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 tst: "EQUIV bla"
sorry
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 thm =
let
val prems_str = prems_of thm
|> map (Syntax.string_of_term ctxt)
|> cat_lines
val _ = tracing (s ^ "\n" ^ prems_str)
in
Seq.single thm
end
fun DT ctxt s tac = EVERY' [tac, K (my_print_tac ctxt ("after " ^ s))]
*}
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 {* 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 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
(* 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')) =>
if x = x'
then rtrm (* FIXME: check whether types corresponds *)
else 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
*)
lemma universal_twice:
assumes *: "\<And>x. (P x \<longrightarrow> Q x)"
shows "(\<forall>x. P x) \<longrightarrow> (\<forall>x. Q x)"
using * by auto
lemma implication_twice:
assumes a: "c \<longrightarrow> a"
assumes b: "b \<longrightarrow> d"
shows "(a \<longrightarrow> b) \<longrightarrow> (c \<longrightarrow> d)"
using a b by auto
(* version of regularize_tac including debugging information *)
ML {*
fun regularize_tac ctxt rel_eqv rel_refl =
(ObjectLogic.full_atomize_tac) THEN'
REPEAT_ALL_NEW (FIRST'
[(K (print_tac "start")) THEN' (K no_tac),
DT ctxt "1" (resolve_tac rel_refl),
DT ctxt "2" atac,
DT ctxt "3" (rtac @{thm universal_twice}),
DT ctxt "4" (rtac @{thm impI} THEN' atac),
DT ctxt "5" (rtac @{thm implication_twice}),
DT ctxt "6" (EqSubst.eqsubst_tac ctxt [0]
[(@{thm ball_reg_eqv} OF [rel_eqv]),
(@{thm ball_reg_eqv} OF [rel_eqv])]),
(* For a = b \<longrightarrow> a \<approx> b *)
DT ctxt "7" (rtac @{thm impI} THEN' (asm_full_simp_tac HOL_ss) THEN' (resolve_tac rel_refl)),
DT ctxt "8" (rtac @{thm ball_reg_right})
]);
*}
lemma move_forall:
"(\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)) \<Longrightarrow> ((\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y))"
by auto
lemma move_exists:
"((\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)) \<Longrightarrow> ((\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y))"
by auto
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 SOLVES' tac = tac THEN_ALL_NEW (fn _ => no_tac)
*}
(*
ML {*
fun regularize_tac ctxt rel_eqvs rel_refl =
let
val subs1 = map (fn x => @{thm equiv_res_forall} OF [x]) rel_eqvs
val subs2 = map (fn x => @{thm equiv_res_exists} OF [x]) rel_eqvs
val subs = map (fn x => @{thm eq_reflection} OF [x]) (subs1 @ subs2)
in
(ObjectLogic.full_atomize_tac) THEN'
(simp_tac ((Simplifier.context ctxt empty_ss) addsimps subs))
THEN'
REPEAT_ALL_NEW (FIRST'
[(rtac @{thm RIGHT_RES_FORALL_REGULAR}),
(rtac @{thm LEFT_RES_EXISTS_REGULAR}),
(resolve_tac (Inductive.get_monos ctxt)),
(rtac @{thm ball_respects_refl} THEN' (RANGE [SOLVES' (equiv_tac rel_eqvs)])),
(rtac @{thm bex_respects_refl} THEN' (RANGE [SOLVES' (equiv_tac rel_eqvs)])),
rtac @{thm move_forall},
rtac @{thm move_exists},
(rtac @{thm impI} THEN' (asm_full_simp_tac HOL_ss) THEN' (resolve_tac rel_refl))])
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 = lambda 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 *}
(*
To prove that the regularised theorem implies the abs/rep injected, we first
atomize it and then 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
10) reflexivity of the relation
11) assumption
(Lambdas under respects may have left us some assumptions)
12) proving obvious higher order equalities by simplifying fun_rel
(not sure if it is still needed?)
