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
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"
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 Pure.equal_elim_rule1} OF [thm'', thm']
end
*}
ML {* atomize_thm @{thm list.induct} *}
section {* REGULARIZE *}
(*
Regularizing a theorem 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
- 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
appropriate relations:
A = B \<Longrightarrow> A \<approx> B
Example with more complicated types of A, B:
A = B \<Longrightarrow> (op = \<Longrightarrow> op \<approx>) A B
Regularizing is done in 3 phases:
- First a regularized term is created
- Next we prove that the original theorem implies the new one
- Finally using MP we get the new theorem.
To prove that the old theorem implies the new one, we first
atomize it and then try:
- 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
*)
text {* tyRel takes a type and builds a relation that a quantifier over this
type needs to respect. *}
ML {*
fun tyRel ty rty rel lthy =
if Sign.typ_instance (ProofContext.theory_of lthy) (ty, rty)
then rel
else (case ty of
Type (s, tys) =>
let
val tys_rel = map (fn ty => ty --> ty --> @{typ bool}) tys;
val ty_out = ty --> ty --> @{typ bool};
val tys_out = tys_rel ---> ty_out;
in
(case (maps_lookup (ProofContext.theory_of lthy) s) of
SOME (info) => list_comb (Const (#relfun info, tys_out),
map (fn ty => tyRel ty rty rel lthy) tys)
| NONE => HOLogic.eq_const ty
)
end
| _ => HOLogic.eq_const ty)
*}
(*
ML {* cterm_of @{theory}
(tyRel @{typ "'a \<Rightarrow> 'a list \<Rightarrow> 't \<Rightarrow> 't"} (Logic.varifyT @{typ "'a list"})
@{term "f::('a list \<Rightarrow> 'a list \<Rightarrow> bool)"} @{context})
*}
*)
definition
Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
where
"(x \<in> p) \<Longrightarrow> (Babs p m x = m x)"
(* TODO: Consider defining it with an "if"; sth like:
Babs p m = \<lambda>x. if x \<in> p then m x else undefined
*)
ML {*
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 mk_babs ty ty' = Const (@{const_name "Babs"}, [ty' --> @{typ bool}, ty] ---> ty)
fun mk_ball ty = Const (@{const_name "Ball"}, [ty, ty] ---> @{typ bool})
fun mk_bex ty = Const (@{const_name "Bex"}, [ty, ty] ---> @{typ bool})
fun mk_resp ty = Const (@{const_name Respects}, [[ty, ty] ---> @{typ bool}, ty] ---> @{typ bool})
*}
(* applies f to the subterm of an abstractions, otherwise to the given term *)
ML {*
fun apply_subt f trm =
case trm of
Abs (x, T, t) =>
let
val (x', t') = Term.dest_abs (x, T, t)
in
Term.absfree (x', T, f t')
end
| _ => f trm
*}
(* FIXME: if there are more than one quotient, then you have to look up the relation *)
ML {*
fun my_reg lthy rel rty trm =
case trm of
Abs (x, T, t) =>
if (needs_lift rty T) then
let
val rrel = tyRel T rty rel lthy
in
(mk_babs (fastype_of trm) T) $ (mk_resp T $ rrel) $ (apply_subt (my_reg lthy rel rty) trm)
end
else
Abs(x, T, (apply_subt (my_reg lthy rel rty) t))
| Const (@{const_name "All"}, ty) $ (t as Abs (x, T, _)) =>
let
val ty1 = domain_type ty
val ty2 = domain_type ty1
val rrel = tyRel T rty rel lthy
in
if (needs_lift rty T) then
(mk_ball ty1) $ (mk_resp ty2 $ rrel) $ (apply_subt (my_reg lthy rel rty) t)
else
Const (@{const_name "All"}, ty) $ apply_subt (my_reg lthy rel rty) t
end
| Const (@{const_name "Ex"}, ty) $ (t as Abs (x, T, _)) =>
let
val ty1 = domain_type ty
val ty2 = domain_type ty1
val rrel = tyRel T rty rel lthy
in
if (needs_lift rty T) then
(mk_bex ty1) $ (mk_resp ty2 $ rrel) $ (apply_subt (my_reg lthy rel rty) t)
else
