fv_names - 1))+ −
val (fv_names_fst, fv_eqs_fst) = apsnd flat (split_list (map fst (filter_out filter_fun all_fvs)))+ −
val (fv_names_snd, fv_eqs_snd) = apsnd flat (split_list (map fst (filter filter_fun all_fvs)))+ −
val fv_eqs_all = fv_eqs_fst @ (flat fv_bn_eqs);+ −
val fv_names_all = fv_names_fst @ fv_bn_names;+ −
val add_binds = map (fn x => (Attrib.empty_binding, x))+ −
(* Function_Fun.add_fun Function_Common.default_config ... true *)+ −
val (fvs, lthy') = (Primrec.add_primrec+ −
(map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_all) (add_binds fv_eqs_all) lthy)+ −
val (fvs2, lthy'') =+ −
if fv_eqs_snd = [] then (([], []), lthy') else+ −
(Primrec.add_primrec+ −
(map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_snd) (add_binds fv_eqs_snd) 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_bn_names)+ −
(alpha_types @ alpha_bn_types)) []+ −
(add_binds (alpha_eqs @ flat alpha_bn_eqs)) [] lthy'')+ −
val all_fvs = (fst fvs @ fst fvs2, snd fvs @ snd fvs2)+ −
in+ −
((all_fvs, alphas), lthy''')+ −
end+ −
*}+ −
+ −
(*+ −
atom_decl name+ −
datatype lam =+ −
VAR "name"+ −
| APP "lam" "lam"+ −
| LET "bp" "lam"+ −
and bp =+ −
BP "name" "lam"+ −
primrec+ −
bi::"bp \<Rightarrow> atom set"+ −
where+ −
"bi (BP x t) = {atom x}"+ −
setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Fv.lam") 2 *}+ −
local_setup {*+ −
snd o define_fv_alpha (Datatype.the_info @{theory} "Fv.lam")+ −
[[[], [], [(SOME (@{term bi}, true), 0, 1)]], [[]]] [(@{term bi}, 1, [[0]])] *}+ −
print_theorems+ −
*)+ −
+ −
(*atom_decl name+ −
datatype 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"+ −
primrec+ −
bv1+ −
where+ −
"bv1 (BUnit) = {}"+ −
| "bv1 (BVr x) = {atom x}"+ −
| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp1)"+ −
setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Fv.rtrm1") 2 *}+ −
local_setup {*+ −
snd o define_fv_alpha (Datatype.the_info @{theory} "Fv.rtrm1")+ −
[[[], [], [(NONE, 0, 1)], [(SOME (@{term bv1}, false), 0, 2)]],+ −
[[], [], []]] [(@{term bv1}, 1, [[], [0], [0, 1]])] *}+ −
print_theorems+ −
*)+ −
+ −
(*+ −
atom_decl name+ −
datatype rtrm5 =+ −
rVr5 "name"+ −
| rLt5 "rlts" "rtrm5" --"bind (bv5 lts) in (rtrm5)"+ −
and rlts =+ −
rLnil+ −
| rLcons "name" "rtrm5" "rlts"+ −
primrec+ −
rbv5+ −
where+ −
"rbv5 rLnil = {}"+ −
| "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"+ −
setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Fv.rtrm5") 2 *}+ −
local_setup {* snd o define_fv_alpha (Datatype.the_info @{theory} "Fv.rtrm5")+ −
[[[], [(SOME (@{term rbv5}, false), 0, 1)]], [[], []]] [(@{term rbv5}, 1, [[], [0, 2]])] *}+ −
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 thms+ −
end+ −
*}+ −
+ −
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 l+ −
in+ −
(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 split_conv})+ −
*}+ −
+ −
+ −
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 auto+ −
+ −
ML {*+ −
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+ −
TRY o (rtac @{thm alpha_gen_compose_sym2} ORELSE' rtac @{thm alpha_gen_compose_sym}) THEN_ALL_NEW+ −
(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 auto+ −
+ −
ML {*+ −
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 blast+ −
+ −
ML {*+ −
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+ −
end+ −
in+ −
map equivp alphas+ −
end+ −
*}+ −
+ −
(*+ −
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+ −
*}+ −
+ −
lemma not_in_union: "c \<notin> a \<union> b \<equiv> (c \<notin> a \<and> c \<notin> b)"+ −
by auto+ −
+ −
ML {*+ −
fun supports_tac perm =+ −
simp_tac (HOL_ss addsimps @{thms supports_def not_in_union} @ perm) THEN_ALL_NEW (+ −
REPEAT o rtac allI THEN' REPEAT o rtac impI THEN' split_conjs THEN'+ −
asm_full_simp_tac (HOL_ss addsimps @{thms fresh_def[symmetric]+ −
swap_fresh_fresh fresh_atom swap_at_base_simps(3) swap_atom_image_fresh+ −
supp_fset_to_set supp_fmap_atom}))+ −
*}+ −
+ −
ML {*+ −
fun mk_supp ty x =+ −
Const (@{const_name supp}, ty --> @{typ "atom set"}) $ x+ −
*}+ −
+ −
ML {*+ −
fun mk_supports_eq thy cnstr =+ −
let+ −
val (tys, ty) = (strip_type o fastype_of) cnstr+ −
val names = Datatype_Prop.make_tnames tys+ −
val frees = map Free (names ~~ tys)+ −
val rhs = list_comb (cnstr, frees)+ −
+ −
fun mk_supp_arg (x, ty) =+ −
if is_atom thy ty then mk_supp @{typ atom} (mk_atom ty $ x) else+ −
if is_atom_set thy ty then mk_supp @{typ "atom set"} (mk_atom_set x) else+ −
if is_atom_fset thy ty then mk_supp @{typ "atom set"} (mk_atom_fset x)+ −
else mk_supp ty x+ −
val lhss = map mk_supp_arg (frees ~~ tys)+ −
val supports = Const(@{const_name "supports"}, @{typ "atom set"} --> ty --> @{typ bool})+ −
val eq = HOLogic.mk_Trueprop (supports $ mk_union lhss $ rhs)+ −
in+ −
(names, eq)+ −
end+ −
*}+ −
+ −
ML {*+ −
fun prove_supports ctxt perms cnst =+ −
let+ −
val (names, eq) = mk_supports_eq (ProofContext.theory_of ctxt) cnst+ −
in+ −
Goal.prove ctxt names [] eq (fn _ => supports_tac perms 1)+ −
end+ −
*}+ −
+ −
ML {*+ −
fun mk_fs tys =+ −
let+ −
val names = Datatype_Prop.make_tnames tys+ −
val frees = map Free (names ~~ tys)+ −
val supps = map2 mk_supp tys frees+ −
val fin_supps = map (fn x => @{term "finite :: atom set \<Rightarrow> bool"} $ x) supps+ −
in+ −
(names, HOLogic.mk_Trueprop (mk_conjl fin_supps))+ −
end+ −
*}+ −
+ −
ML {*+ −
fun fs_tac induct supports = ind_tac induct THEN_ALL_NEW (+ −
rtac @{thm supports_finite} THEN' resolve_tac supports) THEN_ALL_NEW+ −
asm_full_simp_tac (HOL_ss addsimps @{thms supp_atom supp_atom_image supp_fset_to_set+ −
supp_fmap_atom finite_insert finite.emptyI finite_Un finite_supp})+ −
*}+ −
+ −
ML {*+ −
fun prove_fs ctxt induct supports tys =+ −
let+ −
val (names, eq) = mk_fs tys+ −
in+ −
Goal.prove ctxt names [] eq (fn _ => fs_tac induct supports 1)+ −
end+ −
*}+ −
+ −
end+ −