theory QuotMain+ −
imports QuotScript QuotList Prove+ −
uses ("quotient_info.ML")+ −
("quotient.ML")+ −
("quotient_def.ML")+ −
begin+ −
+ −
locale QUOT_TYPE =+ −
fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"+ −
and Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"+ −
and Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"+ −
assumes equiv: "equivp R"+ −
and rep_prop: "\<And>y. \<exists>x. Rep y = R x"+ −
and rep_inverse: "\<And>x. Abs (Rep x) = x"+ −
and abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)"+ −
and rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"+ −
begin+ −
+ −
definition+ −
ABS::"'a \<Rightarrow> 'b"+ −
where+ −
"ABS x \<equiv> Abs (R x)"+ −
+ −
definition+ −
REP::"'b \<Rightarrow> 'a"+ −
where+ −
"REP a = Eps (Rep a)"+ −
+ −
lemma lem9:+ −
shows "R (Eps (R x)) = R x"+ −
proof -+ −
have a: "R x x" using equiv by (simp add: equivp_reflp_symp_transp reflp_def)+ −
then have "R x (Eps (R x))" by (rule someI)+ −
then show "R (Eps (R x)) = R x"+ −
using equiv unfolding equivp_def by simp+ −
qed+ −
+ −
theorem thm10:+ −
shows "ABS (REP a) \<equiv> a"+ −
apply (rule eq_reflection)+ −
unfolding ABS_def REP_def+ −
proof -+ −
from rep_prop+ −
obtain x where eq: "Rep a = R x" by auto+ −
have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp+ −
also have "\<dots> = Abs (R x)" using lem9 by simp+ −
also have "\<dots> = Abs (Rep a)" using eq by simp+ −
also have "\<dots> = a" using rep_inverse by simp+ −
finally+ −
show "Abs (R (Eps (Rep a))) = a" by simp+ −
qed+ −
+ −
lemma REP_refl:+ −
shows "R (REP a) (REP a)"+ −
unfolding REP_def+ −
by (simp add: equiv[simplified equivp_def])+ −
+ −
lemma lem7:+ −
shows "(R x = R y) = (Abs (R x) = Abs (R y))"+ −
apply(rule iffI)+ −
apply(simp)+ −
apply(drule rep_inject[THEN iffD2])+ −
apply(simp add: abs_inverse)+ −
done+ −
+ −
theorem thm11:+ −
shows "R r r' = (ABS r = ABS r')"+ −
unfolding ABS_def+ −
by (simp only: equiv[simplified equivp_def] lem7)+ −
+ −
+ −
lemma REP_ABS_rsp:+ −
shows "R f (REP (ABS g)) = R f g"+ −
and "R (REP (ABS g)) f = R g f"+ −
by (simp_all add: thm10 thm11)+ −
+ −
lemma Quotient:+ −
"Quotient R ABS REP"+ −
apply(unfold Quotient_def)+ −
apply(simp add: thm10)+ −
apply(simp add: REP_refl)+ −
apply(subst thm11[symmetric])+ −
apply(simp add: equiv[simplified equivp_def])+ −
done+ −
+ −
lemma R_trans:+ −
assumes ab: "R a b"+ −
and bc: "R b c"+ −
shows "R a c"+ −
proof -+ −
have tr: "transp R" using equiv equivp_reflp_symp_transp[of R] by simp+ −
moreover have ab: "R a b" by fact+ −
moreover have bc: "R b c" by fact+ −
ultimately show "R a c" unfolding transp_def by blast+ −
qed+ −
+ −
lemma R_sym:+ −
assumes ab: "R a b"+ −
shows "R b a"+ −
proof -+ −
have re: "symp R" using equiv equivp_reflp_symp_transp[of R] by simp+ −
then show "R b a" using ab unfolding symp_def by blast+ −
qed+ −
+ −
lemma R_trans2:+ −
assumes ac: "R a c"+ −
and bd: "R b d"+ −
shows "R a b = R c d"+ −
using ac bd+ −
by (blast intro: R_trans R_sym)+ −
+ −
lemma REPS_same:+ −
shows "R (REP a) (REP b) \<equiv> (a = b)"+ −
proof -+ −
have "R (REP a) (REP b) = (a = b)"+ −
proof+ −
assume as: "R (REP a) (REP b)"+ −
from rep_prop+ −
obtain x y+ −
where eqs: "Rep a = R x" "Rep b = R y" by blast+ −
from eqs have "R (Eps (R x)) (Eps (R y))" using as unfolding REP_def by simp+ −
then have "R x (Eps (R y))" using lem9 by simp+ −
then have "R (Eps (R y)) x" using R_sym by blast+ −
then have "R y x" using lem9 by simp+ −
then have "R x y" using R_sym by blast+ −
then have "ABS x = ABS y" using thm11 by simp+ −
then have "Abs (Rep a) = Abs (Rep b)" using eqs unfolding ABS_def by simp+ −
then show "a = b" using rep_inverse by simp+ −
next+ −
assume ab: "a = b"+ −
have "reflp R" using equiv equivp_reflp_symp_transp[of R] by simp+ −
then show "R (REP a) (REP b)" unfolding reflp_def using ab by auto+ −
qed+ −
then show "R (REP a) (REP b) \<equiv> (a = b)" by simp+ −
qed+ −
+ −
end+ −
+ −
section {* type definition for the quotient type *}+ −
+ −
(* the auxiliary data for the quotient types *)+ −
use "quotient_info.ML"+ −
+ −
declare [[map list = (map, list_rel)]]+ −
declare [[map * = (prod_fun, prod_rel)]]+ −
declare [[map "fun" = (fun_map, fun_rel)]]+ −
+ −
lemmas [quotient_thm] =+ −
fun_quotient list_quotient+ −
+ −
lemmas [quotient_rsp] =+ −
quot_rel_rsp nil_rsp cons_rsp+ −
+ −
ML {* maps_lookup @{theory} "List.list" *}+ −
ML {* maps_lookup @{theory} "*" *}+ −
ML {* maps_lookup @{theory} "fun" *}+ −
+ −
+ −
(* definition of the quotient types *)+ −
(* FIXME: should be called quotient_typ.ML *)+ −
use "quotient.ML"+ −
+ −
+ −
(* lifting of constants *)+ −
use "quotient_def.ML"+ −
+ −
(* TODO: Consider defining it with an "if"; sth like:+ −
Babs p m = \<lambda>x. if x \<in> p then m x else undefined+ −
*)+ −
definition+ −
Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"+ −
where+ −
"(x \<in> p) \<Longrightarrow> (Babs p m x = m x)"+ −
+ −
section {* atomize *}+ −