Const (@{const_name "Ex"}, ty) $ apply_subt (my_reg lthy rel rty) t
end
| Const (@{const_name "op ="}, ty) $ t =>
if needs_lift rty (fastype_of t) then
(tyRel (fastype_of t) rty rel lthy) $ t (* FIXME: t should be regularised too *)
else Const (@{const_name "op ="}, ty) $ (my_reg lthy rel rty t)
| t1 $ t2 => (my_reg lthy rel rty t1) $ (my_reg lthy rel rty t2)
| _ => trm
*}
(* For polymorphic types we need to find the type of the Relation term. *)
(* TODO: we assume that the relation is a Constant. Is this always true? *)
ML {*
fun my_reg_inst lthy rel rty trm =
case rel of
Const (n, _) => Syntax.check_term lthy
(my_reg lthy (Const (n, dummyT)) (Logic.varifyT rty) trm)
*}
(*
ML {*
val r = Free ("R", dummyT);
val t = (my_reg_inst @{context} r @{typ "'a list"} @{term "\<forall>(x::'b list). P x"});
val t2 = Syntax.check_term @{context} t;
cterm_of @{theory} t2
*}
*)
text {* Assumes that the given theorem is atomized *}
ML {*
fun build_regularize_goal thm rty rel lthy =
Logic.mk_implies
((prop_of thm),
(my_reg_inst lthy rel rty (prop_of thm)))
*}
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: "a \<Longrightarrow> b \<longrightarrow> d"
shows "(a \<longrightarrow> b) \<longrightarrow> (c \<longrightarrow> d)"
using a b by auto
ML {*
fun regularize thm rty rel rel_eqv rel_refl lthy =
let
val goal = build_regularize_goal thm rty rel lthy;
fun tac ctxt =
(ObjectLogic.full_atomize_tac) THEN'
REPEAT_ALL_NEW (FIRST' [
rtac rel_refl,
atac,
rtac @{thm universal_twice},
(rtac @{thm impI} THEN' atac),
rtac @{thm implication_twice},
EqSubst.eqsubst_tac ctxt [0]
[(@{thm equiv_res_forall} OF [rel_eqv]),
(@{thm equiv_res_exists} OF [rel_eqv])],
(* For a = b \<longrightarrow> a \<approx> b *)
(rtac @{thm impI} THEN' (asm_full_simp_tac HOL_ss) THEN' rtac rel_refl),
(rtac @{thm RIGHT_RES_FORALL_REGULAR})
]);
val cthm = Goal.prove lthy [] [] goal
(fn {context, ...} => tac context 1);
in
cthm OF [thm]
end
*}
section {* RepAbs injection *}
(*
RepAbs injection is done in the following phases:
1) build_repabs_term inserts rep-abs pairs in the term
2) we prove the equality between the original theorem and this one
3) we use Pure.equal_elim_rule1 to get the new theorem.
build_repabs_term does:
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.
To prove that the old theorem implies the new one, 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
*)
(* changes (?'a ?'b raw) (?'a ?'b quo) (int 'b raw \<Rightarrow> bool) to (int 'b quo \<Rightarrow> bool) *)
ML {*
fun exchange_ty lthy rty qty ty =
let
val thy = ProofContext.theory_of lthy
in
if Sign.typ_instance thy (ty, rty) then
let
val inst = Sign.typ_match thy (rty, ty) Vartab.empty
in
Envir.subst_type inst qty
end
else
let
val (s, tys) = dest_Type ty
in
Type (s, map (exchange_ty lthy rty qty) tys)
end
end
handle TYPE _ => ty (* for dest_Type *)
*}
(*consts Rl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
axioms Rl_eq: "EQUIV Rl"
quotient ql = "'a list" / "Rl"
by (rule Rl_eq)
ML {*
ctyp_of @{theory} (exchange_ty @{context} (Logic.varifyT @{typ "'a list"}) (Logic.varifyT @{typ "'a ql"}) @{typ "nat list \<Rightarrow> (nat \<times> nat) list"});
ctyp_of @{theory} (exchange_ty @{context} (Logic.varifyT @{typ "nat \<times> nat"}) (Logic.varifyT @{typ "int"}) @{typ "nat list \<Rightarrow> (nat \<times> nat) list"})
*}
*)
ML {*
fun negF absF = repF
| negF repF = absF
fun get_fun_noexchange flag (rty, qty) lthy ty =
let
fun get_fun_aux s fs_tys =
let
val (fs, tys) = split_list fs_tys
val (otys, ntys) = split_list tys
val oty = Type (s, otys)
