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
(* EQUALS_RSP is stronger *)
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 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')) =>
(* 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 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 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
| _ =>
let
val (rhead, rargs) = strip_comb rtrm
val (qhead, qargs) = strip_comb qtrm
val rargs' = map (inj_repabs_trm lthy) (rargs ~~ qargs)
in
case (rhead, qhead) of
(Const _, Const _) =>
if rty = qty
then list_comb (rhead, rargs')
else mk_repabs lthy (rty, qty) (list_comb (rhead, rargs'))
| (Free (x, T), Free (x', T')) =>
if T = T'
then list_comb (rhead, rargs')
else list_comb (mk_repabs lthy (T, T') rhead, rargs')
| (Abs _, Abs _) =>
list_comb (inj_repabs_trm lthy (rhead, qhead), rargs')
| _ => raise (LIFT_MATCH "injection")
end
end
*}
section {* RepAbs Injection Tactic *}
(* TODO: check if it still works with first_order_match *)
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 quot_thms =
REPEAT_ALL_NEW (FIRST'
[rtac @{thm FUN_QUOTIENT},
resolve_tac quot_thms,
rtac @{thm IDENTITY_QUOTIENT},
rtac @{thm LIST_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 ((Simplifier.context ctxt empty_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
ML {*
val lambda_rsp_tac =
SUBGOAL (fn (goal, i) =>
case HOLogic.dest_Trueprop (Logic.strip_assums_concl goal) of
(_ $ (Abs _) $ (Abs _)) => rtac @{thm FUN_REL_I} i
| _ => no_tac)
*}
ML {*
val ball_rsp_tac =
SUBGOAL (fn (goal, i) =>
case HOLogic.dest_Trueprop (Logic.strip_assums_concl goal) of
(_ $ (Const (@{const_name Ball}, _) $ _)
$ (Const (@{const_name Ball}, _) $ _)) => rtac @{thm FUN_REL_I} i
|_ => no_tac)
*}
ML {*
val bex_rsp_tac =
SUBGOAL (fn (goal, i) =>
case HOLogic.dest_Trueprop (Logic.strip_assums_concl goal) of
(_ $ (Const (@{const_name Bex}, _) $ _)
$ (Const (@{const_name Bex}, _) $ _)) => rtac @{thm FUN_REL_I} i
| _ => 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
*}
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
*}
ML {*
fun APPLY_RSP_TAC rty =
Subgoal.FOCUS (fn {concl, asms, ...} =>
case ((HOLogic.dest_Trueprop (term_of concl))) of
((_ $ (f $ _) $ (_ $ _))) =>
let
val (asmf, asma) = find_qt_asm (map term_of asms);
val pat = Drule.strip_imp_concl (cprop_of @{thm APPLY_RSP});
val insts = Thm.first_order_match (pat, concl)
in
if (fastype_of asmf) = (fastype_of f)
then no_tac
else rtac (Drule.instantiate insts @{thm APPLY_RSP}) 1
end
| _ => no_tac)
*}
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 ty_d = range_type (type_of asmf);
val thy = ProofContext.theory_of context;
val ty_inst = map (fn x => SOME (ctyp_of thy x)) [ty_a, ty_b, ty_c, ty_d];
val [R2, f, g, x, y] = map (cterm_of thy) [R2, f, g, x, y];
val t_inst = [NONE, NONE, NONE, SOME R2, NONE, NONE, SOME f, SOME g, SOME x, SOME y];
val thm = Drule.instantiate' ty_inst t_inst @{thm APPLY_RSP}
(*val _ = tracing (Syntax.string_of_term @{context} (prop_of thm))*)
in
rtac thm 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 inj_repabs_tac ctxt 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 THEN' RANGE [
quot_true_tac ctxt (fst o dest_bcomb), quot_true_tac ctxt (snd o dest_bcomb)]),
(* (R1 ===> R2) (\<lambda>x\<dots>) (\<lambda>y\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (\<dots>x) (\<dots>y) *)
NDT ctxt "2" (lambda_rsp_tac THEN' quot_true_tac ctxt unlam),
(* (op =) (Ball\<dots>) (Ball\<dots>) ----> (op =) (\<dots>) (\<dots>) *)
NDT ctxt "3" (rtac @{thm ball_rsp} THEN' dtac @{thm QT_all}),
(* R2 is always equality *)
(* (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 THEN' quot_true_tac ctxt unlam),
(* (op =) (Bex\<dots>) (Bex\<dots>) ----> (op =) (\<dots>) (\<dots>) *)
NDT ctxt "5" (rtac @{thm bex_rsp} THEN' dtac @{thm QT_ex}),
(* (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 THEN' dtac @{thm QT_ext}),
(* 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} ctxt
THEN' (RANGE [SOLVES' (quotient_tac ctxt quot_thms)]))),
(* R (t $ \<dots>) (t' $ \<dots>) ----> APPLY_RSP provided type of t needs lifting *)
NDT ctxt "A" (APPLY_RSP_TAC' ctxt THEN'
(RANGE [SOLVES' (quotient_tac ctxt quot_thms), SOLVES' (quotient_tac ctxt quot_thms),
