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: "EQUIV 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: EQUIV_REFL_SYM_TRANS REFL_def)+ −
then have "R x (Eps (R x))" by (rule someI)+ −
then show "R (Eps (R x)) = R x"+ −
using equiv unfolding EQUIV_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 EQUIV_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 EQUIV_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 EQUIV_def])+ −
done+ −
+ −
lemma R_trans:+ −
assumes ab: "R a b"+ −
and bc: "R b c"+ −
shows "R a c"+ −
proof -+ −
have tr: "TRANS R" using equiv EQUIV_REFL_SYM_TRANS[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 TRANS_def by blast+ −
qed+ −
+ −
lemma R_sym:+ −
assumes ab: "R a b"+ −
shows "R b a"+ −
proof -+ −
have re: "SYM R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp+ −
then show "R b a" using ab unfolding SYM_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 "REFL R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp+ −
then show "R (REP a) (REP b)" unfolding REFL_def using ab by auto+ −
qed+ −
then show "R (REP a) (REP b) \<equiv> (a = b)" by simp+ −
qed+ −
+ −
end+ −
+ −
lemma equiv_trans2:+ −
assumes e: "EQUIV R"+ −
and ac: "R a c"+ −
and bd: "R b d"+ −
shows "R a b = R c d"+ −
using ac bd e+ −
unfolding EQUIV_REFL_SYM_TRANS TRANS_def SYM_def+ −
by (blast)+ −
+ −
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)]]+ −
+ −
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_I[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 about definitions *}+ −
+ −
(* Does the same as 'subst' in a given theorem *)+ −
ML {*+ −
fun eqsubst_thm ctxt thms thm =+ −
let+ −
val goalstate = Goal.init (Thm.cprop_of thm)+ −
val a' = case (SINGLE (EqSubst.eqsubst_tac ctxt [0] thms 1) goalstate) of+ −
NONE => error "eqsubst_thm"+ −
| SOME th => cprem_of th 1+ −
val tac = (EqSubst.eqsubst_tac ctxt [0] thms 1) THEN simp_tac HOL_ss 1+ −
val goal = Logic.mk_equals (term_of (Thm.cprop_of thm), term_of a');+ −
val cgoal = cterm_of (ProofContext.theory_of ctxt) goal+ −
val rt = Goal.prove_internal [] cgoal (fn _ => tac);+ −
in+ −
@{thm equal_elim_rule1} OF [rt, thm]+ −
end+ −
*}+ −
+ −
(* expects atomized definitions *)+ −
ML {*+ −
fun add_lower_defs_aux lthy thm =+ −
let+ −
val e1 = @{thm fun_cong} OF [thm];+ −
val f = eqsubst_thm lthy @{thms fun_map.simps} e1;+ −
val g = simp_ids f+ −
in+ −
(simp_ids thm) :: (add_lower_defs_aux lthy g)+ −
end+ −
handle _ => [simp_ids thm]+ −
*}+ −
+ −
ML {*+ −
fun add_lower_defs lthy def =+ −
let+ −
val def_pre_sym = symmetric def+ −
val def_atom = atomize_thm def_pre_sym+ −
val defs_all = add_lower_defs_aux lthy def_atom+ −
in+ −
map Thm.varifyT defs_all+ −
end+ −
*}+ −
+ −
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+ −
(* cu: Changed the lookup\<dots>not sure whether this works *)+ −
