# File ‹Tools/reification.ML›

```(*  Title:      HOL/Tools/reification.ML
Author:     Amine Chaieb, TU Muenchen

A trial for automatical reification.
*)

signature REIFICATION =
sig
val conv: Proof.context -> thm list -> conv
val tac: Proof.context -> thm list -> term option -> int -> tactic
val lift_conv: Proof.context -> conv -> term option -> int -> tactic
val dereify: Proof.context -> thm list -> conv
end;

structure Reification : REIFICATION =
struct

fun dest_listT (Type (\<^type_name>‹list›, [T])) = T;

val FWD = curry (op OF);

fun rewrite_with ctxt eqs = Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps eqs);

val pure_subst = @{lemma "x == y ==> PROP P y ==> PROP P x" by simp}

fun lift_conv ctxt conv some_t = Subgoal.FOCUS (fn {context = ctxt', concl, ...} =>
let
val ct =
(case some_t of
NONE => Thm.dest_arg concl
| SOME t => Thm.cterm_of ctxt' t)
val thm = conv ct;
in
if Thm.is_reflexive thm then no_tac
else ALLGOALS (resolve_tac ctxt [pure_subst OF [thm]])
end) ctxt;

(* Make a congruence rule out of a defining equation for the interpretation

th is one defining equation of f,
i.e. th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)"
Cp is a constructor pattern and P is a pattern

The result is:
[|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn)
+ the a list of names of the A1 .. An, Those are fresh in the ctxt *)

fun mk_congeq ctxt fs th =
let
val Const (fN, _) = th |> Thm.prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
|> fst |> strip_comb |> fst;
val ((_, [th']), ctxt') = Variable.import true [th] ctxt;
val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'));
fun add_fterms (t as t1 \$ t2) =
if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs
then insert (op aconv) t
| add_fterms (t as Abs _) =
if exists_Const (fn (c, _) => c = fN) t
then K [t]
else K []
val fterms = add_fterms rhs [];
val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt';
val tys = map fastype_of fterms;
val vs = map Free (xs ~~ tys);
val env = fterms ~~ vs; (*FIXME*)
fun replace_fterms (t as t1 \$ t2) =
(case AList.lookup (op aconv) env t of
SOME v => v
| NONE => replace_fterms t1 \$ replace_fterms t2)
| replace_fterms t =
(case AList.lookup (op aconv) env t of
SOME v => v
| NONE => t);
fun mk_def (Abs (x, xT, t), v) =
HOLogic.mk_Trueprop (HOLogic.all_const xT \$ Abs (x, xT, HOLogic.mk_eq (v \$ Bound 0, t)))
| mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t));
fun tryext x =
(x RS @{lemma "(∀x. f x = g x) ⟹ f = g" by blast} handle THM _ => x);
val cong =
(Goal.prove ctxt'' [] (map mk_def env)
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
(fn {context, prems, ...} =>
Local_Defs.unfold0_tac context (map tryext prems) THEN resolve_tac ctxt'' [th'] 1)) RS sym;
val (cong' :: vars') =
Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o Thm.cterm_of ctxt'') vs);
val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars';

in (vs', cong') end;

(* congs is a list of pairs (P,th) where th is a theorem for
[| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)

fun rearrange congs =
let
fun P (_, th) =
let val \<^term>‹Trueprop› \$ (Const (\<^const_name>‹HOL.eq›, _) \$ l \$ _) = Thm.concl_of th
in can dest_Var l end;
val (yes, no) = List.partition P congs;
in no @ yes end;

fun dereify ctxt eqs =
rewrite_with ctxt (eqs @ @{thms nth_Cons_0 nth_Cons_Suc});

fun index_of t bds =
let
val tt = HOLogic.listT (fastype_of t);
in
(case AList.lookup Type.could_unify bds tt of