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),
(* 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),
(* (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 quot_thms rel_refl trans2 =
REPEAT_ALL_NEW (inj_repabs_tac ctxt quot_thms rel_refl trans2)
*}
section {* Cleaning of the theorem *}
(* TODO: This is slow *)
ML {*
fun allex_prs_tac ctxt quot =
(EqSubst.eqsubst_tac ctxt [0] @{thms all_prs ex_prs}) THEN' (quotient_tac ctxt quot)
*}
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;
*}
(* It proves the QUOTIENT assumptions by calling quotient_tac *)
ML {*
fun solve_quotient_assum i quot_thms ctxt thm =
let
val tac =
(compose_tac (false, thm, i)) THEN_ALL_NEW
(quotient_tac ctxt quot_thms);
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 lambda_allex_prs_simple_conv quot_thms ctxt 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 te = @{thm eq_reflection} OF [solve_quotient_assum 2 quot_thms 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
| (Const (@{const_name Ball}, _) $ (Const (@{const_name Respects}, _) $ R) $
(((Const (@{const_name fun_map}, _) $ absf $ (Const (@{const_name id}, _)))) $ f)) =>
let
val (_, [ty_a, ty_b]) = dest_Type (fastype_of absf);
val thy = ProofContext.theory_of ctxt;
val (cty_a, cty_b) = (ctyp_of thy ty_a, ctyp_of thy ty_b);
val tyinst = [SOME cty_a, SOME cty_b];
val tinst = [SOME (cterm_of thy R), SOME (cterm_of thy absf), NONE, SOME (cterm_of thy f)];
val thm = Drule.instantiate' tyinst tinst @{thm all_prs};
val ti = @{thm eq_reflection} OF [solve_quotient_assum 1 quot_thms ctxt thm];
in
Conv.rewr_conv ti ctrm
end
| (Const (@{const_name Bex}, _) $ (Const (@{const_name Respects}, _) $ R) $
(((Const (@{const_name fun_map}, _) $ absf $ (Const (@{const_name id}, _)))) $ f)) =>
let
val (_, [ty_a, ty_b]) = dest_Type (fastype_of absf);
val thy = ProofContext.theory_of ctxt;
val (cty_a, cty_b) = (ctyp_of thy ty_a, ctyp_of thy ty_b);
val tyinst = [SOME cty_a, SOME cty_b];
val tinst = [SOME (cterm_of thy R), SOME (cterm_of thy absf), NONE, SOME (cterm_of thy f)];
val thm = Drule.instantiate' tyinst tinst @{thm ex_prs};
val ti = @{thm eq_reflection} OF [solve_quotient_assum 1 quot_thms ctxt thm];
in
Conv.rewr_conv ti ctrm
end
| _ => Conv.all_conv ctrm
*}
(* quot stands for the QUOTIENT theorems: *)
(* could be potentially all of them *)
ML {*
fun lambda_allex_prs_conv quot =
More_Conv.top_conv (lambda_allex_prs_simple_conv quot)
*}
ML {*
fun lambda_allex_prs_tac quot ctxt = CONVERSION (lambda_allex_prs_conv quot 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
- FORALL_PRS (and the same for exists: EXISTS_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 clean_tac lthy quot defs =
let
val absrep = map (fn x => @{thm QUOTIENT_ABS_REP} OF [x]) quot
val meta_absrep = map (fn x => @{thm eq_reflection} OF [x]) absrep;
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
(@{thm eq_reflection[OF fun_map.simps]} :: @{thms id_simps} @ meta_absrep @ meta_reps_same @ defs)
in
EVERY' [
(* (Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) ----> f *)
(* \<forall>x\<in>Respects R. (abs ---> id) f ----> \<forall>x. f *)
NDT lthy "a" (TRY o lambda_allex_prs_tac quot lthy),
(* NewConst ----> (rep ---> abs) oldConst *)
(* Abs (Rep x) ----> x *)
(* id_simps; fun_map.simps *)
NDT lthy "c" (TRY o simp_tac simp_ctxt),
(* 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 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)
val thm = Drule.instantiate' [] [SOME (snd (Thm.dest_comb concl))] @{thm QUOT_TRUE_i}
in
EVERY1 [rtac rule, rtac rthm'] THEN RANGE [(fn _ => all_tac), rtac thm] 1
end) lthy
*}
ML {*
(* FIXME/TODO should only get as arguments the rthm like procedure_tac *)
fun lift_tac lthy rthm rel_eqv 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 rel_refl = map (fn x => @{thm EQUIV_REFL} OF [x]) rel_eqv
val trans2 = map (fn x => @{thm equiv_trans2} OF [x]) rel_eqv
val thm = Drule.instantiate' [] [SOME (snd (Thm.dest_comb concl))] @{thm QUOT_TRUE_i}
in
EVERY1
[rtac rule,
RANGE [rtac rthm',
regularize_tac lthy rel_eqv,
rtac thm THEN' all_inj_repabs_tac lthy quot rel_refl trans2,
clean_tac lthy quot defs]]
end) lthy
*}
end