val nty = Type (s, ntys)
val ftys = map (op -->) tys
in
(case (maps_lookup (ProofContext.theory_of lthy) s) of
SOME info => (list_comb (Const (#mapfun info, ftys ---> (oty --> nty)), fs), (oty, nty))
| NONE => error ("no map association for type " ^ s))
end
fun get_fun_fun fs_tys =
let
val (fs, tys) = split_list fs_tys
val ([oty1, oty2], [nty1, nty2]) = split_list tys
val oty = nty1 --> oty2
val nty = oty1 --> nty2
val ftys = map (op -->) tys
in
(list_comb (Const (@{const_name "fun_map"}, ftys ---> oty --> nty), fs), (oty, nty))
end
val thy = ProofContext.theory_of lthy
fun get_const flag (rty, qty) =
let
val qty_name = Long_Name.base_name (fst (dest_Type qty))
in
case flag of
absF => (Const (Sign.full_bname thy ("ABS_" ^ qty_name), rty --> qty), (rty, qty))
| repF => (Const (Sign.full_bname thy ("REP_" ^ qty_name), qty --> rty), (qty, rty))
end
fun mk_identity ty = Abs ("", ty, Bound 0)
in
if (Sign.typ_instance thy (ty, rty))
then (get_const flag (ty, (exchange_ty lthy rty qty ty)))
else (case ty of
TFree _ => (mk_identity ty, (ty, ty))
| Type (_, []) => (mk_identity ty, (ty, ty))
| Type ("fun" , [ty1, ty2]) =>
get_fun_fun [get_fun_noexchange (negF flag) (rty, qty) lthy ty1,
get_fun_noexchange flag (rty, qty) lthy ty2]
| Type (s, tys) => get_fun_aux s (map (get_fun_noexchange flag (rty, qty) lthy) tys)
| _ => raise ERROR ("no type variables"))
end
fun get_fun_noex flag (rty, qty) lthy ty =
fst (get_fun_noexchange flag (rty, qty) lthy ty)
*}
ML {*
fun find_matching_types rty ty =
if Type.raw_instance (Logic.varifyT ty, rty)
then [ty]
else
let val (s, tys) = dest_Type ty in
flat (map (find_matching_types rty) tys)
end
handle TYPE _ => []
*}
ML {*
fun negF absF = repF
| negF repF = absF
fun get_fun flag qenv lthy ty =
let
fun get_fun_aux s fs =
(case (maps_lookup (ProofContext.theory_of lthy) s) of
SOME info => list_comb (Const (#mapfun info, dummyT), fs)
| NONE => error ("no map association for type " ^ s))
fun get_const flag qty =
let
val thy = ProofContext.theory_of lthy
val qty_name = Long_Name.base_name (fst (dest_Type qty))
in
case flag of
absF => Const (Sign.full_bname thy ("ABS_" ^ qty_name), dummyT)
| repF => Const (Sign.full_bname thy ("REP_" ^ qty_name), dummyT)
end
fun mk_identity ty = Abs ("", ty, Bound 0)
in
if (AList.defined (op=) qenv ty)
then (get_const flag ty)
else (case ty of
TFree _ => mk_identity ty
| Type (_, []) => mk_identity ty
| Type ("fun" , [ty1, ty2]) =>
let
val fs_ty1 = get_fun (negF flag) qenv lthy ty1
val fs_ty2 = get_fun flag qenv lthy ty2
in
get_fun_aux "fun" [fs_ty1, fs_ty2]
end
| Type (s, tys) => get_fun_aux s (map (get_fun flag qenv lthy) tys)
| _ => error ("no type variables allowed"))
end
(* returns all subterms where two types differ *)
fun diff (T, S) Ds =
case (T, S) of
(TVar v, TVar u) => if v = u then Ds else (T, S)::Ds
| (TFree x, TFree y) => if x = y then Ds else (T, S)::Ds
| (Type (a, Ts), Type (b, Us)) =>
if a = b then diffs (Ts, Us) Ds else (T, S)::Ds
| _ => (T, S)::Ds
and diffs (T::Ts, U::Us) Ds = diffs (Ts, Us) (diff (T, U) Ds)
| diffs ([], []) Ds = Ds
| diffs _ _ = error "Unequal length of type arguments"
*}
ML {*
fun get_fun_OLD flag (rty, qty) lthy ty =
let
val tys = find_matching_types rty ty;
val qenv = map (fn t => (exchange_ty lthy rty qty t, t)) tys;
val xchg_ty = exchange_ty lthy rty qty ty
in
get_fun flag qenv lthy xchg_ty
end
*}
(*
consts Rl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
axioms Rl_eq: "EQUIV Rl"
quotient ql = "'a list" / "Rl"
by (rule Rl_eq)
ML {* val al = snd (dest_Free (term_of @{cpat "f :: ?'a list"})) *}