(* 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 EQUIV_REFL} 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+ −
*}+ −
+ −
ML {*+ −
fun lookup_quot_consts defs =+ −
let+ −
fun dest_term (a $ b) = (a, b);+ −
val def_terms = map (snd o Logic.dest_equals o concl_of) defs;+ −
in+ −
map (fst o dest_Const o snd o dest_term) def_terms+ −
end+ −
*}+ −
+ −
section {* Infrastructure for special quotient identity *}+ −
+ −
definition+ −
"qid TYPE('a) TYPE('b) x \<equiv> x"+ −
+ −
ML {*+ −
fun get_typ_aux (Type ("itself", [T])) = T + −
fun get_typ (Const ("TYPE", T)) = get_typ_aux T+ −
fun get_tys (Const (@{const_name "qid"},_) $ ty1 $ ty2) =+ −
(get_typ ty1, get_typ ty2)+ −
+ −
fun is_qid (Const (@{const_name "qid"},_) $ _ $ _) = true+ −
| is_qid _ = false+ −
+ −
+ −
fun mk_itself ty = Type ("itself", [ty])+ −
fun mk_TYPE ty = Const ("TYPE", mk_itself ty)+ −
fun mk_qid (rty, qty, trm) = + −
Const (@{const_name "qid"}, [mk_itself rty, mk_itself qty, dummyT] ---> dummyT) + −
$ mk_TYPE rty $ mk_TYPE qty $ trm+ −
*}+ −
+ −
ML {*+ −
fun insertion_aux rtrm qtrm =+ −
case (rtrm, qtrm) of+ −
(Abs (x, ty, t), Abs (x', ty', t')) =>+ −
let+ −
val (y, s) = Term.dest_abs (x, ty, t)+ −
val (_, s') = Term.dest_abs (x', ty', t')+ −
val yvar = Free (y, ty)+ −
val result = Term.lambda_name (y, yvar) (insertion_aux s s')+ −
in + −
if ty = ty'+ −
then result+ −
else mk_qid (ty, ty', result)+ −
end+ −
| (t1 $ t2, t1' $ t2') =>+ −
let + −
val rty = fastype_of rtrm+ −
val qty = fastype_of qtrm + −
val subtrm1 = insertion_aux t1 t1' + −
val subtrm2 = insertion_aux t2 t2'+ −
in+ −
if rty = qty+ −
then subtrm1 $ subtrm2+ −
else mk_qid (rty, qty, subtrm1 $ subtrm2)+ −
end+ −
| (Free (_, ty), Free (_, ty')) =>+ −
if ty = ty'+ −
then rtrm + −
else mk_qid (ty, ty', rtrm)+ −
| (Const (s, ty), Const (s', ty')) =>+ −
if s = s' andalso ty = ty'+ −
then rtrm+ −
else mk_qid (ty, ty', rtrm) + −
| (_, _) => raise (LIFT_MATCH "insertion")+ −
*}+ −
+ −
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 {*+ −
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 trm =+ −
case trm of+ −
(Abs (x, T, t)) => Abs (x, T, f t)+ −
| _ => f trm+ −
+ −
(* the major type of All and Ex quantifiers *)+ −
fun qnt_typ ty = domain_type (domain_type ty) + −
*}+ −
+ −
ML {*+ −
(* produces a regularized version of trm *)+ −
(* - the result is still not completely typed *)+ −
(* - does not need any special treatment of *)+ −
(* bound variables *)+ −
+ −
fun regularize_trm lthy trm =+ −
case trm of+ −
(Const (@{const_name "qid"},_) $ rty' $ qty' $ Abs (x, ty, t)) =>+ −
let+ −
val rty = get_typ rty'+ −
val qty = get_typ qty'+ −
val subtrm = regularize_trm lthy t+ −
in + −
mk_qid (rty, qty, mk_babs $ (mk_resp $ mk_resp_arg lthy (rty, qty)) $ subtrm)+ −
end+ −
| (Const (@{const_name "qid"},_) $ rty' $ qty' $ (Const (@{const_name "All"}, ty) $ t)) => + −
let+ −