+ −
lemma atomize_eqv[atomize]:+ −
shows "(Trueprop A \<equiv> Trueprop B) \<equiv> (A \<equiv> B)"+ −
proof+ −
assume "A \<equiv> B"+ −
then show "Trueprop A \<equiv> Trueprop B" by unfold+ −
next+ −
assume *: "Trueprop A \<equiv> Trueprop B"+ −
have "A = B"+ −
proof (cases A)+ −
case True+ −
have "A" by fact+ −
then show "A = B" using * by simp+ −
next+ −
case False+ −
have "\<not>A" by fact+ −
then show "A = B" using * by auto+ −
qed+ −
then show "A \<equiv> B" by (rule eq_reflection)+ −
qed+ −
+ −
ML {*+ −
fun atomize_thm thm =+ −
let+ −
val thm' = Thm.freezeT (forall_intr_vars thm)+ −
val thm'' = ObjectLogic.atomize (cprop_of thm')+ −
in+ −
@{thm equal_elim_rule1} OF [thm'', thm']+ −
end+ −
*}+ −
+ −
section {* infrastructure about id *}+ −
+ −
lemma prod_fun_id: "prod_fun id id \<equiv> id"+ −
by (rule eq_reflection) (simp add: prod_fun_def)+ −
+ −
lemma map_id: "map id \<equiv> id"+ −
apply (rule eq_reflection)+ −
apply (rule ext)+ −
apply (rule_tac list="x" in list.induct)+ −
apply (simp_all)+ −
done+ −
+ −
lemmas id_simps =+ −
fun_map_id[THEN eq_reflection]+ −
id_apply[THEN eq_reflection]+ −
id_def[THEN eq_reflection,symmetric]+ −
prod_fun_id map_id+ −
+ −
ML {*+ −
fun simp_ids thm =+ −
MetaSimplifier.rewrite_rule @{thms id_simps} thm+ −
*}+ −
+ −
section {* Debugging infrastructure for testing tactics *}+ −
+ −
ML {*+ −
fun my_print_tac ctxt s i thm =+ −
let+ −
val prem_str = nth (prems_of thm) (i - 1)+ −
|> Syntax.string_of_term ctxt+ −
handle Subscript => "no subgoal"+ −
val _ = tracing (s ^ "\n" ^ prem_str)+ −
in+ −
Seq.single thm+ −
end *}+ −
+ −
+ −
ML {*+ −
fun DT ctxt s tac i thm =+ −
let+ −
val before_goal = nth (prems_of thm) (i - 1)+ −
|> Syntax.string_of_term ctxt+ −
val before_msg = ["before: " ^ s, before_goal, "after: " ^ s]+ −
|> cat_lines+ −
in + −
EVERY [tac i, my_print_tac ctxt before_msg i] thm+ −
end+ −
+ −
fun NDT ctxt s tac thm = tac thm + −
*}+ −
+ −
+ −
section {* Infrastructure for collecting theorems for starting the lifting *}+ −
+ −
ML {*+ −
fun lookup_quot_data lthy qty =+ −
let+ −
val qty_name = fst (dest_Type qty)+ −
val SOME quotdata = quotdata_lookup lthy qty_name+ −
(* TODO: Should no longer be needed *)+ −
val rty = Logic.unvarifyT (#rtyp quotdata)+ −
val rel = #rel quotdata+ −
val rel_eqv = #equiv_thm quotdata+ −
val rel_refl = @{thm equivp_reflp} OF [rel_eqv]+ −
in+ −
(rty, rel, rel_refl, rel_eqv)+ −
end+ −
*}+ −
+ −
ML {*+ −
fun lookup_quot_thms lthy qty_name =+ −
let+ −
val thy = ProofContext.theory_of lthy;+ −
val trans2 = PureThy.get_thm thy ("QUOT_TYPE_I_" ^ qty_name ^ ".R_trans2")+ −
val reps_same = PureThy.get_thm thy ("QUOT_TYPE_I_" ^ qty_name ^ ".REPS_same")+ −
val absrep = PureThy.get_thm thy ("QUOT_TYPE_I_" ^ qty_name ^ ".thm10")+ −
val quot = PureThy.get_thm thy ("Quotient_" ^ qty_name)+ −
in+ −
(trans2, reps_same, absrep, quot)+ −
end+ −
*}+ −
+ −
section {* Regularization *} + −
+ −
(*+ −
Regularizing an rtrm means:+ −
- quantifiers over a type that needs lifting are replaced by+ −
bounded quantifiers, for example:+ −
\<forall>x. P \<Longrightarrow> \<forall>x \<in> (Respects R). P / All (Respects R) P+ −
+ −
the relation R is given by the rty and qty;+ −
+ −
- abstractions over a type that needs lifting are replaced+ −
by bounded abstractions:+ −
\<lambda>x. P \<Longrightarrow> Ball (Respects R) (\<lambda>x. P)+ −
+ −
- equalities over the type being lifted are replaced by+ −
corresponding relations:+ −
A = B \<Longrightarrow> A \<approx> B+ −
+ −
example with more complicated types of A, B:+ −
A = B \<Longrightarrow> (op = \<Longrightarrow> op \<approx>) A B+ −
*)+ −
+ −
ML {*+ −
(* builds the relation that is the argument of respects *)+ −
fun mk_resp_arg lthy (rty, qty) =+ −
let+ −
val thy = ProofContext.theory_of lthy+ −
in + −
if rty = qty+ −
then HOLogic.eq_const rty+ −
else+ −
case (rty, qty) of+ −
(Type (s, tys), Type (s', tys')) =>+ −
if s = s' + −
then let+ −
val SOME map_info = maps_lookup thy s+ −
val args = map (mk_resp_arg lthy) (tys ~~ tys')+ −
in+ −
list_comb (Const (#relfun map_info, dummyT), args) + −
end + −
else let + −
val SOME qinfo = quotdata_lookup_thy thy s'+ −
(* FIXME: check in this case that the rty and qty *)+ −
(* FIXME: correspond to each other *)+ −
val (s, _) = dest_Const (#rel qinfo)+ −
(* FIXME: the relation should only be the string *)+ −
(* FIXME: and the type needs to be calculated as below; *)+ −
(* FIXME: maybe one should actually have a term *)+ −
(* FIXME: and one needs to force it to have this type *)+ −
in+ −
Const (s, rty --> rty --> @{typ bool})+ −
end+ −
| _ => HOLogic.eq_const dummyT + −
(* FIXME: check that the types correspond to each other? *)+ −
end+ −
*}+ −
+ −
ML {*+ −
fun matches_typ (ty, ty') =+ −
case (ty, ty') of+ −
(_, TVar _) => true+ −
| (TFree x, TFree x') => x = x'+ −
| (Type (s, tys), Type (s', tys')) => + −
s = s' andalso + −
if (length tys = length tys') + −
then (List.all matches_typ (tys ~~ tys')) + −