ML {* val aq = snd (dest_Free (term_of @{cpat "f :: ?'a ql"})) *}
ML {* val ttt = term_of @{cterm "f :: bool list \<Rightarrow> nat list"} *}
ML {*
get_fun_noexchange absF (al, aq) @{context} (fastype_of ttt)
*}
ML {*
get_fun_new absF al aq @{context} (fastype_of ttt)
*}
ML {*
fun mk_abs tm =
let
val ty = fastype_of tm
in (get_fun_noexchange absF (al, aq) @{context} ty) $ tm end
fun mk_repabs tm =
let
val ty = fastype_of tm
in (get_fun_noexchange repF (al, aq) @{context} ty) $ (mk_abs tm) end
*}
ML {*
cterm_of @{theory} (mk_repabs ttt)
*}
*)
text {* Does the same as 'subst' in a given prop or theorem *}
ML {*
fun eqsubst_prop ctxt thms t =
let
val goalstate = Goal.init (cterm_of (ProofContext.theory_of ctxt) t)
val a' = case (SINGLE (EqSubst.eqsubst_tac ctxt [0] thms 1) goalstate) of
NONE => error "eqsubst_prop"
| SOME th => cprem_of th 1
in term_of a' end
*}
ML {*
fun repeat_eqsubst_prop ctxt thms t =
repeat_eqsubst_prop ctxt thms (eqsubst_prop ctxt thms t)
handle _ => t
*}
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 Pure.equal_elim_rule1} OF [rt, thm]
end
*}
ML {*
fun repeat_eqsubst_thm ctxt thms thm =
repeat_eqsubst_thm ctxt thms (eqsubst_thm ctxt thms thm)
handle _ => thm
*}
(* Needed to have a meta-equality *)
lemma id_def_sym: "(\<lambda>x. x) \<equiv> id"
by (simp add: id_def)
(* TODO: can be also obtained with: *)
ML {* symmetric (eq_reflection OF @{thms id_def}) *}
ML {*
fun build_repabs_term lthy thm consts rty qty =
let
(* TODO: The rty and qty stored in the quotient_info should
be varified, so this will soon not be needed *)
val rty = Logic.varifyT rty;
val qty = Logic.varifyT qty;
fun mk_abs tm =
let
val ty = fastype_of tm
in Syntax.check_term lthy ((get_fun_OLD absF (rty, qty) lthy ty) $ tm) end
fun mk_repabs tm =
let
val ty = fastype_of tm
in Syntax.check_term lthy ((get_fun_OLD repF (rty, qty) lthy ty) $ (mk_abs tm)) end
fun is_lifted_const (Const (x, _)) = member (op =) consts x
| is_lifted_const _ = false;
fun build_aux lthy tm =
case tm of
Abs (a as (_, vty, _)) =>
let
val (vs, t) = Term.dest_abs a;
val v = Free(vs, vty);
val t' = lambda v (build_aux lthy t)
in
if (not (needs_lift rty (fastype_of tm))) then t'
else mk_repabs (
if not (needs_lift rty vty) then t'
else
let
val v' = mk_repabs v;
(* TODO: I believe 'beta' is not needed any more *)
val t1 = (* Envir.beta_norm *) (t' $ v')
in
lambda v t1
end)
end
| x =>
case Term.strip_comb tm of
(Const(@{const_name Respects}, _), _) => tm
| (opp, tms0) =>
let
val tms = map (build_aux lthy) tms0
val ty = fastype_of tm
in
if (is_lifted_const opp andalso needs_lift rty ty) then
mk_repabs (list_comb (opp, tms))
else if ((Term.is_Free opp) andalso (length tms > 0) andalso (needs_lift rty ty)) then
mk_repabs (list_comb (opp, tms))
else if tms = [] then opp
else list_comb(opp, tms)
end
in
repeat_eqsubst_prop lthy @{thms id_def_sym}
(build_aux lthy (Thm.prop_of thm))
end
*}
text {* Builds provable goals for regularized theorems *}
ML {*
fun build_repabs_goal ctxt thm cons rty qty =
Logic.mk_equals ((Thm.prop_of thm), (build_repabs_term ctxt thm cons rty qty))
*}
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 CHANGED' tac = (fn i => CHANGED (tac i))
*}
ML {*
fun quotient_tac quot_thm =
REPEAT_ALL_NEW (FIRST' [
rtac @{thm FUN_QUOTIENT},
rtac quot_thm,
rtac @{thm IDENTITY_QUOTIENT},
(* For functional identity quotients, (op = ---> op =) *)
CHANGED' (
(simp_tac (HOL_ss addsimps @{thms FUN_MAP_I})) THEN'
rtac @{thm IDENTITY_QUOTIENT}
)
])
*}
ML {*
fun LAMBDA_RES_TAC ctxt i st =