| SOME (tbs, tats) =>
let
val i = find_index (fn t' => t' = t) tats;
val j = find_index (fn t' => t' = t) tbs;
in
if j = ~1 then
if i = ~1
then (length tbs + length tats, AList.update Type.could_unify (tt, (tbs, tats @ [t])) bds)
else (i, bds)
else (j, bds)
end)
end;

(* Generic decomp for reification : matches the actual term with the
rhs of one cong rule. The result of the matching guides the
proof synthesis: The matches of the introduced Variables A1 .. An are
processed recursively
The rest is instantiated in the cong rule,i.e. no reification is needed *)

(* da is the decomposition for atoms, ie. it returns ([],g) where g
returns the right instance f (AtC n) = t , where AtC is the Atoms
constructor and n is the number of the atom corresponding to t *)
fun decomp_reify da cgns (ct, ctxt) bds =
let
val thy = Proof_Context.theory_of ctxt;
fun tryabsdecomp (ct, ctxt) bds =
(case Thm.term_of ct of
Abs (_, xT, _) =>
let
val ((cx, cta), ctxt') = Variable.dest_abs_cterm ct ctxt;
val x = Thm.term_of cx;
val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT) of
| SOME (bsT, atsT) => AList.update Type.could_unify (HOLogic.listT xT,
(x :: bsT, atsT)) bds);
in (([(cta, ctxt')],
fn ([th], bds) =>
(hd (Variable.export ctxt' ctxt [(Thm.forall_intr cx th) COMP allI]),
let
val (bsT, asT) = the (AList.lookup Type.could_unify bds (HOLogic.listT xT));
in
AList.update Type.could_unify (HOLogic.listT xT, (tl bsT, asT)) bds
end)),
bds)
end
| _ => da (ct, ctxt) bds)
in
(case cgns of
[] => tryabsdecomp (ct, ctxt) bds
| ((vns, cong) :: congs) =>
(let
val (tyenv, tmenv) =
Pattern.match thy
((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (Thm.concl_of cong), Thm.term_of ct)
(Vartab.empty, Vartab.empty);
val (fnvs, invs) = List.partition (fn ((vn, _),_) => member (op =) vns vn) (Vartab.dest tmenv);
val (fts, its) =
(map (snd o snd) fnvs,
map (fn ((vn, vi), (tT, t)) => (((vn, vi), tT), Thm.cterm_of ctxt t)) invs);
val ctyenv =
map (fn ((vn, vi), (s, ty)) => (((vn, vi), s), Thm.ctyp_of ctxt ty))
(Vartab.dest tyenv);
in
((map (Thm.cterm_of ctxt) fts ~~ replicate (length fts) ctxt,
apfst (FWD (Drule.instantiate_normalize (TVars.make ctyenv, Vars.make its) cong))), bds)
end handle Pattern.MATCH => decomp_reify da congs (ct, ctxt) bds))
end;

fun get_nths (t as (Const (\<^const_name>‹List.nth›, _) \$ vs \$ n)) =
AList.update (op aconv) (t, (vs, n))
| get_nths (t1 \$ t2) = get_nths t1 #> get_nths t2
| get_nths (Abs (_, _, t')) = get_nths t'
| get_nths _ = I;

fun tryeqs [] (ct, ctxt) bds = error "Cannot find the atoms equation"
| tryeqs (eq :: eqs) (ct, ctxt) bds = ((
let
val rhs = eq |> Thm.prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd;
val nths = get_nths rhs [];
val (vss, _) = fold_rev (fn (_, (vs, n)) => fn (vss, ns) =>
(insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([], []);
val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt;
val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt';
val thy = Proof_Context.theory_of ctxt'';
val vsns_map = vss ~~ vsns;
val xns_map = fst (split_list nths) ~~ xns;
val subst = map (fn (nt, xn) => (nt, Var ((xn, 0), fastype_of nt))) xns_map;
val rhs_P = subst_free subst rhs;
val (tyenv, tmenv) = Pattern.match thy (rhs_P, Thm.term_of ct) (Vartab.empty, Vartab.empty);
val sbst = Envir.subst_term (tyenv, tmenv);
val sbsT = Envir.subst_type tyenv;
val subst_ty =
map (fn (n, (s, t)) => ((n, s), Thm.ctyp_of ctxt'' t)) (Vartab.dest tyenv)
val tml = Vartab.dest tmenv;