else false+ −
| _ => false+ −
*}+ −
ML {*+ −
fun matches_term (trm, trm') = + −
case (trm, trm') of + −
(_, Var _) => true+ −
| (Const (s, ty), Const (s', ty')) => s = s' andalso matches_typ (ty, ty')+ −
| (Free (x, ty), Free (x', ty')) => x = x' andalso matches_typ (ty, ty')+ −
| (Bound i, Bound j) => i = j+ −
| (Abs (_, T, t), Abs (_, T', t')) => matches_typ (T, T') andalso matches_term (t, t')+ −
| (t $ s, t' $ s') => matches_term (t, t') andalso matches_term (s, s') + −
| _ => false+ −
*}+ −
+ −
ML {*+ −
val mk_babs = Const (@{const_name Babs}, dummyT)+ −
val mk_ball = Const (@{const_name Ball}, dummyT)+ −
val mk_bex = Const (@{const_name Bex}, dummyT)+ −
val mk_resp = Const (@{const_name Respects}, dummyT)+ −
*}+ −
+ −
ML {*+ −
(* - applies f to the subterm of an abstraction, *)+ −
(* otherwise to the given term, *)+ −
(* - used by regularize, therefore abstracted *)+ −
(* variables do not have to be treated specially *)+ −
+ −
fun apply_subt f trm1 trm2 =+ −
case (trm1, trm2) of+ −
(Abs (x, T, t), Abs (x', T', t')) => Abs (x, T, f t t')+ −
| _ => f trm1 trm2+ −
+ −
(* the major type of All and Ex quantifiers *)+ −
fun qnt_typ ty = domain_type (domain_type ty) + −
*}+ −
+ −
ML {*+ −
(* produces a regularized version of rtm *)+ −
(* - the result is still not completely typed *)+ −
(* - does not need any special treatment of *)+ −
(* bound variables *)+ −
+ −
fun regularize_trm lthy rtrm qtrm =+ −
case (rtrm, qtrm) of+ −
(Abs (x, ty, t), Abs (x', ty', t')) =>+ −
let+ −
val subtrm = regularize_trm lthy t t'+ −
in + −
if ty = ty'+ −
then Abs (x, ty, subtrm)+ −
else mk_babs $ (mk_resp $ mk_resp_arg lthy (ty, ty')) $ subtrm+ −
end+ −
+ −
| (Const (@{const_name "All"}, ty) $ t, Const (@{const_name "All"}, ty') $ t') =>+ −
let+ −
val subtrm = apply_subt (regularize_trm lthy) t t'+ −
in+ −
if ty = ty'+ −
then Const (@{const_name "All"}, ty) $ subtrm+ −
else mk_ball $ (mk_resp $ mk_resp_arg lthy (qnt_typ ty, qnt_typ ty')) $ subtrm+ −
end+ −
+ −
| (Const (@{const_name "Ex"}, ty) $ t, Const (@{const_name "Ex"}, ty') $ t') =>+ −
let+ −
val subtrm = apply_subt (regularize_trm lthy) t t'+ −
in+ −
if ty = ty'+ −
then Const (@{const_name "Ex"}, ty) $ subtrm+ −
else mk_bex $ (mk_resp $ mk_resp_arg lthy (qnt_typ ty, qnt_typ ty')) $ subtrm+ −
end+ −
+ −
| (* equalities need to be replaced by appropriate equivalence relations *) + −
(Const (@{const_name "op ="}, ty), Const (@{const_name "op ="}, ty')) =>+ −
if ty = ty'+ −
then rtrm+ −
else mk_resp_arg lthy (domain_type ty, domain_type ty') + −
+ −
| (* in this case we check whether the given equivalence relation is correct *) + −
(rel, Const (@{const_name "op ="}, ty')) =>+ −
let + −
val exc = LIFT_MATCH "regularise (relation mismatch)"+ −
val rel_ty = (fastype_of rel) handle TERM _ => raise exc + −
val rel' = mk_resp_arg lthy (domain_type rel_ty, domain_type ty') + −
in + −
if rel' = rel+ −
then rtrm+ −
else raise exc+ −
end + −
| (_, Const (s, _)) =>+ −
let + −
fun same_name (Const (s, _)) (Const (s', _)) = (s = s')+ −
| same_name _ _ = false+ −
in+ −
if same_name rtrm qtrm + −
then rtrm + −
else + −
let + −
fun exc1 s = LIFT_MATCH ("regularize (constant " ^ s ^ " not found)")+ −
val exc2 = LIFT_MATCH ("regularize (constant mismatch)")+ −
val thy = ProofContext.theory_of lthy+ −
val rtrm' = (#rconst (qconsts_lookup thy s)) handle NotFound => raise (exc1 s) + −
in + −
if matches_term (rtrm, rtrm')+ −
then rtrm+ −
else raise exc2+ −
end+ −
end + −
+ −
| (t1 $ t2, t1' $ t2') =>+ −
(regularize_trm lthy t1 t1') $ (regularize_trm lthy t2 t2')+ −
+ −
| (Free (x, ty), Free (x', ty')) => + −
(* this case cannot arrise as we start with two fully atomized terms *)+ −
raise (LIFT_MATCH "regularize (frees)")+ −
+ −
| (Bound i, Bound i') =>+ −
if i = i' + −
then rtrm + −
else raise (LIFT_MATCH "regularize (bounds mismatch)")+ −
+ −
| (rt, qt) =>+ −
raise (LIFT_MATCH "regularize (default)")+ −
*}+ −
+ −
(*+ −
FIXME / TODO: needs to be adapted+ −
+ −
To prove that the raw theorem implies the regularised one, + −
we try in order:+ −
+ −
- Reflexivity of the relation+ −
- Assumption+ −
- Elimnating quantifiers on both sides of toplevel implication+ −
- Simplifying implications on both sides of toplevel implication+ −
- Ball (Respects ?E) ?P = All ?P+ −
- (\<And>x. ?R x \<Longrightarrow> ?P x \<longrightarrow> ?Q x) \<Longrightarrow> All ?P \<longrightarrow> Ball ?R ?Q+ −
+ −
*)+ −
+ −
(* FIXME: they should be in in the new Isabelle *)+ −
lemma [mono]: + −
"(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (Ex P) \<longrightarrow> (Ex Q)"+ −
by blast+ −
+ −
lemma [mono]: "P \<longrightarrow> Q \<Longrightarrow> \<not>Q \<longrightarrow> \<not>P"+ −
by auto+ −
+ −
(* FIXME: OPTION_equivp, PAIR_equivp, ... *)+ −
ML {*+ −
fun equiv_tac rel_eqvs =+ −
REPEAT_ALL_NEW (FIRST' + −
[resolve_tac rel_eqvs,+ −
rtac @{thm identity_equivp},+ −
rtac @{thm list_equivp}])+ −
*}+ −
+ −
ML {*+ −
