(*  Title:      Pure/Proof/reconstruct.ML

    Author:     Stefan Berghofer, TU Muenchen

Reconstruction of partial proof terms.

*)

signature RECONSTRUCT =

sig

  val quiet_mode : bool Unsynchronized.ref

  val reconstruct_proof : theory -> term -> Proofterm.proof -> Proofterm.proof

  val prop_of' : term list -> Proofterm.proof -> term

  val prop_of : Proofterm.proof -> term

  val expand_proof : theory -> (string * term option) list ->

    Proofterm.proof -> Proofterm.proof

end;

structure Reconstruct : RECONSTRUCT =

struct

open Proofterm;

val quiet_mode = Unsynchronized.ref true;

fun message s = if !quiet_mode then () else writeln s;

fun vars_of t = map Var (rev (Term.add_vars t []));

fun frees_of t = map Free (rev (Term.add_frees t []));

fun forall_intr_vfs prop = fold_rev Logic.all

  (vars_of prop @ frees_of prop) prop;

fun forall_intr_vfs_prf prop prf = fold_rev forall_intr_proof'

  (vars_of prop @ frees_of prop) prf;

(**** generate constraints for proof term ****)

fun mk_var env Ts T =

  let val (env', v) = Envir.genvar "a" (env, rev Ts ---> T)

  in (list_comb (v, map Bound (length Ts - 1 downto 0)), env') end;

fun mk_tvar S (Envir.Envir {maxidx, tenv, tyenv}) =

  (TVar (("'t", maxidx + 1), S),

    Envir.Envir {maxidx = maxidx + 1, tenv = tenv, tyenv = tyenv});

val mk_abs = fold (fn T => fn u => Abs ("", T, u));

fun unifyT thy env T U =

  let

    val Envir.Envir {maxidx, tenv, tyenv} = env;

    val (tyenv', maxidx') = Sign.typ_unify thy (T, U) (tyenv, maxidx);

  in Envir.Envir {maxidx = maxidx', tenv = tenv, tyenv = tyenv'} end;

fun chaseT env (T as TVar v) =

      (case Type.lookup (Envir.type_env env) v of

        NONE => T

      | SOME T' => chaseT env T')

  | chaseT _ T = T;

fun infer_type thy (env as Envir.Envir {maxidx, tenv, tyenv}) Ts vTs

      (t as Const (s, T)) = if T = dummyT then

        (case Sign.const_type thy s of

          NONE => error ("reconstruct_proof: No such constant: " ^ quote s)

        | SOME T =>

            let val T' = Type.strip_sorts (Logic.incr_tvar (maxidx + 1) T)

            in (Const (s, T'), T', vTs,

              Envir.Envir {maxidx = maxidx + 1, tenv = tenv, tyenv = tyenv})

            end)

      else (t, T, vTs, env)

  | infer_type thy env Ts vTs (t as Free (s, T)) =

      if T = dummyT then (case Symtab.lookup vTs s of

          NONE =>

            let val (T, env') = mk_tvar [] env

            in (Free (s, T), T, Symtab.update_new (s, T) vTs, env') end

        | SOME T => (Free (s, T), T, vTs, env))

      else (t, T, vTs, env)

  | infer_type thy env Ts vTs (Var _) = error "reconstruct_proof: internal error"

  | infer_type thy env Ts vTs (Abs (s, T, t)) =

      let

        val (T', env') = if T = dummyT then mk_tvar [] env else (T, env);

        val (t', U, vTs', env'') = infer_type thy env' (T' :: Ts) vTs t

      in (Abs (s, T', t'), T' --> U, vTs', env'') end

  | infer_type thy env Ts vTs (t $ u) =

      let

        val (t', T, vTs1, env1) = infer_type thy env Ts vTs t;

        val (u', U, vTs2, env2) = infer_type thy env1 Ts vTs1 u;

      in (case chaseT env2 T of

          Type ("fun", [U', V]) => (t' $ u', V, vTs2, unifyT thy env2 U U')