(case (term_of o #concl o fst) (Subgoal.focus ctxt i st) of
(_ $ (_ $ (Abs(_, _, _)) $ (Abs(_, _, _)))) =>
(EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN'
(rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI})
| _ => fn _ => no_tac) i st
*}
ML {*
fun WEAK_LAMBDA_RES_TAC ctxt i st =
(case (term_of o #concl o fst) (Subgoal.focus ctxt i st) of
(_ $ (_ $ _ $ (Abs(_, _, _)))) =>
(EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN'
(rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI})
| (_ $ (_ $ (Abs(_, _, _)) $ _)) =>
(EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN'
(rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI})
| _ => fn _ => no_tac) i st
*}
ML {*
fun APPLY_RSP_TAC rty = Subgoal.FOCUS (fn {concl, ...} =>
let
val (_ $ (R $ (f $ _) $ (_ $ _))) = term_of concl;
val pat = Drule.strip_imp_concl (cprop_of @{thm APPLY_RSP});
val insts = Thm.match (pat, concl)
in
if needs_lift rty (type_of f) then
rtac (Drule.instantiate insts @{thm APPLY_RSP}) 1
else no_tac
end
handle _ => no_tac)
*}
ML {*
val ball_rsp_tac = Subgoal.FOCUS (fn {concl, context = ctxt, ...} =>
let
val _ $ (_ $ (Const (@{const_name Ball}, _) $ _) $
(Const (@{const_name Ball}, _) $ _)) = term_of concl
in
((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
end
handle _ => no_tac)
*}
ML {*
val bex_rsp_tac = Subgoal.FOCUS (fn {concl, context = ctxt, ...} =>
let
val _ $ (_ $ (Const (@{const_name Bex}, _) $ _) $
(Const (@{const_name Bex}, _) $ _)) = term_of concl
in
((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
end
handle _ => no_tac)
*}
ML {*
fun SOLVES' tac = tac THEN_ALL_NEW (fn _ => no_tac)
*}
ML {*
fun r_mk_comb_tac ctxt rty quot_thm reflex_thm trans_thm rsp_thms =
(FIRST' [
rtac trans_thm,
LAMBDA_RES_TAC ctxt,
ball_rsp_tac ctxt,
bex_rsp_tac ctxt,
FIRST' (map rtac rsp_thms),
(instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN' (RANGE [quotient_tac quot_thm])),
rtac refl,
(APPLY_RSP_TAC rty ctxt THEN' (RANGE [quotient_tac quot_thm, quotient_tac quot_thm])),
Cong_Tac.cong_tac @{thm cong},
rtac @{thm ext},
rtac reflex_thm,
atac,
SOLVES' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps})),
WEAK_LAMBDA_RES_TAC ctxt,
CHANGED' (asm_full_simp_tac (HOL_ss addsimps @{thms FUN_REL_EQ}))
])
*}
ML {*
fun repabs lthy thm consts rty qty quot_thm reflex_thm trans_thm rsp_thms =
let
val rt = build_repabs_term lthy thm consts rty qty;
val rg = Logic.mk_equals ((Thm.prop_of thm), rt);
fun tac ctxt = (ObjectLogic.full_atomize_tac) THEN'
(REPEAT_ALL_NEW (r_mk_comb_tac ctxt rty quot_thm reflex_thm trans_thm rsp_thms));
val cthm = Goal.prove lthy [] [] rg (fn x => tac (#context x) 1);
in
@{thm Pure.equal_elim_rule1} OF [cthm, thm]
end
*}
section {* Cleaning the goal *}
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
ML {*
fun simp_ids lthy thm =
MetaSimplifier.rewrite_rule @{thms eq_reflection[OF FUN_MAP_I] eq_reflection[OF id_apply] id_def_sym prod_fun_id map_id} thm
*}
ML {*
fun simp_ids_trm trm =
trm |>
MetaSimplifier.rewrite false @{thms eq_reflection[OF FUN_MAP_I] eq_reflection[OF id_apply] id_def_sym prod_fun_id map_id}
|> cprop_of |> Thm.dest_equals |> snd
*}
text {* expects atomized definition *}
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 lthy f
in
(simp_ids lthy thm) :: (add_lower_defs_aux lthy g)
end
handle _ => [simp_ids lthy 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
*}
(* TODO: Check if it behaves properly with varifyed rty *)
ML {*
fun findabs_all rty tm =
case tm of
Abs(_, T, b) =>
let
val b' = subst_bound ((Free ("x", T)), b);