fun ball_reg_eqv_simproc rel_eqvs ss redex =+ −
let+ −
val ctxt = Simplifier.the_context ss+ −
val thy = ProofContext.theory_of ctxt+ −
in+ −
case redex of+ −
(ogl as ((Const (@{const_name "Ball"}, _)) $+ −
((Const (@{const_name "Respects"}, _)) $ R) $ P1)) =>+ −
(let+ −
val gl = Const (@{const_name "equivp"}, dummyT) $ R;+ −
val glc = HOLogic.mk_Trueprop (Syntax.check_term ctxt gl);+ −
val eqv = Goal.prove ctxt [] [] glc (fn _ => equiv_tac rel_eqvs 1);+ −
val thm = (@{thm eq_reflection} OF [@{thm ball_reg_eqv} OF [eqv]]);+ −
(* val _ = tracing (Syntax.string_of_term ctxt (prop_of thm)); *)+ −
in+ −
SOME thm+ −
end+ −
handle _ => NONE+ −
)+ −
| _ => NONE+ −
end+ −
*}+ −
+ −
ML {*+ −
fun bex_reg_eqv_simproc rel_eqvs ss redex =+ −
let+ −
val ctxt = Simplifier.the_context ss+ −
val thy = ProofContext.theory_of ctxt+ −
in+ −
case redex of+ −
(ogl as ((Const (@{const_name "Bex"}, _)) $+ −
((Const (@{const_name "Respects"}, _)) $ R) $ P1)) =>+ −
(let+ −
val gl = Const (@{const_name "equivp"}, dummyT) $ R;+ −
val glc = HOLogic.mk_Trueprop (Syntax.check_term ctxt gl);+ −
val eqv = Goal.prove ctxt [] [] glc (fn _ => equiv_tac rel_eqvs 1);+ −
val thm = (@{thm eq_reflection} OF [@{thm bex_reg_eqv} OF [eqv]]);+ −
(* val _ = tracing (Syntax.string_of_term ctxt (prop_of thm)); *)+ −
in+ −
SOME thm+ −
end+ −
handle _ => NONE+ −
)+ −
| _ => NONE+ −
end+ −
*}+ −
+ −
ML {*+ −
fun prep_trm thy (x, (T, t)) =+ −
(cterm_of thy (Var (x, T)), cterm_of thy t)+ −
+ −
fun prep_ty thy (x, (S, ty)) =+ −
(ctyp_of thy (TVar (x, S)), ctyp_of thy ty)+ −
*}+ −
+ −
ML {*+ −
fun matching_prs thy pat trm =+ −
let+ −
val univ = Unify.matchers thy [(pat, trm)]+ −
val SOME (env, _) = Seq.pull univ+ −
val tenv = Vartab.dest (Envir.term_env env)+ −
val tyenv = Vartab.dest (Envir.type_env env)+ −
in+ −
(map (prep_ty thy) tyenv, map (prep_trm thy) tenv)+ −
end+ −
*}+ −
+ −
ML {*+ −
fun ball_reg_eqv_range_simproc rel_eqvs ss redex =+ −
let+ −
val ctxt = Simplifier.the_context ss+ −
val thy = ProofContext.theory_of ctxt+ −
in+ −
case redex of+ −
(ogl as ((Const (@{const_name "Ball"}, _)) $+ −
((Const (@{const_name "Respects"}, _)) $ ((Const (@{const_name "fun_rel"}, _)) $ R1 $ R2)) $ _)) =>+ −
(let+ −
val gl = Const (@{const_name "equivp"}, dummyT) $ R2;+ −
val glc = HOLogic.mk_Trueprop (Syntax.check_term ctxt gl);+ −
(* val _ = tracing (Syntax.string_of_term ctxt glc);*)+ −
val eqv = Goal.prove ctxt [] [] glc (fn _ => equiv_tac rel_eqvs 1);+ −
val thm = (@{thm eq_reflection} OF [@{thm ball_reg_eqv_range} OF [eqv]]);+ −
val R1c = cterm_of thy R1;+ −
val thmi = Drule.instantiate' [] [SOME R1c] thm;+ −
(* val _ = tracing (Syntax.string_of_term ctxt (prop_of thmi));*)+ −
val inst = matching_prs thy (term_of (Thm.lhs_of thmi)) ogl+ −
val thm2 = Drule.eta_contraction_rule (Drule.instantiate inst thmi);+ −
(* val _ = tracing (Syntax.string_of_term ctxt (prop_of thm2)); *)+ −
in+ −
SOME thm2+ −
end+ −
handle _ => NONE+ −
+ −
)+ −
| _ => NONE+ −
end+ −
*}+ −
+ −
ML {*+ −
fun bex_reg_eqv_range_simproc rel_eqvs ss redex =+ −
let+ −
val ctxt = Simplifier.the_context ss+ −
val thy = ProofContext.theory_of ctxt+ −
in+ −
case redex of+ −
(ogl as ((Const (@{const_name "Bex"}, _)) $+ −
((Const (@{const_name "Respects"}, _)) $ ((Const (@{const_name "fun_rel"}, _)) $ R1 $ R2)) $ _)) =>+ −
(let+ −
val gl = Const (@{const_name "equivp"}, dummyT) $ R2;+ −
val glc = HOLogic.mk_Trueprop (Syntax.check_term ctxt gl);+ −
(* val _ = tracing (Syntax.string_of_term ctxt glc); *)+ −
val eqv = Goal.prove ctxt [] [] glc (fn _ => equiv_tac rel_eqvs 1);+ −
val thm = (@{thm eq_reflection} OF [@{thm bex_reg_eqv_range} OF [eqv]]);+ −
val R1c = cterm_of thy R1;+ −
val thmi = Drule.instantiate' [] [SOME R1c] thm;+ −
(* val _ = tracing (Syntax.string_of_term ctxt (prop_of thmi)); *)+ −
val inst = matching_prs thy (term_of (Thm.lhs_of thmi)) ogl+ −
val thm2 = Drule.eta_contraction_rule (Drule.instantiate inst thmi);+ −
(* val _ = tracing (Syntax.string_of_term ctxt (prop_of thm2));*)+ −
in+ −
SOME thm2+ −
end+ −
handle _ => NONE+ −
+ −
)+ −
| _ => NONE+ −
end+ −
*}+ −
+ −
lemma eq_imp_rel: "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b"+ −
by (simp add: equivp_reflp)+ −
+ −
ML {*+ −
fun regularize_tac ctxt rel_eqvs =+ −
let+ −
val pat1 = [@{term "Ball (Respects R) P"}];+ −
val pat2 = [@{term "Bex (Respects R) P"}];+ −
val pat3 = [@{term "Ball (Respects (R1 ===> R2)) P"}];+ −
val pat4 = [@{term "Bex (Respects (R1 ===> R2)) P"}];+ −
val thy = ProofContext.theory_of ctxt+ −
val simproc1 = Simplifier.simproc_i thy "" pat1 (K (ball_reg_eqv_simproc rel_eqvs))+ −
val simproc2 = Simplifier.simproc_i thy "" pat2 (K (bex_reg_eqv_simproc rel_eqvs))+ −
val simproc3 = Simplifier.simproc_i thy "" pat3 (K (ball_reg_eqv_range_simproc rel_eqvs))+ −
val simproc4 = Simplifier.simproc_i thy "" pat4 (K (bex_reg_eqv_range_simproc rel_eqvs))+ −
val simp_ctxt = (Simplifier.context ctxt empty_ss) addsimprocs [simproc1, simproc2, simproc3, simproc4]+ −