13) unfolding lambda on one side
14) simplifying (= ===> =) for simpler respectfullness
*)
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 (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 (term_of concl) of
(_ $ (_ $ _ $ (Abs _))) => rtac @{thm FUN_REL_I} 1
| (_ $ (_ $ (Abs _) $ _)) => 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 (term_of concl) of
(_ $ (R $ (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 {*
val ball_rsp_tac =
Subgoal.FOCUS (fn {concl, context = ctxt, ...} =>
case (term_of concl) of
(_ $ (_ $ (Const (@{const_name Ball}, _) $ _) $ (Const (@{const_name Ball}, _) $ _))) =>
((simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))
THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' rtac @{thm impI}
THEN' instantiate_tac @{thm RES_FORALL_RSP} ctxt THEN'
(simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))) 1
|_ => no_tac)
*}
ML {*
val bex_rsp_tac =
Subgoal.FOCUS (fn {concl, context = ctxt, ...} =>
case (term_of concl) of
(_ $ (_ $ (Const (@{const_name Bex}, _) $ _) $ (Const (@{const_name Bex}, _) $ _))) =>
((simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))
THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' rtac @{thm impI}
THEN' instantiate_tac @{thm RES_EXISTS_RSP} ctxt THEN'
(simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))) 1
| _ => no_tac)
*}
ML {*
fun r_mk_comb_tac ctxt rty quot_thms rel_refl trans2 rsp_thms =
(FIRST' [resolve_tac trans2,
LAMBDA_RES_TAC ctxt,
rtac @{thm RES_FORALL_RSP},
ball_rsp_tac ctxt,
rtac @{thm RES_EXISTS_RSP},
bex_rsp_tac ctxt,
resolve_tac rsp_thms,
rtac @{thm refl},
(instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN'
(RANGE [SOLVES' (quotient_tac quot_thms)])),
(APPLY_RSP_TAC rty ctxt THEN'
(RANGE [SOLVES' (quotient_tac quot_thms), SOLVES' (quotient_tac quot_thms)])),
Cong_Tac.cong_tac @{thm cong},
rtac @{thm ext},
resolve_tac rel_refl,
atac,
SOLVES' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps})),
WEAK_LAMBDA_RES_TAC ctxt,
CHANGED o (asm_full_simp_tac (HOL_ss addsimps @{thms FUN_REL_EQ}))])
fun all_r_mk_comb_tac ctxt rty quot_thms rel_refl trans2 rsp_thms =
REPEAT_ALL_NEW (r_mk_comb_tac ctxt rty quot_thms rel_refl trans2 rsp_thms)
*}
(*
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 r_mk_comb_tac' ctxt rty quot_thms rel_refl trans2 rsp_thms =
(FIRST' [
(K (print_tac "start")) THEN' (K no_tac),
(* "cong" rule of the of the relation *)
(* \<lbrakk>a \<approx> c; b \<approx> d\<rbrakk> \<Longrightarrow> (a \<approx> b) = (c \<approx> d) *)
DT ctxt "1" (resolve_tac trans2),
(* (R1 ===> R2) (\<lambda>x\<dots>) (\<lambda>\<dots>) \<Longrightarrow> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (\<dots>x) (\<dots>y) *)
DT ctxt "2" (LAMBDA_RES_TAC ctxt),
(* R (Ball\<dots>) (Ball\<dots>) \<Longrightarrow> R (\<dots>) (\<dots>) *)
DT ctxt "3" (rtac @{thm RES_FORALL_RSP}),
DT ctxt "4" (ball_rsp_tac ctxt),
DT ctxt "5" (rtac @{thm RES_EXISTS_RSP}),
DT ctxt "6" (bex_rsp_tac ctxt),
(* respectfulness of ops *)
DT ctxt "7" (resolve_tac rsp_thms),
(* reflexivity of operators arising from Cong_tac *)
DT ctxt "8" (rtac @{thm refl}),
(* R (\<dots>) (Rep (Abs \<dots>)) \<Longrightarrow> R (\<dots>) (\<dots>) *)
(* observe ---> *)
DT ctxt "9" ((instantiate_tac @{thm REP_ABS_RSP(1)} ctxt
THEN' (RANGE [SOLVES' (quotient_tac quot_thms)]))),
(* R (t $ \<dots>) (t' $ \<dots>) \<Longrightarrow> APPLY_RSP provided type of t needs lifting *)
DT ctxt "A" ((APPLY_RSP_TAC rty ctxt THEN'
(RANGE [SOLVES' (quotient_tac quot_thms), SOLVES' (quotient_tac quot_thms)]))),
(* R (t $ \<dots>) (t' $ \<dots>) \<Longrightarrow> Cong provided R = (op =) and t does not need lifting *)
DT ctxt "B" (Cong_Tac.cong_tac @{thm cong}),
(* = (\<lambda>x\<dots>) (\<lambda>x\<dots>) \<Longrightarrow> = (\<dots>) (\<dots>) *)
DT ctxt "C" (rtac @{thm ext}),
(* reflexivity of the basic relations *)
DT ctxt "D" (resolve_tac rel_refl),
(* resolving with R x y assumptions *)
DT ctxt "E" (atac),
DT ctxt "F" (SOLVES' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))),
DT ctxt "G" (WEAK_LAMBDA_RES_TAC ctxt),
DT ctxt "H" (CHANGED o (asm_full_simp_tac (HOL_ss addsimps @{thms FUN_REL_EQ})))])
fun all_r_mk_comb_tac' ctxt rty quot_thms rel_refl trans2 rsp_thms =
REPEAT_ALL_NEW (r_mk_comb_tac' ctxt rty quot_thms rel_refl trans2 rsp_thms)
*}
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)
*}
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
*}
(* 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 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 _ = tracing (Syntax.string_of_term @{context} (term_of ctrm));*)
(* val _ = tracing (Syntax.string_of_term @{context} (term_of tl));*)
val insts = matching_prs (ProofContext.theory_of ctxt) (term_of tl) (term_of ctrm);
val ti = Drule.eta_contraction_rule (Drule.instantiate insts ts);
(* val _ = tracing (Syntax.string_of_term @{context} (term_of (cprop_of ti)));*)
in
Conv.rewr_conv ti ctrm
end
(* TODO: We can add a proper error message... *)
handle Bind => Conv.all_conv ctrm
*}
(* 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 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 aps_thms = map (applic_prs lthy absrep) aps
in
EVERY' [TRY o REPEAT_ALL_NEW (allex_prs_tac lthy quot),
lambda_prs_tac lthy quot,
TRY o REPEAT_ALL_NEW (EqSubst.eqsubst_tac lthy [0] aps_thms),
TRY o REPEAT_ALL_NEW (EqSubst.eqsubst_tac lthy [0] lower),
simp_tac (HOL_ss addsimps reps_same)]
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 rsp_thms 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 (hd rel_eqv) rel_refl, (* temporary hd *)
REPEAT_ALL_NEW (r_mk_comb_tac lthy rty quot rel_refl trans2 rsp_thms),
clean_tac lthy quot defs aps]]
end) lthy
*}
end