val subtrm = apply_subt (regularize_trm lthy) 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+ −
(* FIXME: Should = only be replaced, when fully applied? *) + −
(* Then there must be a 2nd argument *)+ −
| (Const (@{const_name "op ="}, ty) $ t, Const (@{const_name "op ="}, ty') $ t') =>+ −
let+ −
val subtrm = regularize_trm lthy t t'+ −
in+ −
if ty = ty'+ −
then Const (@{const_name "op ="}, ty) $ subtrm+ −
else mk_resp_arg lthy (domain_type ty, domain_type ty') $ subtrm+ −
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)")+ −
| (Const (s, ty), Const (s', ty')) =>+ −
if s = s' andalso ty = ty'+ −
then rtrm+ −
else rtrm (* FIXME: check correspondence according to definitions *) + −
| (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_EQUIV, PAIR_EQUIV, ... *)+ −
ML {*+ −
fun equiv_tac rel_eqvs =+ −
REPEAT_ALL_NEW (FIRST' + −
[resolve_tac rel_eqvs,+ −
rtac @{thm IDENTITY_EQUIV},+ −
rtac @{thm LIST_EQUIV}])+ −
*}+ −
+ −
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 "EQUIV"}, 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 "EQUIV"}, 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 "EQUIV"}, 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 _ = tracing "AAA";+ −
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 "EQUIV"}, 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 _ = tracing "AAA";+ −
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: "EQUIV R \<Longrightarrow> a = b \<longrightarrow> R a b"+ −
by (simp add: EQUIV_REFL)+ −
+ −
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 *)+ −
+ −
fun inj_repabs_trm lthy (rtrm, qtrm) =+ −
let+ −
val rty = fastype_of rtrm+ −
val qty = fastype_of qtrm+ −
in+ −
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') =>+ −
Const (@{const_name "Babs"}, T) $ r $ (inj_repabs_trm lthy (t, t'))+ −
| (Abs (x, T, t), Abs (x', T', t')) =>+ −
let+ −
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+ −
| _ =>+ −
(* FIXME / TODO: this is a case that needs to be looked at *)+ −
(* - variables get a rep-abs insde and outside an application *)+ −
(* - constants only get a rep-abs on the outside of the application *)+ −
(* - applications get a rep-abs insde and outside an application *)+ −
let+ −
val (rhead, rargs) = strip_comb rtrm+ −
val (qhead, qargs) = strip_comb qtrm+ −
val rargs' = map (inj_repabs_trm lthy) (rargs ~~ qargs)+ −
in+ −
if rty = qty+ −
then+ −
case (rhead, qhead) of+ −
(Free (_, T), Free (_, T')) =>+ −
if T = T' then list_comb (rhead, rargs')+ −
else list_comb (mk_repabs lthy (T, T') rhead, rargs')+ −
| _ => list_comb (rhead, rargs')+ −
else+ −
case (rhead, qhead, length rargs') of+ −
(Const _, Const _, 0) => mk_repabs lthy (rty, qty) rhead+ −
| (Free (_, T), Free (_, T'), 0) => mk_repabs lthy (T, T') rhead+ −
| (Const _, Const _, _) => mk_repabs lthy (rty, qty) (list_comb (rhead, rargs')) + −
| (Free (x, T), Free (x', T'), _) => + −