        | _ =>

          let val (V, env3) = mk_tvar [] env2

          in (t' $ u', V, vTs2, unifyT thy env3 T (U --> V)) end)

      end

  | infer_type thy env Ts vTs (t as Bound i) = ((t, nth Ts i, vTs, env)

      handle Subscript => error ("infer_type: bad variable index " ^ string_of_int i));

fun cantunify thy (t, u) = error ("Non-unifiable terms:\n" ^

  Syntax.string_of_term_global thy t ^ "\n\n" ^ Syntax.string_of_term_global thy u);

fun decompose thy Ts (p as (t, u)) env =

  let

    fun rigrig (a, T) (b, U) uT ts us =

      if a <> b then cantunify thy p

      else apfst flat (fold_map (decompose thy Ts) (ts ~~ us) (uT env T U))

  in

    case pairself (strip_comb o Envir.head_norm env) p of

      ((Const c, ts), (Const d, us)) => rigrig c d (unifyT thy) ts us

    | ((Free c, ts), (Free d, us)) => rigrig c d (unifyT thy) ts us

    | ((Bound i, ts), (Bound j, us)) =>

        rigrig (i, dummyT) (j, dummyT) (K o K) ts us

    | ((Abs (_, T, t), []), (Abs (_, U, u), [])) =>

        decompose thy (T::Ts) (t, u) (unifyT thy env T U)

    | ((Abs (_, T, t), []), _) =>

        decompose thy (T::Ts) (t, incr_boundvars 1 u $ Bound 0) env

    | (_, (Abs (_, T, u), [])) =>

        decompose thy (T::Ts) (incr_boundvars 1 t $ Bound 0, u) env

    | _ => ([(mk_abs Ts t, mk_abs Ts u)], env)

  end;

fun make_constraints_cprf thy env cprf =

  let

    fun add_cnstrt Ts prop prf cs env vTs (t, u) =

      let

        val t' = mk_abs Ts t;

        val u' = mk_abs Ts u

      in

        (prop, prf, cs, Pattern.unify thy (t', u') env, vTs)

        handle Pattern.Pattern =>

            let val (cs', env') = decompose thy [] (t', u') env

            in (prop, prf, cs @ cs', env', vTs) end

        | Pattern.Unif =>

            cantunify thy (Envir.norm_term env t', Envir.norm_term env u')

      end;

    fun mk_cnstrts_atom env vTs prop opTs prf =

          let

            val tvars = Term.add_tvars prop [] |> rev;

            val tfrees = Term.add_tfrees prop [] |> rev;

            val (Ts, env') =

              (case opTs of

                NONE => fold_map mk_tvar (map snd tvars @ map snd tfrees) env

              | SOME Ts => (Ts, env));

            val prop' = subst_atomic_types (map TVar tvars @ map TFree tfrees ~~ Ts)

              (forall_intr_vfs prop) handle Library.UnequalLengths =>

                error ("Wrong number of type arguments for " ^

                  quote (get_name [] prop prf))

          in (prop', change_type (SOME Ts) prf, [], env', vTs) end;

    fun head_norm (prop, prf, cnstrts, env, vTs) =

      (Envir.head_norm env prop, prf, cnstrts, env, vTs);

    fun mk_cnstrts env _ Hs vTs (PBound i) = ((nth Hs i, PBound i, [], env, vTs)

          handle Subscript => error ("mk_cnstrts: bad variable index " ^ string_of_int i))

      | mk_cnstrts env Ts Hs vTs (Abst (s, opT, cprf)) =

          let

            val (T, env') =

              (case opT of

                NONE => mk_tvar [] env

              | SOME T => (T, env));

            val (t, prf, cnstrts, env'', vTs') =

              mk_cnstrts env' (T::Ts) (map (incr_boundvars 1) Hs) vTs cprf;

          in (Const ("all", (T --> propT) --> propT) $ Abs (s, T, t), Abst (s, SOME T, prf),

            cnstrts, env'', vTs')