(* TODO: Make sure that there are no list_rel, pair_rel etc involved *)+ −
val eq_eqvs = map (fn x => @{thm eq_imp_rel} OF [x]) rel_eqvs+ −
in+ −
ObjectLogic.full_atomize_tac THEN'+ −
simp_tac simp_ctxt THEN'+ −
REPEAT_ALL_NEW (FIRST' [+ −
rtac @{thm ball_reg_right},+ −
rtac @{thm bex_reg_left},+ −
resolve_tac (Inductive.get_monos ctxt),+ −
rtac @{thm ball_all_comm},+ −
rtac @{thm bex_ex_comm},+ −
resolve_tac eq_eqvs,+ −
simp_tac simp_ctxt+ −
])+ −
end+ −
*}+ −
+ −
section {* Injections of rep and abses *}+ −
+ −
(*+ −
Injecting repabs means:+ −
+ −
For abstractions:+ −
* If the type of the abstraction doesn't need lifting we recurse.+ −
* If it does we add RepAbs around the whole term and check if the+ −
variable needs lifting.+ −
* If it doesn't then we recurse+ −
* If it does we recurse and put 'RepAbs' around all occurences+ −
of the variable in the obtained subterm. This in combination+ −
with the RepAbs above will let us change the type of the+ −
abstraction with rewriting.+ −
For applications:+ −
* If the term is 'Respects' applied to anything we leave it unchanged+ −
* If the term needs lifting and the head is a constant that we know+ −
how to lift, we put a RepAbs and recurse+ −
* If the term needs lifting and the head is a free applied to subterms+ −
(if it is not applied we treated it in Abs branch) then we+ −
put RepAbs and recurse+ −
* Otherwise just recurse.+ −
*)+ −
+ −
ML {*+ −
fun mk_repabs lthy (T, T') trm = + −
Quotient_Def.get_fun repF lthy (T, T') + −
$ (Quotient_Def.get_fun absF lthy (T, T') $ trm)+ −
*}+ −
+ −
ML {*+ −
(* bound variables need to be treated properly, *)+ −
(* as the type of subterms need to be calculated *)+ −
(* in the abstraction case *)+ −
+ −
fun inj_repabs_trm lthy (rtrm, qtrm) =+ −
case (rtrm, qtrm) of+ −
(Const (@{const_name "Ball"}, T) $ r $ t, Const (@{const_name "All"}, _) $ t') =>+ −
Const (@{const_name "Ball"}, T) $ r $ (inj_repabs_trm lthy (t, t'))+ −
+ −
| (Const (@{const_name "Bex"}, T) $ r $ t, Const (@{const_name "Ex"}, _) $ t') =>+ −
Const (@{const_name "Bex"}, T) $ r $ (inj_repabs_trm lthy (t, t'))+ −
+ −
| (Const (@{const_name "Babs"}, T) $ r $ t, t' as (Abs _)) =>+ −
Const (@{const_name "Babs"}, T) $ r $ (inj_repabs_trm lthy (t, t'))+ −
+ −
| (Abs (x, T, t), Abs (x', T', t')) =>+ −
let+ −
val rty = fastype_of rtrm+ −
val qty = fastype_of qtrm+ −
val (y, s) = Term.dest_abs (x, T, t)+ −
val (_, s') = Term.dest_abs (x', T', t')+ −
val yvar = Free (y, T)+ −
val result = Term.lambda_name (y, yvar) (inj_repabs_trm lthy (s, s'))+ −
in+ −
if rty = qty + −
then result+ −
else mk_repabs lthy (rty, qty) result+ −
end+ −
+ −
| (t $ s, t' $ s') => + −
(inj_repabs_trm lthy (t, t')) $ (inj_repabs_trm lthy (s, s'))+ −
+ −
| (Free (_, T), Free (_, T')) => + −
if T = T' + −
then rtrm + −
else mk_repabs lthy (T, T') rtrm+ −
+ −
| (Const (_, T), Const (_, T')) =>+ −
if T = T' + −
then rtrm+ −
else mk_repabs lthy (T, T') rtrm+ −
+ −
| (_, Const (@{const_name "op ="}, _)) => rtrm+ −
+ −
| _ => raise (LIFT_MATCH "injection")+ −
*}+ −
+ −
section {* RepAbs Injection Tactic *}+ −
+ −
(* Not used anymore *)+ −
(* FIXME/TODO: do not handle everything *)+ −
ML {*+ −
fun instantiate_tac thm = + −
Subgoal.FOCUS (fn {concl, ...} =>+ −
let+ −
val pat = Drule.strip_imp_concl (cprop_of thm)+ −
val insts = Thm.first_order_match (pat, concl)+ −
in+ −
rtac (Drule.instantiate insts thm) 1+ −
end+ −
handle _ => no_tac)+ −
*}+ −
+ −
ML {*+ −
fun quotient_tac ctxt =+ −
REPEAT_ALL_NEW (FIRST'+ −
[rtac @{thm identity_quotient},+ −
resolve_tac (quotient_rules_get ctxt)])+ −
*}+ −
+ −
definition+ −
"QUOT_TRUE x \<equiv> True"+ −
+ −
ML {*+ −
fun find_qt_asm asms =+ −
let+ −
fun find_fun trm =+ −
case trm of+ −
(Const(@{const_name Trueprop}, _) $ (Const (@{const_name QUOT_TRUE}, _) $ _)) => true+ −
| _ => false+ −
in+ −
case find_first find_fun asms of+ −
SOME (_ $ (_ $ (f $ a))) => (f, a)+ −
| _ => error "find_qt_asm"+ −
end+ −
*}+ −
+ −
(* It proves the Quotient assumptions by calling quotient_tac *)+ −
ML {*+ −
fun solve_quotient_assum i ctxt thm =+ −
let+ −
val tac =+ −
(compose_tac (false, thm, i)) THEN_ALL_NEW+ −
(quotient_tac ctxt);+ −
val gc = Drule.strip_imp_concl (cprop_of thm);+ −
in+ −
Goal.prove_internal [] gc (fn _ => tac 1)+ −
end+ −
handle _ => error "solve_quotient_assum"+ −
*}+ −
+ −
ML {*+ −
fun solve_quotient_assums ctxt thm =+ −
let val gl = hd (Drule.strip_imp_prems (cprop_of thm)) in+ −
thm OF [Goal.prove_internal [] gl (fn _ => quotient_tac ctxt 1)]+ −
end+ −
handle _ => error "solve_quotient_assums"+ −
*}+ −
+ −
ML {*+ −
val apply_rsp_tac =+ −
Subgoal.FOCUS (fn {concl, asms, context,...} =>+ −
case ((HOLogic.dest_Trueprop (term_of concl))) of+ −
((R2 $ (f $ x) $ (g $ y))) =>+ −
let+ −
val (asmf, asma) = find_qt_asm (map term_of asms);+ −
in+ −
if (fastype_of asmf) = (fastype_of f) then no_tac else let+ −
val ty_a = fastype_of x;+ −