mk_repabs lthy (rty, qty) (list_comb (mk_repabs lthy (T, T') rhead, rargs'))+ −
| (Abs _, Abs _, _ ) =>+ −
mk_repabs lthy (rty, qty) (list_comb (inj_repabs_trm lthy (rhead, qhead), rargs')) + −
| _ => raise (LIFT_MATCH "injection")+ −
end+ −
end+ −
*}+ −
+ −
section {* RepAbs Injection Tactic *}+ −
+ −
ML {*+ −
fun stripped_term_of ct =+ −
ct |> term_of |> HOLogic.dest_Trueprop+ −
*}+ −
+ −
ML {*+ −
fun instantiate_tac thm = + −
Subgoal.FOCUS (fn {concl, ...} =>+ −
let+ −
val pat = Drule.strip_imp_concl (cprop_of thm)+ −
val insts = Thm.match (pat, concl)+ −
in+ −
rtac (Drule.instantiate insts thm) 1+ −
end+ −
handle _ => no_tac)+ −
*}+ −
+ −
ML {*+ −
fun quotient_tac quot_thms =+ −
REPEAT_ALL_NEW (FIRST' + −
[rtac @{thm FUN_QUOTIENT},+ −
resolve_tac quot_thms,+ −
rtac @{thm IDENTITY_QUOTIENT},+ −
(* For functional identity quotients, (op = ---> op =) *)+ −
(* TODO: think about the other way around, if we need to shorten the relation *)+ −
CHANGED o (simp_tac (HOL_ss addsimps @{thms id_simps}))])+ −
*}+ −
+ −
lemma FUN_REL_I:+ −
assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"+ −
shows "(R1 ===> R2) f g"+ −
using a by (simp add: FUN_REL.simps)+ −
+ −
ML {*+ −
val lambda_res_tac =+ −
Subgoal.FOCUS (fn {concl, ...} =>+ −
case (stripped_term_of concl) of+ −
(_ $ (Abs _) $ (Abs _)) => rtac @{thm FUN_REL_I} 1+ −
| _ => no_tac)+ −
*}+ −
+ −
ML {*+ −
val weak_lambda_res_tac =+ −
Subgoal.FOCUS (fn {concl, ...} =>+ −
case (stripped_term_of concl) of+ −
(_ $ _ $ (Abs _)) => rtac @{thm FUN_REL_I} 1+ −
| (_ $ (Abs _) $ _) => rtac @{thm FUN_REL_I} 1+ −
| _ => no_tac)+ −
*}+ −
+ −
ML {*+ −
val ball_rsp_tac = + −
Subgoal.FOCUS (fn {concl, ...} =>+ −
case (stripped_term_of concl) of+ −
(_ $ (Const (@{const_name Ball}, _) $ _) + −
$ (Const (@{const_name Ball}, _) $ _)) => rtac @{thm FUN_REL_I} 1+ −
|_ => no_tac)+ −
*}+ −
+ −
ML {*+ −
val bex_rsp_tac = + −
Subgoal.FOCUS (fn {concl, context = ctxt, ...} =>+ −
case (stripped_term_of concl) of+ −
(_ $ (Const (@{const_name Bex}, _) $ _) + −
$ (Const (@{const_name Bex}, _) $ _)) => rtac @{thm FUN_REL_I} 1+ −
| _ => no_tac)+ −
*}+ −
+ −
ML {* (* Legacy *)+ −
fun needs_lift (rty as Type (rty_s, _)) ty =+ −
case ty of+ −
Type (s, tys) => (s = rty_s) orelse (exists (needs_lift rty) tys)+ −
| _ => false+ −
+ −
*}+ −
+ −
ML {*+ −
fun APPLY_RSP_TAC rty = + −
Subgoal.FOCUS (fn {concl, ...} =>+ −
case (stripped_term_of concl) of+ −
(_ $ (f $ _) $ (_ $ _)) =>+ −
let+ −
val pat = Drule.strip_imp_concl (cprop_of @{thm APPLY_RSP});+ −
val insts = Thm.match (pat, concl)+ −
in+ −
if needs_lift rty (fastype_of f) + −
then rtac (Drule.instantiate insts @{thm APPLY_RSP}) 1+ −
else no_tac+ −
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_res_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+ −
+ −
*)+ −
+ −
ML {*+ −
fun inj_repabs_tac ctxt rty quot_thms rel_refl trans2 =+ −
(FIRST' [+ −