val tys = findabs_all rty b'
val ty = fastype_of tm
in if needs_lift rty ty then (ty :: tys) else tys
end
| f $ a => (findabs_all rty f) @ (findabs_all rty a)
| _ => [];
fun findabs rty tm = distinct (op =) (findabs_all rty tm)
*}
ML {*
fun findaps_all rty tm =
case tm of
Abs(_, T, b) =>
findaps_all rty (subst_bound ((Free ("x", T)), b))
| (f $ a) => (findaps_all rty f @ findaps_all rty a)
| Free (_, (T as (Type ("fun", (_ :: _))))) =>
(if needs_lift rty T then [T] else [])
| _ => [];
fun findaps rty tm = distinct (op =) (findaps_all rty tm)
*}
(* Currently useful only for LAMBDA_PRS *)
ML {*
fun make_simp_prs_thm lthy quot_thm thm typ =
let
val (_, [lty, rty]) = dest_Type typ;
val thy = ProofContext.theory_of lthy;
val (lcty, rcty) = (ctyp_of thy lty, ctyp_of thy rty)
val inst = [SOME lcty, NONE, SOME rcty];
val lpi = Drule.instantiate' inst [] thm;
val tac =
(compose_tac (false, lpi, 2)) THEN_ALL_NEW
(quotient_tac quot_thm);
val gc = Drule.strip_imp_concl (cprop_of lpi);
val t = Goal.prove_internal [] gc (fn _ => tac 1)
in
MetaSimplifier.rewrite_rule [@{thm eq_reflection} OF @{thms id_apply}] t
end
*}
ML {*
fun findallex_all rty qty tm =
case tm of
Const (@{const_name All}, T) $ (s as (Abs(_, _, b))) =>
let
val (tya, tye) = findallex_all rty qty s
in if needs_lift rty T then
((T :: tya), tye)
else (tya, tye) end
| Const (@{const_name Ex}, T) $ (s as (Abs(_, _, b))) =>
let
val (tya, tye) = findallex_all rty qty s
in if needs_lift rty T then
(tya, (T :: tye))
else (tya, tye) end
| Abs(_, T, b) =>
findallex_all rty qty (subst_bound ((Free ("x", T)), b))
| f $ a =>
let
val (a1, e1) = findallex_all rty qty f;
val (a2, e2) = findallex_all rty qty a;
in (a1 @ a2, e1 @ e2) end
| _ => ([], []);
*}
ML {*
fun findallex lthy rty qty tm =
let
val (a, e) = findallex_all rty qty tm;
val (ad, ed) = (map domain_type a, map domain_type e);
val (au, eu) = (distinct (op =) ad, distinct (op =) ed);
val (rty, qty) = (Logic.varifyT rty, Logic.varifyT qty)
in
(map (exchange_ty lthy rty qty) au, map (exchange_ty lthy rty qty) eu)
end
*}
ML {*
fun make_allex_prs_thm lthy quot_thm thm typ =
let
val (_, [lty, rty]) = dest_Type typ;
val thy = ProofContext.theory_of lthy;
val (lcty, rcty) = (ctyp_of thy lty, ctyp_of thy rty)
val inst = [NONE, SOME lcty];
val lpi = Drule.instantiate' inst [] thm;
val tac =
(compose_tac (false, lpi, 1)) THEN_ALL_NEW
(quotient_tac quot_thm);
val gc = Drule.strip_imp_concl (cprop_of lpi);
val t = Goal.prove_internal [] gc (fn _ => tac 1)
val t_noid = MetaSimplifier.rewrite_rule
[@{thm eq_reflection} OF @{thms id_apply}] t;
val t_sym = @{thm "HOL.sym"} OF [t_noid];
val t_eq = @{thm "eq_reflection"} OF [t_sym]
in
t_eq
end
*}
ML {*
fun applic_prs lthy rty qty absrep ty =
let
val rty = Logic.varifyT rty;
val qty = Logic.varifyT qty;
fun absty ty =
exchange_ty lthy rty qty ty
fun mk_rep tm =
let
val ty = exchange_ty lthy qty rty (fastype_of tm)
in Syntax.check_term lthy ((get_fun_OLD repF (rty, qty) lthy ty) $ tm) end;
fun mk_abs tm =
let
val ty = fastype_of tm
in Syntax.check_term lthy ((get_fun_OLD absF (rty, qty) lthy ty) $ tm) end
val (l, ltl) = Term.strip_type ty;
val nl = map absty l;
val vs = map (fn _ => "x") l;
val ((fname :: vfs), lthy') = Variable.variant_fixes ("f" :: vs) lthy;
val args = map Free (vfs ~~ nl);
val lhs = list_comb((Free (fname, nl ---> ltl)), args);
val rargs = map mk_rep args;
val f = Free (fname, nl ---> ltl);
val rhs = mk_abs (list_comb((mk_rep f), rargs));
val eq = Logic.mk_equals (rhs, lhs);
val ceq = cterm_of (ProofContext.theory_of lthy') eq;