          end

      | mk_cnstrts env Ts Hs vTs (AbsP (s, SOME t, cprf)) =

          let

            val (t', _, vTs', env') = infer_type thy env Ts vTs t;

            val (u, prf, cnstrts, env'', vTs'') = mk_cnstrts env' Ts (t'::Hs) vTs' cprf;

          in (Logic.mk_implies (t', u), AbsP (s, SOME t', prf), cnstrts, env'', vTs'')

          end

      | mk_cnstrts env Ts Hs vTs (AbsP (s, NONE, cprf)) =

          let

            val (t, env') = mk_var env Ts propT;

            val (u, prf, cnstrts, env'', vTs') = mk_cnstrts env' Ts (t::Hs) vTs cprf;

          in (Logic.mk_implies (t, u), AbsP (s, SOME t, prf), cnstrts, env'', vTs')

          end

      | mk_cnstrts env Ts Hs vTs (cprf1 %% cprf2) =

          let val (u, prf2, cnstrts, env', vTs') = mk_cnstrts env Ts Hs vTs cprf2

          in (case head_norm (mk_cnstrts env' Ts Hs vTs' cprf1) of

              (Const ("==>", _) $ u' $ t', prf1, cnstrts', env'', vTs'') =>

                add_cnstrt Ts t' (prf1 %% prf2) (cnstrts' @ cnstrts)

                  env'' vTs'' (u, u')

            | (t, prf1, cnstrts', env'', vTs'') =>

                let val (v, env''') = mk_var env'' Ts propT

                in add_cnstrt Ts v (prf1 %% prf2) (cnstrts' @ cnstrts)

                  env''' vTs'' (t, Logic.mk_implies (u, v))

                end)

          end

      | mk_cnstrts env Ts Hs vTs (cprf % SOME t) =

          let val (t', U, vTs1, env1) = infer_type thy env Ts vTs t

          in (case head_norm (mk_cnstrts env1 Ts Hs vTs1 cprf) of

             (Const ("all", Type ("fun", [Type ("fun", [T, _]), _])) $ f,

                 prf, cnstrts, env2, vTs2) =>

               let val env3 = unifyT thy env2 T U

               in (betapply (f, t'), prf % SOME t', cnstrts, env3, vTs2)

               end

           | (u, prf, cnstrts, env2, vTs2) =>

               let val (v, env3) = mk_var env2 Ts (U --> propT);

               in

                 add_cnstrt Ts (v $ t') (prf % SOME t') cnstrts env3 vTs2

                   (u, Const ("all", (U --> propT) --> propT) $ v)

               end)

          end

      | mk_cnstrts env Ts Hs vTs (cprf % NONE) =

          (case head_norm (mk_cnstrts env Ts Hs vTs cprf) of

             (Const ("all", Type ("fun", [Type ("fun", [T, _]), _])) $ f,

                 prf, cnstrts, env', vTs') =>

               let val (t, env'') = mk_var env' Ts T

               in (betapply (f, t), prf % SOME t, cnstrts, env'', vTs')

               end

           | (u, prf, cnstrts, env', vTs') =>

               let

                 val (T, env1) = mk_tvar [] env';

                 val (v, env2) = mk_var env1 Ts (T --> propT);

                 val (t, env3) = mk_var env2 Ts T

               in

                 add_cnstrt Ts (v $ t) (prf % SOME t) cnstrts env3 vTs'

                   (u, Const ("all", (T --> propT) --> propT) $ v)

               end)

      | mk_cnstrts env _ _ vTs (prf as PThm (_, ((_, prop, opTs), _))) =

          mk_cnstrts_atom env vTs prop opTs prf

      | mk_cnstrts env _ _ vTs (prf as PAxm (_, prop, opTs)) =

          mk_cnstrts_atom env vTs prop opTs prf

      | mk_cnstrts env _ _ vTs (prf as OfClass (T, c)) =

          mk_cnstrts_atom env vTs (Logic.mk_of_class (T, c)) NONE prf

      | mk_cnstrts env _ _ vTs (prf as Oracle (_, prop, opTs)) =

          mk_cnstrts_atom env vTs prop opTs prf

      | mk_cnstrts env _ _ vTs (Hyp t) = (t, Hyp t, [], env, vTs)

      | mk_cnstrts _ _ _ _ _ = error "reconstruct_proof: minimal proof object"

  in mk_cnstrts env [] [] Symtab.empty cprf end;