(* "cong" rule of the of the relation / transitivity*)+ −
(* (op =) (R a b) (R c d) ----> \<lbrakk>R a c; R b d\<rbrakk> *)+ −
NDT ctxt "1" (resolve_tac trans2),+ −
+ −
(* (R1 ===> R2) (\<lambda>x\<dots>) (\<lambda>y\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (\<dots>x) (\<dots>y) *) + −
NDT ctxt "2" (lambda_res_tac ctxt),+ −
+ −
(* (op =) (Ball\<dots>) (Ball\<dots>) ----> (op =) (\<dots>) (\<dots>) *)+ −
NDT ctxt "3" (rtac @{thm RES_FORALL_RSP}),+ −
+ −
(* (R1 ===> R2) (Ball\<dots>) (Ball\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (Ball\<dots>x) (Ball\<dots>y) *)+ −
NDT ctxt "4" (ball_rsp_tac ctxt),+ −
+ −
(* (op =) (Bex\<dots>) (Bex\<dots>) ----> (op =) (\<dots>) (\<dots>) *)+ −
NDT ctxt "5" (rtac @{thm RES_EXISTS_RSP}),+ −
+ −
(* (R1 ===> R2) (Bex\<dots>) (Bex\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (Bex\<dots>x) (Bex\<dots>y) *)+ −
NDT ctxt "6" (bex_rsp_tac ctxt),+ −
+ −
(* respectfulness of constants *)+ −
NDT ctxt "7" (resolve_tac (rsp_rules_get ctxt)),+ −
+ −
(* reflexivity of operators arising from Cong_tac *)+ −
NDT ctxt "8" (rtac @{thm refl}),+ −
+ −
(* R (\<dots>) (Rep (Abs \<dots>)) ----> R (\<dots>) (\<dots>) *)+ −
(* observe ---> *) + −
NDT ctxt "9" ((instantiate_tac @{thm REP_ABS_RSP(1)} ctxt + −
THEN' (RANGE [SOLVES' (quotient_tac quot_thms)]))),+ −
+ −
(* R (t $ \<dots>) (t' $ \<dots>) ----> APPLY_RSP provided type of t needs lifting *)+ −
NDT ctxt "A" ((APPLY_RSP_TAC rty ctxt THEN' + −
(RANGE [SOLVES' (quotient_tac quot_thms), SOLVES' (quotient_tac quot_thms)]))),+ −
+ −
(* (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}),+ −
+ −
(* (op =) (\<lambda>x\<dots>) (\<lambda>x\<dots>) ----> (op =) (\<dots>) (\<dots>) *)+ −
NDT ctxt "C" (rtac @{thm ext}),+ −
+ −
(* reflexivity of the basic relations *)+ −
(* R \<dots> \<dots> *)+ −
NDT ctxt "D" (resolve_tac rel_refl),+ −
+ −
(* resolving with R x y assumptions *)+ −
NDT ctxt "E" (atac),+ −
+ −
(* seems not necessay:: NDT ctxt "F" (SOLVES' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))),*)+ −
+ −
(* (R1 ===> R2) (\<dots>) (\<lambda>y\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (\<dots>x) (\<dots>y) *) + −
(* (R1 ===> R2) (\<lambda>x\<dots>) (\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (\<dots>x) (\<dots>y) *) + −
(*NDT ctxt "G" (weak_lambda_res_tac ctxt),*)+ −
+ −
(* (op =) ===> (op =) \<Longrightarrow> (op =), needed in order to apply respectfulness theorems *)+ −
(* global simplification *)+ −
NDT ctxt "H" (CHANGED o (asm_full_simp_tac ((Simplifier.context ctxt empty_ss) addsimps @{thms eq_reflection[OF FUN_REL_EQ]})))])+ −
*}+ −
+ −
ML {*+ −
fun all_inj_repabs_tac ctxt rty quot_thms rel_refl trans2 =+ −
REPEAT_ALL_NEW (inj_repabs_tac ctxt rty quot_thms rel_refl trans2)+ −
*}+ −
+ −
+ −
section {* Cleaning of the theorem *}+ −
+ −
ML {*+ −
fun applic_prs lthy absrep (rty, qty) =+ −