val (subst_ns, bds) = fold_map
(fn (Const _ \$ _ \$ n, Var (xn0, _)) => fn bds =>
let
val name = snd (the (AList.lookup (op =) tml xn0));
val (idx, bds) = index_of name bds;
in (apply2 (Thm.cterm_of ctxt'') (n, idx |> HOLogic.mk_nat), bds) end) subst bds;
val subst_vs =
let
fun h (Const _ \$ (vs as Var (_, lT)) \$ _, Var (_, T)) =
let
val cns = sbst (Const (\<^const_name>‹List.Cons›, T --> lT --> lT));
val lT' = sbsT lT;
val (bsT, _) = the (AList.lookup Type.could_unify bds lT);
val vsn = the (AList.lookup (op =) vsns_map vs);
val vs' = fold_rev (fn x => fn xs => cns \$ x \$xs) bsT (Free (vsn, lT'));
in apply2 (Thm.cterm_of ctxt'') (vs, vs') end;
in map h subst end;
val cts =
map (fn ((vn, vi), (tT, t)) => apply2 (Thm.cterm_of ctxt'') (Var ((vn, vi), tT), t))
(fold (AList.delete (fn (((a : string), _), (b, _)) => a = b))
(map (fn n => (n, 0)) xns) tml);
val substt =
let
val ih = Drule.cterm_rule (Thm.instantiate (TVars.make subst_ty, Vars.empty));
in map (apply2 ih) (subst_ns @ subst_vs @ cts) end;
val th =
(Drule.instantiate_normalize
(TVars.make subst_ty, Vars.make (map (apfst (dest_Var o Thm.term_of)) substt)) eq)
RS sym;
in (hd (Variable.export ctxt'' ctxt [th]), bds) end)
handle Pattern.MATCH => tryeqs eqs (ct, ctxt) bds);

(* looks for the atoms equation and instantiates it with the right number *)

fun mk_decompatom eqs (ct, ctxt) bds = (([], fn (_, bds) =>
let
val tT = fastype_of (Thm.term_of ct);
fun isat eq =
let
val rhs = eq |> Thm.prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd;
in exists_Const
(fn (n, ty) => n = \<^const_name>‹List.nth›
andalso AList.defined Type.could_unify bds (domain_type ty)) rhs
andalso Type.could_unify (fastype_of rhs, tT)
end;
in tryeqs (filter isat eqs) (ct, ctxt) bds end), bds);

(* Generic reification procedure: *)
(* creates all needed cong rules and then just uses the theorem synthesis *)

fun mk_congs ctxt eqs =
let
val fs = fold_rev (fn eq => insert (op =) (eq |> Thm.prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> strip_comb
|> fst)) eqs [];
val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)) fs [];
val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt;
val subst =
the o AList.lookup (op =)
(map2 (fn T => fn v => (T, Thm.cterm_of ctxt' (Free (v, T)))) tys vs);
fun prep_eq eq =
let
val (_, _ :: vs) = eq |> Thm.prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> strip_comb;
val subst = map_filter (fn Var v => SOME (v, subst (#2 v)) | _ => NONE) vs;
in Thm.instantiate (TVars.empty, Vars.make subst) eq end;
val (ps, congs) = map_split (mk_congeq ctxt' fs o prep_eq) eqs;
val bds = AList.make (K ([], [])) tys;
in (ps ~~ Variable.export ctxt' ctxt congs, bds) end

fun conv ctxt eqs ct =
let
val (congs, bds) = mk_congs ctxt eqs;
val congs = rearrange congs;
val (th, bds') =
apfst mk_eq (divide_and_conquer' (decomp_reify (mk_decompatom eqs) congs) (ct, ctxt) bds);
fun is_list_var (Var (_, t)) = can dest_listT t
| is_list_var _ = false;
val vars = th |> Thm.prop_of |> Logic.dest_equals |> snd
|> strip_comb |> snd |> filter is_list_var;
val vs = map (fn Var (v as (_, T)) =>
(v, the (AList.lookup Type.could_unify bds' T) |> snd |> HOLogic.mk_list (dest_listT T))) vars;
val th' =
Drule.instantiate_normalize (TVars.empty, Vars.make (map (apsnd (Thm.cterm_of ctxt)) vs)) th;
val th'' = Thm.symmetric (dereify ctxt [] (Thm.lhs_of th'));
in Thm.transitive th'' th' end;

fun tac ctxt eqs =
lift_conv ctxt (conv ctxt eqs);

end;
```