val ty_b = fastype_of asma;+ −
val ty_c = range_type (type_of f);+ −
val thy = ProofContext.theory_of context;+ −
val ty_inst = map (SOME o (ctyp_of thy)) [ty_a, ty_b, ty_c];+ −
val thm = Drule.instantiate' ty_inst [] @{thm apply_rsp}+ −
val te = solve_quotient_assums context thm+ −
val t_inst = map (SOME o (cterm_of thy)) [R2, f, g, x, y];+ −
val thm = Drule.instantiate' [] t_inst te+ −
in+ −
compose_tac (false, thm, 2) 1+ −
end+ −
end+ −
| _ => no_tac)+ −
*}+ −
+ −
ML {*+ −
fun SOLVES' tac = tac THEN_ALL_NEW (fn _ => no_tac)+ −
*}+ −
+ −
(*+ −
To prove that the regularised theorem implies the abs/rep injected, + −
we try:+ −
+ −
1) theorems 'trans2' from the appropriate QUOT_TYPE+ −
2) remove lambdas from both sides: lambda_rsp_tac+ −
3) remove Ball/Bex from the right hand side+ −
4) use user-supplied RSP theorems+ −
5) remove rep_abs from the right side+ −
6) reflexivity of equality+ −
7) split applications of lifted type (apply_rsp)+ −
8) split applications of non-lifted type (cong_tac)+ −
9) apply extentionality+ −
A) reflexivity of the relation+ −
B) assumption+ −
(Lambdas under respects may have left us some assumptions)+ −
C) proving obvious higher order equalities by simplifying fun_rel+ −
(not sure if it is still needed?)+ −
D) unfolding lambda on one side+ −
E) simplifying (= ===> =) for simpler respectfulness+ −
+ −
*)+ −
+ −
lemma quot_true_dests:+ −
shows QT_all: "QUOT_TRUE (All P) \<Longrightarrow> QUOT_TRUE P"+ −
and QT_ex: "QUOT_TRUE (Ex P) \<Longrightarrow> QUOT_TRUE P"+ −
and QT_lam: "QUOT_TRUE (\<lambda>x. P x) \<Longrightarrow> (\<And>x. QUOT_TRUE (P x))"+ −
and QT_ext: "(\<And>x. QUOT_TRUE (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (QUOT_TRUE a \<Longrightarrow> f = g)"+ −
apply(simp_all add: QUOT_TRUE_def ext)+ −
done+ −
+ −
lemma QUOT_TRUE_i: "(QUOT_TRUE (a :: bool) \<Longrightarrow> P) \<Longrightarrow> P"+ −
by (simp add: QUOT_TRUE_def)+ −
+ −
lemma QUOT_TRUE_imp: "QUOT_TRUE a \<equiv> QUOT_TRUE b"+ −
by (simp add: QUOT_TRUE_def)+ −
+ −
ML {*+ −
fun quot_true_conv1 ctxt fnctn ctrm =+ −
case (term_of ctrm) of+ −
(Const (@{const_name QUOT_TRUE}, _) $ x) =>+ −
let+ −
val fx = fnctn x;+ −
val thy = ProofContext.theory_of ctxt;+ −
val cx = cterm_of thy x;+ −
val cfx = cterm_of thy fx;+ −
val cxt = ctyp_of thy (fastype_of x);+ −
val cfxt = ctyp_of thy (fastype_of fx);+ −
val thm = Drule.instantiate' [SOME cxt, SOME cfxt] [SOME cx, SOME cfx] @{thm QUOT_TRUE_imp}+ −
in+ −
Conv.rewr_conv thm ctrm+ −
end+ −
*}+ −
+ −
ML {*+ −
fun quot_true_conv ctxt fnctn ctrm =+ −
case (term_of ctrm) of+ −
(Const (@{const_name QUOT_TRUE}, _) $ _) =>+ −
quot_true_conv1 ctxt fnctn ctrm+ −
| _ $ _ => Conv.comb_conv (quot_true_conv ctxt fnctn) ctrm+ −
| Abs _ => Conv.abs_conv (fn (_, ctxt) => quot_true_conv ctxt fnctn) ctxt ctrm+ −
| _ => Conv.all_conv ctrm+ −
*}+ −
+ −
ML {*+ −
fun quot_true_tac ctxt fnctn = CONVERSION+ −
((Conv.params_conv ~1 (fn ctxt =>+ −
(Conv.prems_conv ~1 (quot_true_conv ctxt fnctn)))) ctxt)+ −
*}+ −
+ −
ML {* fun dest_comb (f $ a) = (f, a) *}+ −
ML {* fun dest_bcomb ((_ $ l) $ r) = (l, r) *}+ −
(* TODO: Can this be done easier? *)+ −
ML {*+ −
fun unlam t =+ −
case t of+ −
(Abs a) => snd (Term.dest_abs a)+ −
| _ => unlam (Abs("", domain_type (fastype_of t), (incr_boundvars 1 t) $ (Bound 0)))+ −
*}+ −
+ −
ML {*+ −
fun dest_fun_type (Type("fun", [T, S])) = (T, S)+ −
| dest_fun_type _ = error "dest_fun_type"+ −
*}+ −
+ −
ML {*+ −
val bare_concl = HOLogic.dest_Trueprop o Logic.strip_assums_concl+ −
*}+ −
+ −
ML {*+ −
fun rep_abs_rsp_tac ctxt =+ −
SUBGOAL (fn (goal, i) =>+ −
case (bare_concl goal) of + −
(rel $ _ $ (rep $ (abs $ _))) =>+ −
(let+ −
val thy = ProofContext.theory_of ctxt;+ −
val (ty_a, ty_b) = dest_fun_type (fastype_of abs);+ −
val ty_inst = map (SOME o (ctyp_of thy)) [ty_a, ty_b];+ −
val t_inst = map (SOME o (cterm_of thy)) [rel, abs, rep];+ −
val thm = Drule.instantiate' ty_inst t_inst @{thm rep_abs_rsp}+ −
val te = solve_quotient_assums ctxt thm+ −
in+ −
rtac te i+ −
end+ −
handle _ => no_tac)+ −
| _ => no_tac)+ −
*}+ −
+ −
ML {*+ −
fun inj_repabs_tac_match ctxt trans2 = SUBGOAL (fn (goal, i) =>+ −
(case (bare_concl goal) of+ −
(* (R1 ===> R2) (\<lambda>x\<dots>) (\<lambda>y\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (\<dots>x) (\<dots>y) *)+ −
((Const (@{const_name fun_rel}, _) $ _ $ _) $ (Abs _) $ (Abs _))+ −
=> rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam+ −
(* (op =) (Ball\<dots>) (Ball\<dots>) ----> (op =) (\<dots>) (\<dots>) *)+ −
| (Const (@{const_name "op ="},_) $+ −
(Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $+ −
(Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _))+ −
=> rtac @{thm ball_rsp} THEN' dtac @{thm QT_all}+ −
(* (R1 ===> op =) (Ball\<dots>) (Ball\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> (Ball\<dots>x) = (Ball\<dots>y) *)+ −
| (Const (@{const_name fun_rel}, _) $ _ $ _) $+ −
(Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $+ −
(Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _)+ −
=> rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam+ −
(* (op =) (Bex\<dots>) (Bex\<dots>) ----> (op =) (\<dots>) (\<dots>) *)+ −
| Const (@{const_name "op ="},_) $+ −
(Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $+ −
(Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _)+ −
=> rtac @{thm bex_rsp} THEN' dtac @{thm QT_ex}+ −
(* (R1 ===> op =) (Bex\<dots>) (Bex\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> (Bex\<dots>x) = (Bex\<dots>y) *)+ −
| (Const (@{const_name fun_rel}, _) $ _ $ _) $+ −
(Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $+ −
(Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _)+ −
=> rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam+ −
| Const (@{const_name "op ="},_) $ _ $ _ => + −
(* reflexivity of operators arising from Cong_tac *)+ −
rtac @{thm refl} ORELSE'+ −
(resolve_tac trans2 THEN' RANGE [+ −
quot_true_tac ctxt (fst o dest_bcomb), quot_true_tac ctxt (snd o dest_bcomb)])+ −
| (Const (@{const_name fun_rel}, _) $ _ $ _) $+ −
(Const (@{const_name fun_rel}, _) $ _ $ _) $+ −
(Const (@{const_name fun_rel}, _) $ _ $ _)+ −
=> rtac @{thm quot_rel_rsp} THEN_ALL_NEW quotient_tac ctxt+ −
| _ $ (Const _) $ (Const _) => (* fun_rel, list_rel, etc but not equality *)+ −
(* respectfulness of constants; in particular of a simple relation *)+ −
resolve_tac (rsp_rules_get ctxt) THEN_ALL_NEW quotient_tac ctxt+ −
| _ $ _ $ _ =>+ −
(* R (\<dots>) (Rep (Abs \<dots>)) ----> R (\<dots>) (\<dots>) *)+ −
(* observe ---> *)+ −
rep_abs_rsp_tac ctxt+ −
| _ => error "inj_repabs_tac not a relation"+ −
) i)+ −
*}+ −
+ −
ML {*+ −
fun inj_repabs_tac ctxt rel_refl trans2 =+ −
(FIRST' [+ −
inj_repabs_tac_match ctxt trans2,+ −
(* R (t $ \<dots>) (t' $ \<dots>) ----> apply_rsp provided type of t needs lifting *)+ −
NDT ctxt "A" (apply_rsp_tac ctxt THEN'+ −
(RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)])),+ −
(* (op =) (t $ \<dots>) (t' $ \<dots>) ----> Cong provided type of t does not need lifting *)+ −
(* merge with previous tactic *)+ −
NDT ctxt "B" (Cong_Tac.cong_tac @{thm cong} THEN'+ −
(RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)])),+ −
(* (op =) (\<lambda>x\<dots>) (\<lambda>x\<dots>) ----> (op =) (\<dots>) (\<dots>) *)+ −
NDT ctxt "C" (rtac @{thm ext} THEN' quot_true_tac ctxt unlam),+ −
(* resolving with R x y assumptions *)+ −
NDT ctxt "E" (atac),+ −
(* reflexivity of the basic relations *)+ −
(* R \<dots> \<dots> *)+ −
NDT ctxt "D" (resolve_tac rel_refl)+ −
])+ −
*}+ −
+ −
ML {*+ −
fun all_inj_repabs_tac ctxt rel_refl trans2 =+ −
REPEAT_ALL_NEW (inj_repabs_tac ctxt rel_refl trans2)+ −
*}+ −
+ −
+ −
+ −
section {* Cleaning of the theorem *}+ −
+ −
ML {*+ −
fun make_inst lhs t =+ −
let+ −
val _ $ (Abs (_, _, (f as Var (_, Type ("fun", [T, _]))) $ u)) = lhs;+ −
val _ $ (Abs (_, _, g)) = t;+ −
fun mk_abs i t =+ −
if incr_boundvars i u aconv t then Bound i+ −
else (case t of+ −
t1 $ t2 => mk_abs i t1 $ mk_abs i t2+ −
| Abs (s, T, t') => Abs (s, T, mk_abs (i + 1) t')+ −
| Bound j => if i = j then error "make_inst" else t+ −
| _ => t);+ −
in (f, Abs ("x", T, mk_abs 0 g)) end;+ −
*}+ −
+ −
ML {*+ −
fun lambda_allex_prs_simple_conv ctxt ctrm =+ −
case (term_of ctrm) of+ −
((Const (@{const_name fun_map}, _) $ r1 $ a2) $ (Abs _)) =>+ −
let+ −
val (ty_b, ty_a) = dest_fun_type (fastype_of r1);+ −
val (ty_c, ty_d) = dest_fun_type (fastype_of a2);+ −
val thy = ProofContext.theory_of ctxt;+ −
val tyinst = map (SOME o (ctyp_of thy)) [ty_a, ty_b, ty_c, ty_d]+ −
val tinst = [NONE, NONE, SOME (cterm_of thy r1), NONE, SOME (cterm_of thy a2)]+ −
val lpi = Drule.instantiate' tyinst tinst @{thm lambda_prs};+ −
val te = @{thm eq_reflection} OF [solve_quotient_assums ctxt (solve_quotient_assums ctxt lpi)]+ −
val ts = MetaSimplifier.rewrite_rule @{thms id_simps} te+ −
val tl = Thm.lhs_of ts;+ −
val (insp, inst) = make_inst (term_of tl) (term_of ctrm);+ −
val ti = Drule.instantiate ([], [(cterm_of thy insp, cterm_of thy inst)]) ts;+ −
in+ −
Conv.rewr_conv ti ctrm+ −
end+ −
| _ => Conv.all_conv ctrm+ −
*}+ −
+ −
ML {*+ −
val lambda_allex_prs_conv =+ −
More_Conv.top_conv lambda_allex_prs_simple_conv+ −
+ −
fun lambda_allex_prs_tac ctxt = CONVERSION (lambda_allex_prs_conv ctxt)+ −
*}+ −
+ −
(*+ −
Cleaning the theorem consists of 6 kinds of rewrites.+ −
The first two need to be done before fun_map is unfolded+ −
+ −
- lambda_prs:+ −
(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) ----> f+ −
+ −
- all_prs (and the same for exists: ex_prs)+ −
\<forall>x\<in>Respects R. (abs ---> id) f ----> \<forall>x. f+ −
+ −
- Rewriting with definitions from the argument defs+ −
NewConst ----> (rep ---> abs) oldConst+ −
+ −
- Quotient_rel_rep:+ −
Rel (Rep x) (Rep y) ----> x = y+ −
+ −
- ABS_REP+ −
Abs (Rep x) ----> x+ −
+ −
- id_simps; fun_map.simps+ −
+ −
The first one is implemented as a conversion (fast).+ −