val sctxt = HOL_ss addsimps (absrep :: @{thms fun_map.simps});
val t = Goal.prove_internal [] ceq (fn _ => simp_tac sctxt 1)
val t_id = MetaSimplifier.rewrite_rule @{thms id_def_sym} t;
in
singleton (ProofContext.export lthy' lthy) t_id
end
*}
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_pre = @{thm EQUIV_REFL} OF [rel_eqv]
val rel_refl = @{thm spec} OF [MetaSimplifier.rewrite_rule [@{thm REFL_def}] rel_refl_pre]
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
*}
ML {*
fun lift_thm lthy qty qty_name rsp_thms defs rthm =
let
val _ = tracing ("raw theorem:\n" ^ Syntax.string_of_term lthy (prop_of rthm))
val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data lthy qty;
val (trans2, reps_same, absrep, quot) = lookup_quot_thms lthy qty_name;
val consts = lookup_quot_consts defs;
val t_a = atomize_thm rthm;
val _ = tracing ("raw atomized theorem:\n" ^ Syntax.string_of_term lthy (prop_of t_a))
val t_r = regularize t_a rty rel rel_eqv rel_refl lthy;
val _ = tracing ("regularised theorem:\n" ^ Syntax.string_of_term lthy (prop_of t_r))
val t_t = repabs lthy t_r consts rty qty quot rel_refl trans2 rsp_thms;
val _ = tracing ("rep/abs injected theorem:\n" ^ Syntax.string_of_term lthy (prop_of t_t))
val (alls, exs) = findallex lthy rty qty (prop_of t_a);
val allthms = map (make_allex_prs_thm lthy quot @{thm FORALL_PRS}) alls
val exthms = map (make_allex_prs_thm lthy quot @{thm EXISTS_PRS}) exs
val t_a = MetaSimplifier.rewrite_rule (allthms @ exthms) t_t
val _ = tracing ("??:\n" ^ Syntax.string_of_term lthy (prop_of t_a))
val abs = findabs rty (prop_of t_a);
val aps = findaps rty (prop_of t_a);
val app_prs_thms = map (applic_prs lthy rty qty absrep) aps;
val lam_prs_thms = map (make_simp_prs_thm lthy quot @{thm LAMBDA_PRS}) abs;
val t_l = repeat_eqsubst_thm lthy (lam_prs_thms @ app_prs_thms) t_a;
val _ = tracing ("??:\n" ^ Syntax.string_of_term lthy (prop_of t_l))
val defs_sym = flat (map (add_lower_defs lthy) defs);
val defs_sym_eq = map (fn x => eq_reflection OF [x]) defs_sym;
val t_id = simp_ids lthy t_l;
val _ = tracing ("??:\n" ^ Syntax.string_of_term lthy (prop_of t_id))
val t_d0 = MetaSimplifier.rewrite_rule defs_sym_eq t_id;
val _ = tracing ("??:\n" ^ Syntax.string_of_term lthy (prop_of t_d0))
val t_d = repeat_eqsubst_thm lthy defs_sym t_d0;
val _ = tracing ("??:\n" ^ Syntax.string_of_term lthy (prop_of t_d))
val t_r = MetaSimplifier.rewrite_rule [reps_same] t_d;
val _ = tracing ("??:\n" ^ Syntax.string_of_term lthy (prop_of t_r))
val t_rv = ObjectLogic.rulify t_r
val _ = tracing ("lifted theorem:\n" ^ Syntax.string_of_term lthy (prop_of t_rv))
in
Thm.varifyT t_rv
end
*}
ML {*
fun lift_thm_note qty qty_name rsp_thms defs thm name lthy =
let
val lifted_thm = lift_thm lthy qty qty_name rsp_thms defs thm;
val (_, lthy2) = note (name, lifted_thm) lthy;
in
lthy2
end
*}
(******************************************)
(******************************************)
(* version with explicit qtrm *)
(******************************************)
(******************************************)
ML {*
fun mk_resp_arg lthy (rty, qty) =
let
val thy = ProofContext.theory_of lthy
in
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'
in
#rel qinfo
end
| _ => HOLogic.eq_const dummyT
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 {*
fun trm_lift_error lthy rtrm qtrm =
let
val rtrm_str = quote (Syntax.string_of_term lthy rtrm)
val qtrm_str = quote (Syntax.string_of_term lthy qtrm)
val msg = ["The quotient theorem", qtrm_str, "and lifted theorem", rtrm_str, "do not match."]