let+ −
fun mk_rep (T, T') tm = (Quotient_Def.get_fun repF lthy (T, T')) $ tm;+ −
fun mk_abs (T, T') tm = (Quotient_Def.get_fun absF lthy (T, T')) $ tm;+ −
val (raty, rgty) = Term.strip_type rty;+ −
val (qaty, qgty) = Term.strip_type qty;+ −
val vs = map (fn _ => "x") qaty;+ −
val ((fname :: vfs), lthy') = Variable.variant_fixes ("f" :: vs) lthy;+ −
val f = Free (fname, qaty ---> qgty);+ −
val args = map Free (vfs ~~ qaty);+ −
val rhs = list_comb(f, args);+ −
val largs = map2 mk_rep (raty ~~ qaty) args;+ −
val lhs = mk_abs (rgty, qgty) (list_comb((mk_rep (raty ---> rgty, qaty ---> qgty) f), largs));+ −
val llhs = Syntax.check_term lthy lhs;+ −
val eq = Logic.mk_equals (llhs, rhs);+ −
val ceq = cterm_of (ProofContext.theory_of lthy') eq;+ −
val sctxt = HOL_ss addsimps (@{thms fun_map.simps id_simps} @ absrep);+ −
val t = Goal.prove_internal [] ceq (fn _ => simp_tac sctxt 1)+ −
val t_id = MetaSimplifier.rewrite_rule @{thms id_simps} t;+ −
in+ −
singleton (ProofContext.export lthy' lthy) t_id+ −
end+ −
*}+ −
+ −
ML {*+ −
fun find_aps_all rtm qtm =+ −
case (rtm, qtm) of+ −
(Abs(_, T1, s1), Abs(_, T2, s2)) =>+ −
find_aps_all (subst_bound ((Free ("x", T1)), s1)) (subst_bound ((Free ("x", T2)), s2))+ −
| (((f1 as (Free (_, T1))) $ a1), ((f2 as (Free (_, T2))) $ a2)) =>+ −
let+ −
val sub = (find_aps_all f1 f2) @ (find_aps_all a1 a2)+ −
in+ −
if T1 = T2 then sub else (T1, T2) :: sub+ −
end+ −
| ((f1 $ a1), (f2 $ a2)) => (find_aps_all f1 f2) @ (find_aps_all a1 a2)+ −
| _ => [];+ −
+ −
fun find_aps rtm qtm = distinct (op =) (find_aps_all rtm qtm)+ −
*}+ −
+ −
ML {*+ −
fun allex_prs_tac lthy quot =+ −
(EqSubst.eqsubst_tac lthy [0] @{thms FORALL_PRS[symmetric] EXISTS_PRS[symmetric]})+ −
THEN' (quotient_tac quot)+ −
*}+ −
+ −
(* Rewrites the term with LAMBDA_PRS thm.+ −
+ −
Replaces: (Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x)))+ −
with: f+ −
+ −
It proves the QUOTIENT assumptions by calling quotient_tac+ −
*)+ −
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;+ −
+ −
fun lambda_prs_conv1 ctxt quot_thms ctrm =+ −
case (term_of ctrm) of ((Const (@{const_name "fun_map"}, _) $ r1 $ a2) $ (Abs _)) =>+ −
let+ −
val (_, [ty_b, ty_a]) = dest_Type (fastype_of r1);+ −
val (_, [ty_c, ty_d]) = dest_Type (fastype_of a2);+ −
val thy = ProofContext.theory_of ctxt;+ −
val [cty_a, cty_b, cty_c, cty_d] = map (ctyp_of thy) [ty_a, ty_b, ty_c, ty_d]+ −
val tyinst = [SOME cty_a, SOME cty_b, SOME cty_c, SOME cty_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 tac =+ −
(compose_tac (false, lpi, 2)) THEN_ALL_NEW+ −
(quotient_tac quot_thms);+ −
val gc = Drule.strip_imp_concl (cprop_of lpi);+ −
val t = Goal.prove_internal [] gc (fn _ => tac 1)+ −
val te = @{thm eq_reflection} OF [t]+ −
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;+ −
(* val _ = writeln (Syntax.string_of_term @{context} (term_of (cprop_of ti)));*)+ −