The second one is an EqSubst (slow).+ −
The rest are a simp_tac and are fast.+ −
*)+ −
+ −
ML {*+ −
fun quotient_solver_tac ss = quotient_tac (Simplifier.the_context ss)+ −
val quotient_solver = Simplifier.mk_solver' "Quotient goal solver" quotient_solver_tac+ −
*}+ −
+ −
ML {*+ −
fun clean_tac lthy =+ −
let+ −
val thy = ProofContext.theory_of lthy;+ −
val defs = map (Thm.varifyT o #def) (qconsts_dest thy)+ −
val thms1 = @{thms all_prs ex_prs}+ −
val thms2 = @{thms eq_reflection[OF fun_map.simps]} + −
@ @{thms id_simps Quotient_abs_rep Quotient_rel_rep} + −
@ defs+ −
fun simp_ctxt thms = HOL_basic_ss addsimps thms addSolver quotient_solver+ −
(* FIXME: use of someting smaller than HOL_basic_ss *)+ −
in+ −
EVERY' [+ −
(* (rep1 ---> abs2) (\<lambda>x. rep2 (f (abs1 x))) ----> f *)+ −
NDT lthy "a" (TRY o lambda_allex_prs_tac lthy),+ −
+ −
(* Ball (Respects R) ((abs ---> id) f) ----> All f *)+ −
NDT lthy "b" (simp_tac (simp_ctxt thms1)),+ −
+ −
(* NewConst -----> (rep ---> abs) oldConst *)+ −
(* abs (rep x) -----> x *)+ −
(* R (Rep x) (Rep y) -----> x = y *)+ −
(* id_simps; fun_map.simps *)+ −
NDT lthy "c" (simp_tac (simp_ctxt thms2)),+ −
+ −
(* final step *)+ −
NDT lthy "d" (TRY o rtac refl)+ −
]+ −
end+ −
*}+ −
+ −
section {* Genralisation of free variables in a goal *}+ −
+ −
ML {*+ −
fun inst_spec ctrm =+ −
Drule.instantiate' [SOME (ctyp_of_term ctrm)] [NONE, SOME ctrm] @{thm spec}+ −
+ −
fun inst_spec_tac ctrms =+ −
EVERY' (map (dtac o inst_spec) ctrms)+ −
+ −
fun all_list xs trm = + −
fold (fn (x, T) => fn t' => HOLogic.mk_all (x, T, t')) xs trm+ −
+ −
fun apply_under_Trueprop f = + −
HOLogic.dest_Trueprop #> f #> HOLogic.mk_Trueprop+ −
+ −
fun gen_frees_tac ctxt =+ −
SUBGOAL (fn (concl, i) =>+ −
let+ −
val thy = ProofContext.theory_of ctxt+ −
val vrs = Term.add_frees concl []+ −
val cvrs = map (cterm_of thy o Free) vrs+ −
val concl' = apply_under_Trueprop (all_list vrs) concl+ −
val goal = Logic.mk_implies (concl', concl)+ −
val rule = Goal.prove ctxt [] [] goal + −
(K (EVERY1 [inst_spec_tac (rev cvrs), atac]))+ −
in+ −
rtac rule i+ −
end) + −
*}+ −
+ −
section {* General outline of the lifting procedure *}+ −
+ −
(* - A is the original raw theorem *)+ −
(* - B is the regularized theorem *)+ −
(* - C is the rep/abs injected version of B *) + −
(* - D is the lifted theorem *)+ −
(* *)+ −
(* - b is the regularization step *)+ −
(* - c is the rep/abs injection step *)+ −
(* - d is the cleaning part *)+ −
+ −
lemma lifting_procedure:+ −
assumes a: "A"+ −
and b: "A \<Longrightarrow> B"+ −
and c: "B = C"+ −
and d: "C = D"+ −
shows "D"+ −
using a b c d+ −
by simp+ −
+ −
ML {*+ −
fun lift_match_error ctxt fun_str rtrm qtrm =+ −
let+ −
val rtrm_str = Syntax.string_of_term ctxt rtrm+ −
val qtrm_str = Syntax.string_of_term ctxt qtrm+ −
val msg = [enclose "[" "]" fun_str, "The quotient theorem\n", qtrm_str, + −
"and the lifted theorem\n", rtrm_str, "do not match"]+ −
in+ −
error (space_implode " " msg)+ −
end+ −
*}+ −
+ −
ML {* + −
fun procedure_inst ctxt rtrm qtrm =+ −
let+ −
val thy = ProofContext.theory_of ctxt+ −
val rtrm' = HOLogic.dest_Trueprop rtrm+ −
val qtrm' = HOLogic.dest_Trueprop qtrm+ −
val reg_goal = + −
Syntax.check_term ctxt (regularize_trm ctxt rtrm' qtrm')+ −
handle (LIFT_MATCH s) => lift_match_error ctxt s rtrm qtrm+ −
val _ = warning "Regularization done."+ −
val inj_goal = + −
Syntax.check_term ctxt (inj_repabs_trm ctxt (reg_goal, qtrm'))+ −
handle (LIFT_MATCH s) => lift_match_error ctxt s rtrm qtrm+ −
val _ = warning "RepAbs Injection done."+ −
in+ −
Drule.instantiate' []+ −
[SOME (cterm_of thy rtrm'),+ −
SOME (cterm_of thy reg_goal),+ −
SOME (cterm_of thy inj_goal)] @{thm lifting_procedure}+ −
end+ −
*}+ −
+ −
(* Left for debugging *)+ −
ML {*+ −
fun procedure_tac ctxt rthm =+ −
ObjectLogic.full_atomize_tac+ −
THEN' gen_frees_tac ctxt+ −
THEN' CSUBGOAL (fn (gl, i) =>+ −
let+ −
val rthm' = atomize_thm rthm+ −
val rule = procedure_inst ctxt (prop_of rthm') (term_of gl)+ −
val thm = Drule.instantiate' [] [SOME (snd (Thm.dest_comb gl))] @{thm QUOT_TRUE_i}+ −
in+ −
(rtac rule THEN' RANGE [rtac rthm', (fn _ => all_tac), rtac thm]) i+ −
end)+ −
*}+ −
+ −
ML {*+ −
(* FIXME/TODO should only get as arguments the rthm like procedure_tac *)+ −
+ −
fun lift_tac ctxt rthm rel_eqv =+ −
ObjectLogic.full_atomize_tac+ −
THEN' gen_frees_tac ctxt+ −
THEN' CSUBGOAL (fn (gl, i) =>+ −
let+ −
val rthm' = atomize_thm rthm+ −
val rule = procedure_inst ctxt (prop_of rthm') (term_of gl)+ −
val rel_refl = map (fn x => @{thm equivp_reflp} OF [x]) rel_eqv+ −
val quotients = quotient_rules_get ctxt+ −
val trans2 = map (fn x => @{thm equals_rsp} OF [x]) quotients+ −
val thm = Drule.instantiate' [] [SOME (snd (Thm.dest_comb gl))] @{thm QUOT_TRUE_i}+ −
in+ −
(rtac rule THEN'+ −
RANGE [rtac rthm',+ −
regularize_tac ctxt rel_eqv,+ −
rtac thm THEN' all_inj_repabs_tac ctxt rel_refl trans2,+ −
clean_tac ctxt]) i+ −
end)+ −
*}+ −
+ −
end+ −
+ −