(**** update list of free variables of constraints ****)

fun upd_constrs env cs =

  let

    val tenv = Envir.term_env env;

    val tyenv = Envir.type_env env;

    val dom = []

      |> Vartab.fold (cons o #1) tenv

      |> Vartab.fold (cons o #1) tyenv;

    val vran = []

      |> Vartab.fold (Term.add_var_names o #2 o #2) tenv

      |> Vartab.fold (Term.add_tvar_namesT o #2 o #2) tyenv;

    fun check_cs [] = []

      | check_cs ((u, p, vs) :: ps) =

          let val vs' = subtract (op =) dom vs in

            if vs = vs' then (u, p, vs) :: check_cs ps

            else (true, p, fold (insert op =) vs' vran) :: check_cs ps

          end;

  in check_cs cs end;

(**** solution of constraints ****)

fun solve _ [] bigenv = bigenv

  | solve thy cs bigenv =

      let

        fun search env [] = error ("Unsolvable constraints:\n" ^

              Pretty.string_of (Pretty.chunks (map (fn (_, p, _) =>

                Goal_Display.pretty_flexpair (Syntax.init_pretty_global thy) (pairself

                  (Envir.norm_term bigenv) p)) cs)))

          | search env ((u, p as (t1, t2), vs)::ps) =

              if u then

                let

                  val tn1 = Envir.norm_term bigenv t1;

                  val tn2 = Envir.norm_term bigenv t2

                in

                  if Pattern.pattern tn1 andalso Pattern.pattern tn2 then

                    (Pattern.unify thy (tn1, tn2) env, ps) handle Pattern.Unif =>

                       cantunify thy (tn1, tn2)

                  else

                    let val (cs', env') = decompose thy [] (tn1, tn2) env

                    in if cs' = [(tn1, tn2)] then

                         apsnd (cons (false, (tn1, tn2), vs)) (search env ps)

                       else search env' (map (fn q => (true, q, vs)) cs' @ ps)

                    end

                end

              else apsnd (cons (false, p, vs)) (search env ps);

        val Envir.Envir {maxidx, ...} = bigenv;

        val (env, cs') = search (Envir.empty maxidx) cs;

      in

        solve thy (upd_constrs env cs') (Envir.merge (bigenv, env))

      end;

(**** reconstruction of proofs ****)

fun reconstruct_proof thy prop cprf =

  let

    val (cprf' % SOME prop', thawf) = freeze_thaw_prf (cprf % SOME prop);

    val _ = message "Collecting constraints...";

    val (t, prf, cs, env, _) = make_constraints_cprf thy

      (Envir.empty (maxidx_proof cprf ~1)) cprf';

    val cs' = map (fn p => (true, p, uncurry (union (op =)) 

        (pairself (map (fst o dest_Var) o OldTerm.term_vars) p)))

      (map (pairself (Envir.norm_term env)) ((t, prop')::cs));

    val _ = message ("Solving remaining constraints (" ^ string_of_int (length cs') ^ ") ...");

    val env' = solve thy cs' env

  in

    thawf (norm_proof env' prf)

  end;

fun prop_of_atom prop Ts = subst_atomic_types

  (map TVar (Term.add_tvars prop [] |> rev) @ map TFree (Term.add_tfrees prop [] |> rev) ~~ Ts)

