Code for solving symp goals with multiple existentials.
theory Fv+ −
imports "Nominal2_Atoms" "Abs" "Perm" "Rsp"+ −
begin+ −
+ −
(* Bindings are given as a list which has a length being equal+ −
to the length of the number of constructors.+ −
+ −
Each element is a list whose length is equal to the number+ −
of arguents.+ −
+ −
Every element specifies bindings of this argument given as+ −
a tuple: function, bound argument.+ −
+ −
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)]],+ −
[[], [], [(SOME (Const f), 0), (Some (Const g), 1)]]]+ −
+ −
A SOME binding has to have a function returning an atom set,+ −
and a NONE binding has to be on an argument 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.+ −
*)+ −
+ −
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;+ −
*}+ −
+ −
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;+ −
(* Copy from Term *)+ −
fun is_funtype (Type ("fun", [_, _])) = true+ −
| is_funtype _ = false;+ −
(* 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 = bindcs ~~ 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+ −
(* TODO we assume that all can be 'atomized' *)+ −
if (is_funtype o fastype_of) (nth args i) then mk_atoms (nth args i) else+ −
mk_single_atom (nth args i)+ −
| 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+ −
(* TODO: we just assume everything can be 'atomized' *)+ −
if (is_funtype o fastype_of) x then mk_atoms x else+ −
HOLogic.mk_set @{typ atom} [mk_atom (fastype_of x) $ x];+ −
(* 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 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 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+ −
*}+ −
+ −
(* tests+ −
atom_decl name+ −
+ −
datatype ty =+ −
Var "name set"+ −
+ −
ML {* Syntax.check_term @{context} (mk_atoms @{term "a :: name set"}) *}+ −
+ −
local_setup {* define_fv_alpha "Fv.ty" [[[[]]]] *}+ −
print_theorems+ −
+ −
+ −
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"+ −
+ −
(* to be given by the user *)+ −
+ −
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 ["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 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})+ −
*}+ −
+ −
+ −
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+ −
(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 auto+ −
+ −
ML {*+ −
fun transp_tac ctxt induct alpha_inj term_inj distinct cases eqvt =+ −
((rtac impI THEN' etac induct) ORELSE' rtac induct) THEN_ALL_NEW+ −
(TRY o rtac allI THEN' imp_elim_tac cases ctxt) THEN_ALL_NEW+ −
(+ −
asm_full_simp_tac (HOL_ss addsimps alpha_inj @ term_inj @ distinct)+ −
THEN_ALL_NEW (REPEAT o etac conjE THEN' etac @{thm exi_sum} THEN' RANGE [atac]) THEN_ALL_NEW+ −
(REPEAT o etac conjE THEN' (TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI)))+ −
THEN_ALL_NEW (asm_full_simp_tac HOL_ss) THEN_ALL_NEW+ −
(etac @{thm alpha_gen_compose_trans} THEN' RANGE[atac]) THEN_ALL_NEW+ −
(asm_full_simp_tac (HOL_ss addsimps (@{thm atom_eqvt} :: eqvt)))+ −
)+ −
*}+ −
+ −
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+ −
*}+ −
+ −
end+ −