Described automatically created funs.
(* Code for getting the goal *)+ −
apply (tactic {* (ObjectLogic.full_atomize_tac THEN' gen_frees_tac @{context}) 1 *})+ −
ML_prf {* val qtm = #concl (fst (Subgoal.focus @{context} 1 (#goal (Isar.goal ())))) *}+ −
+ −
+ −
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+ −
*}+ −
+ −
+ −
+ −
ML {*+ −
fun repeat_eqsubst_thm ctxt thms thm =+ −
repeat_eqsubst_thm ctxt thms (eqsubst_thm ctxt thms thm)+ −
handle _ => thm+ −
*}+ −
+ −
+ −
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+ −
*}+ −
+ −
+ −
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}) + −
*} + −
*)+ −
+ −
+ −
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)))+ −
*}+ −
+ −
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+ −
*}+ −
+ −
(*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 {*+ −
(* 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 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 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+ −
*}+ −
+ −
+ −
(* 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)+ −
*}+ −
+ −
+ −
(* 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 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+ −
*}+ −
+ −
+ −
ML {*+ −
fun regularize_goal lthy thm rel_eqv rel_refl qtrm =+ −
let+ −
val reg_trm = mk_REGULARIZE_goal lthy (prop_of thm) qtrm;+ −
fun tac lthy = regularize_tac lthy rel_eqv rel_refl;+ −
val cthm = Goal.prove lthy [] [] reg_trm+ −
(fn {context, ...} => tac context 1);+ −
in+ −
cthm OF [thm]+ −
end+ −
*}+ −
+ −
ML {*+ −
fun repabs_goal lthy thm rty quot_thm reflex_thm trans_thm rsp_thms qtrm =+ −
let+ −
val rg = Syntax.check_term lthy (mk_inj_REPABS_goal lthy ((prop_of thm), qtrm));+ −
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+ −
*}+ −
+ −
+ −
ML {*+ −
fun atomize_goal thy gl =+ −
let+ −
val vars = map Free (Term.add_frees gl []);+ −
val all = if fastype_of gl = @{typ bool} then HOLogic.all_const else Term.all;+ −
fun lambda_all (var as Free(_, T)) trm = (all T) $ lambda var trm;+ −
val glv = fold lambda_all vars gl+ −
val gla = (term_of o snd o Thm.dest_equals o cprop_of) (ObjectLogic.atomize (cterm_of thy glv))+ −
val glf = Type.legacy_freeze gla+ −
in+ −
if fastype_of gl = @{typ bool} then @{term Trueprop} $ glf else glf+ −
end+ −
*}+ −
+ −
+ −
ML {* atomize_goal @{theory} @{term "x memb [] = False"} *}+ −
ML {* atomize_goal @{theory} @{term "x = xa ? a # x = a # xa"} *}+ −
+ −
+ −
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 lift_thm_goal lthy qty qty_name rsp_thms defs rthm goal =+ −
let+ −
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 t_a = atomize_thm rthm;+ −
val goal_a = atomize_goal (ProofContext.theory_of lthy) goal;+ −
val t_r = regularize_goal lthy t_a rel_eqv rel_refl goal_a;+ −
val t_t = repabs_goal lthy t_r rty quot rel_refl trans2 rsp_thms goal_a;+ −
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 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 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 t_d0 = MetaSimplifier.rewrite_rule defs_sym_eq t_id;+ −
val t_d = repeat_eqsubst_thm lthy defs_sym t_d0;+ −
val t_r = MetaSimplifier.rewrite_rule [reps_same] t_d;+ −
val t_rv = ObjectLogic.rulify t_r+ −
in+ −
Thm.varifyT t_rv+ −
end+ −
*}+ −
+ −
ML {*+ −
fun lift_thm_goal_note lthy qty qty_name rsp_thms defs thm name goal =+ −
let+ −
val lifted_thm = lift_thm_goal lthy qty qty_name rsp_thms defs thm goal;+ −
val (_, lthy2) = note (name, lifted_thm) lthy;+ −
in+ −
lthy2+ −
end+ −
*}+ −
+ −
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+ −
+ −
*}+ −
+ −
(* Unused part of the locale *)+ −
+ −
lemma R_trans:+ −
assumes ab: "R a b"+ −
and bc: "R b c"+ −
shows "R a c"+ −
proof -+ −
have tr: "transp R" using equivp equivp_reflp_symp_transp[of R] by simp+ −
moreover have ab: "R a b" by fact+ −
moreover have bc: "R b c" by fact+ −
ultimately show "R a c" unfolding transp_def by blast+ −
qed+ −
+ −
lemma R_sym:+ −
assumes ab: "R a b"+ −
shows "R b a"+ −
proof -+ −
have re: "symp R" using equivp equivp_reflp_symp_transp[of R] by simp+ −
then show "R b a" using ab unfolding symp_def by blast+ −
qed+ −
+ −
lemma R_trans2:+ −
assumes ac: "R a c"+ −
and bd: "R b d"+ −
shows "R a b = R c d"+ −
using ac bd+ −
by (blast intro: R_trans R_sym)+ −
+ −
lemma REPS_same:+ −
shows "R (REP a) (REP b) \<equiv> (a = b)"+ −
proof -+ −
have "R (REP a) (REP b) = (a = b)"+ −
proof+ −
assume as: "R (REP a) (REP b)"+ −
from rep_prop+ −
obtain x y+ −
where eqs: "Rep a = R x" "Rep b = R y" by blast+ −
from eqs have "R (Eps (R x)) (Eps (R y))" using as unfolding REP_def by simp+ −
then have "R x (Eps (R y))" using lem9 by simp+ −
then have "R (Eps (R y)) x" using R_sym by blast+ −
then have "R y x" using lem9 by simp+ −
then have "R x y" using R_sym by blast+ −
then have "ABS x = ABS y" using thm11 by simp+ −
then have "Abs (Rep a) = Abs (Rep b)" using eqs unfolding ABS_def by simp+ −
then show "a = b" using rep_inverse by simp+ −
next+ −
assume ab: "a = b"+ −
have "reflp R" using equivp equivp_reflp_symp_transp[of R] by simp+ −
then show "R (REP a) (REP b)" unfolding reflp_def using ab by auto+ −
qed+ −
then show "R (REP a) (REP b) \<equiv> (a = b)" by simp+ −
qed+ −
+ −
+ −
+ −
+ −