  (forall_intr_vfs prop);

val head_norm = Envir.head_norm (Envir.empty 0);

fun prop_of0 Hs (PBound i) = nth Hs i

  | prop_of0 Hs (Abst (s, SOME T, prf)) =

      Term.all T $ (Abs (s, T, prop_of0 Hs prf))

  | prop_of0 Hs (AbsP (s, SOME t, prf)) =

      Logic.mk_implies (t, prop_of0 (t :: Hs) prf)

  | prop_of0 Hs (prf % SOME t) = (case head_norm (prop_of0 Hs prf) of

      Const ("all", _) $ f => f $ t

    | _ => error "prop_of: all expected")

  | prop_of0 Hs (prf1 %% prf2) = (case head_norm (prop_of0 Hs prf1) of

      Const ("==>", _) $ P $ Q => Q

    | _ => error "prop_of: ==> expected")

  | prop_of0 Hs (Hyp t) = t

  | prop_of0 Hs (PThm (_, ((_, prop, SOME Ts), _))) = prop_of_atom prop Ts

  | prop_of0 Hs (PAxm (_, prop, SOME Ts)) = prop_of_atom prop Ts

  | prop_of0 Hs (OfClass (T, c)) = Logic.mk_of_class (T, c)

  | prop_of0 Hs (Oracle (_, prop, SOME Ts)) = prop_of_atom prop Ts

  | prop_of0 _ _ = error "prop_of: partial proof object";

val prop_of' = Envir.beta_eta_contract oo prop_of0;

val prop_of = prop_of' [];

(**** expand and reconstruct subproofs ****)

fun expand_proof thy thms prf =

  let

    fun expand maxidx prfs (AbsP (s, t, prf)) =

          let val (maxidx', prfs', prf') = expand maxidx prfs prf

          in (maxidx', prfs', AbsP (s, t, prf')) end

      | expand maxidx prfs (Abst (s, T, prf)) =

          let val (maxidx', prfs', prf') = expand maxidx prfs prf

          in (maxidx', prfs', Abst (s, T, prf')) end

      | expand maxidx prfs (prf1 %% prf2) =

          let

            val (maxidx', prfs', prf1') = expand maxidx prfs prf1;

            val (maxidx'', prfs'', prf2') = expand maxidx' prfs' prf2;

          in (maxidx'', prfs'', prf1' %% prf2') end

      | expand maxidx prfs (prf % t) =

          let val (maxidx', prfs', prf') = expand maxidx prfs prf

          in (maxidx', prfs', prf' % t) end

      | expand maxidx prfs (prf as PThm (_, ((a, prop, SOME Ts), body))) =

          if not (exists

            (fn (b, NONE) => a = b

              | (b, SOME prop') => a = b andalso prop = prop') thms)

          then (maxidx, prfs, prf) else

          let

            val (maxidx', prf, prfs') =

              (case AList.lookup (op =) prfs (a, prop) of

                NONE =>

                  let

                    val _ = message ("Reconstructing proof of " ^ a);

                    val _ = message (Syntax.string_of_term_global thy prop);

                    val prf' = forall_intr_vfs_prf prop

                      (reconstruct_proof thy prop (join_proof body));

                    val (maxidx', prfs', prf) = expand

                      (maxidx_proof prf' ~1) prfs prf'

                  in (maxidx' + maxidx + 1, incr_indexes (maxidx + 1) prf,

                    ((a, prop), (maxidx', prf)) :: prfs')

                  end

              | SOME (maxidx', prf) => (maxidx' + maxidx + 1,

                  incr_indexes (maxidx + 1) prf, prfs));

            val tfrees = Term.add_tfrees prop [] |> rev;

            val tye = map (fn ((s, j), _) => (s, maxidx + 1 + j))

              (Term.add_tvars prop [] |> rev) @ map (rpair ~1 o fst) tfrees ~~ Ts;

            val varify = map_type_tfree (fn p as (a, S) =>

              if member (op =) tfrees p then TVar ((a, ~1), S) else TFree p)

          in

            (maxidx', prfs', map_proof_types (typ_subst_TVars tye o varify) prf)

          end

      | expand maxidx prfs prf = (maxidx, prfs, prf);

  in #3 (expand (maxidx_proof prf ~1) [] prf) end;

end;