in+ −
Conv.rewr_conv ti ctrm+ −
end+ −
*}+ −
+ −
(* quot stands for the QUOTIENT theorems: *)+ −
(* could be potentially all of them *)+ −
ML {*+ −
fun lambda_prs_conv ctxt quot ctrm =+ −
case (term_of ctrm) of+ −
(Const (@{const_name "fun_map"}, _) $ _ $ _) $ (Abs _) =>+ −
(Conv.arg_conv (Conv.abs_conv (fn (_, ctxt) => lambda_prs_conv ctxt quot) ctxt)+ −
then_conv (lambda_prs_conv1 ctxt quot)) ctrm+ −
| _ $ _ => Conv.comb_conv (lambda_prs_conv ctxt quot) ctrm+ −
| Abs _ => Conv.abs_conv (fn (_, ctxt) => lambda_prs_conv ctxt quot) ctxt ctrm+ −
| _ => Conv.all_conv ctrm+ −
*}+ −
+ −
ML {*+ −
fun lambda_prs_tac ctxt quot = CSUBGOAL (fn (goal, i) =>+ −
CONVERSION+ −
(Conv.params_conv ~1 (fn ctxt =>+ −
(Conv.prems_conv ~1 (lambda_prs_conv ctxt quot) then_conv+ −
Conv.concl_conv ~1 (lambda_prs_conv ctxt quot))) ctxt) i)+ −
*}+ −
+ −
ML {*+ −
fun clean_tac lthy quot defs aps =+ −
let+ −
val lower = flat (map (add_lower_defs lthy) defs)+ −
val meta_lower = map (fn x => @{thm eq_reflection} OF [x]) lower+ −
val absrep = map (fn x => @{thm QUOTIENT_ABS_REP} OF [x]) quot+ −
val reps_same = map (fn x => @{thm QUOTIENT_REL_REP} OF [x]) quot+ −
val meta_reps_same = map (fn x => @{thm eq_reflection} OF [x]) reps_same+ −
val simp_ctxt = (Simplifier.context lthy empty_ss) addsimps (meta_reps_same @ meta_lower)+ −
val aps_thms = map (applic_prs lthy absrep) aps+ −
in+ −
EVERY' [lambda_prs_tac lthy quot,+ −
TRY o simp_tac simp_ctxt,+ −
TRY o REPEAT_ALL_NEW (allex_prs_tac lthy quot),+ −
TRY o REPEAT_ALL_NEW (EqSubst.eqsubst_tac lthy [0] aps_thms),+ −
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 inj_goal = + −
Syntax.check_term ctxt (inj_repabs_trm ctxt (reg_goal, qtrm'))+ −
handle (LIFT_MATCH s) => lift_match_error ctxt s rtrm qtrm+ −
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 lthy rthm =+ −
ObjectLogic.full_atomize_tac+ −
THEN' gen_frees_tac lthy+ −
THEN' Subgoal.FOCUS (fn {context, concl, ...} =>+ −
let+ −
val rthm' = atomize_thm rthm+ −
val rule = procedure_inst context (prop_of rthm') (term_of concl)+ −
in+ −
EVERY1 [rtac rule, rtac rthm']+ −
end) lthy+ −
*}+ −
+ −
ML {*+ −
(* FIXME/TODO should only get as arguments the rthm like procedure_tac *)+ −
fun lift_tac lthy rthm rel_eqv rty quot defs =+ −
ObjectLogic.full_atomize_tac+ −
THEN' gen_frees_tac lthy+ −
THEN' Subgoal.FOCUS (fn {context, concl, ...} =>+ −
let+ −
val rthm' = atomize_thm rthm+ −
val rule = procedure_inst context (prop_of rthm') (term_of concl)+ −
val aps = find_aps (prop_of rthm') (term_of concl)+ −
val rel_refl = map (fn x => @{thm EQUIV_REFL} OF [x]) rel_eqv+ −
val trans2 = map (fn x => @{thm equiv_trans2} OF [x]) rel_eqv+ −
in+ −
EVERY1+ −
[rtac rule,+ −
RANGE [rtac rthm',+ −
regularize_tac lthy rel_eqv,+ −
all_inj_repabs_tac lthy rty quot rel_refl trans2,+ −
clean_tac lthy quot defs aps]]+ −
end) lthy+ −
*}+ −
+ −
end+ −
+ −
+ −