in
raise LIFT_MATCH (space_implode " " msg)
end
*}
ML {*
(* applies f to the subterm of an abstractions, otherwise to the given term *)
(* 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
fun qnt_typ ty = domain_type (domain_type ty)
*}
(*
Regularizing an rterm 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
- 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
appropriate 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 {*
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: Why is there a case for equality? Is it correct? *)
| (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 (ty, 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
else trm_lift_error lthy rtrm qtrm
| (Bound i, Bound i') =>
if i = i'
then rtrm
else trm_lift_error lthy rtrm qtrm
| (Const (s, ty), Const (s', ty')) =>
if s = s' andalso ty = ty'
then rtrm
else rtrm (* FIXME: check correspondence according to definitions *)
| _ => trm_lift_error lthy rtrm qtrm
*}
ML {*
fun mk_REGULARIZE_goal lthy rtrm qtrm =
Logic.mk_implies (rtrm, REGULARIZE_trm lthy rtrm qtrm |> Syntax.check_term lthy)
*}
(*
To prove that the old theorem implies the new one, we first
atomize it and then try:
- 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 my_equiv_res_forall2:
fixes P::"'a \<Rightarrow> bool"
fixes Q::"'b \<Rightarrow> bool"
assumes a: "(All Q) \<longrightarrow> (All P)"
shows "(All Q) \<longrightarrow> Ball (Respects E) P"
using a by auto
lemma my_equiv_res_exists:
fixes P::"'a \<Rightarrow> bool"
fixes Q::"'b \<Rightarrow> bool"
assumes a: "EQUIV E"
and b: "(Ex Q) \<longrightarrow> (Ex P)"
shows "(Ex Q) \<longrightarrow> Bex (Respects E) P"
apply(subst equiv_res_exists)
apply(rule a)
apply(rule b)
done
ML {*
fun REGULARIZE_tac lthy rel_refl rel_eqv =
(REPEAT1 o FIRST1)
[rtac rel_refl,
atac,
rtac @{thm universal_twice},
rtac @{thm impI} THEN' atac,
rtac @{thm implication_twice},
rtac @{thm my_equiv_res_forall2},
rtac (rel_eqv RS @{thm my_equiv_res_exists}),
(* For a = b \<longrightarrow> a \<approx> b *)
rtac @{thm impI} THEN' (asm_full_simp_tac HOL_ss) THEN' rtac rel_refl,
rtac @{thm RIGHT_RES_FORALL_REGULAR}]
*}
(* same version including debugging information *)
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))]
*}
ML {*
fun REGULARIZE_tac' lthy rel_refl rel_eqv =
(REPEAT1 o FIRST1)
[(K (print_tac "start")) THEN' (K no_tac),
DT lthy "1" (rtac rel_refl),
DT lthy "2" atac,
DT lthy "3" (rtac @{thm universal_twice}),
DT lthy "4" (rtac @{thm impI} THEN' atac),
DT lthy "5" (rtac @{thm implication_twice}),
DT lthy "6" (rtac @{thm my_equiv_res_forall2}),
DT lthy "7" (rtac (rel_eqv RS @{thm my_equiv_res_exists})),
(* For a = b \<longrightarrow> a \<approx> b *)
DT lthy "8" (rtac @{thm impI} THEN' (asm_full_simp_tac HOL_ss) THEN' rtac rel_refl),
DT lthy "9" (rtac @{thm RIGHT_RES_FORALL_REGULAR})]
*}
ML {*
fun REGULARIZE_prove rtrm qtrm rel_eqv rel_refl lthy =
let
val goal = mk_REGULARIZE_goal lthy rtrm qtrm
val cthm = Goal.prove lthy [] [] goal
(fn {context, ...} => REGULARIZE_tac' context rel_refl rel_eqv)
in
cthm
end
*}
end