(*  Title:      Pure/Proof/reconstruct.ML

    ID:         $Id$

    Author:     Stefan Berghofer, TU Muenchen

Reconstruction of partial proof terms.

*)

signature RECONSTRUCT =

sig

  val quiet_mode : bool ref

  val reconstruct_proof : Sign.sg -> term -> Proofterm.proof -> Proofterm.proof

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

  val prop_of : Proofterm.proof -> term

  val expand_proof : Sign.sg -> (string * term option) list ->

    Proofterm.proof -> Proofterm.proof

end;

structure Reconstruct : RECONSTRUCT =

struct

open Proofterm;

val quiet_mode = ref true;

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

fun vars_of t = rev (foldl_aterms

  (fn (vs, v as Var _) => v ins vs | (vs, _) => vs) ([], t));

fun forall_intr (t, prop) =

  let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)

  in all T $ Abs (a, T, abstract_over (t, prop)) end;

fun forall_intr_vfs prop = foldr forall_intr prop

  (vars_of prop @ sort (make_ord atless) (term_frees prop));

fun forall_intr_prf (t, prf) =

  let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)

  in Abst (a, SOME T, prf_abstract_over t prf) end;

fun forall_intr_vfs_prf prop prf = foldr forall_intr_prf prf

  (vars_of prop @ sort (make_ord atless) (term_frees prop));

fun merge_envs (Envir.Envir {asol=asol1, iTs=iTs1, maxidx=maxidx1})

  (Envir.Envir {asol=asol2, iTs=iTs2, maxidx=maxidx2}) =

    Envir.Envir {asol=Vartab.merge (op =) (asol1, asol2),

                 iTs=Vartab.merge (op =) (iTs1, iTs2),

                 maxidx=Int.max (maxidx1, maxidx2)};

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

fun mk_var env Ts T = 

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

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

fun mk_tvar (Envir.Envir {iTs, asol, maxidx}, s) =

  (Envir.Envir {iTs = iTs, asol = asol, maxidx = maxidx+1},

   TVar (("'t", maxidx+1), s));

fun mk_abs Ts t = Library.foldl (fn (u, T) => Abs ("", T, u)) (t, Ts);

fun unifyT sg env T U =

  let

    val Envir.Envir {asol, iTs, maxidx} = env;

    val (iTs', maxidx') = Type.unify (Sign.tsig_of sg) (iTs, maxidx) (T, U)

  in Envir.Envir {asol=asol, iTs=iTs', maxidx=maxidx'} end

  handle Type.TUNIFY => error ("Non-unifiable types:\n" ^

    Sign.string_of_typ sg T ^ "\n\n" ^ Sign.string_of_typ sg U);

fun chaseT (env as Envir.Envir {iTs, ...}) (T as TVar ixnS) =

      (case Type.lookup (iTs, ixnS) of NONE => T | SOME T' => chaseT env T')

  | chaseT _ T = T;

fun infer_type sg (env as Envir.Envir {maxidx, asol, iTs}) Ts vTs

      (t as Const (s, T)) = if T = dummyT then (case Sign.const_type sg s of

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

        | SOME T => 

            let val T' = incr_tvar (maxidx + 1) T

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

              Envir.Envir {maxidx = maxidx + 1, asol = asol, iTs = iTs})

            end)

      else (t, T, vTs, env)

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

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

          NONE =>

            let val (env', T) = 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 sg env Ts vTs (Var _) = error "reconstruct_proof: internal error"

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

      let

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

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

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

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

      let

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

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

      in (case chaseT env2 T of

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

        | _ =>

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

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

      end

  | infer_type sg env Ts vTs (t as Bound i) = (t, List.nth (Ts, i), vTs, env);

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

  Sign.string_of_term sg t ^ "\n\n" ^ Sign.string_of_term sg u);

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

  let fun rigrig (a, T) (b, U) uT ts us = if a <> b then cantunify sg p

    else apsnd List.concat (foldl_map (decompose sg Ts) (uT env T U, ts ~~ us))

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

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

    | ((Free c, ts), (Free d, us)) => rigrig c d (unifyT sg) 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 sg (T::Ts) (unifyT sg env T U, (t, u))

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

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

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

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

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

  end;

fun make_constraints_cprf sg 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 (sg, env, [(t', u')]), vTs)

        handle Pattern.Pattern =>

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

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

        | Pattern.Unif =>

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

      end;

    fun mk_cnstrts_atom env vTs prop opTs prf =

          let

            val tvars = term_tvars prop;

            val tfrees = term_tfrees prop;

            val (prop', fmap) = Type.varify (prop, []);

            val (env', Ts) = (case opTs of

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

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

            val prop'' = subst_TVars (map fst tvars @ map snd fmap ~~ Ts)

              (forall_intr_vfs prop') handle UnequalLengths =>

                error ("Wrong number of type arguments for " ^

                  quote (fst (get_name_tags [] 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) = (List.nth (Hs, i), PBound i, [], env, vTs)

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

          let

            val (env', T) = (case opT of

              NONE => mk_tvar (env, []) | SOME T => (env, T));

            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 sg 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 (env', t) = 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 (env''', v) = 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 sg 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 sg env2 T U

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

               end

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

               let val (env3, v) = 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 (env'', t) = mk_var env' Ts T

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

               end

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

               let

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

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

                 val (env3, t) = 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 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;

fun add_term_ixns (is, Var (i, T)) = add_typ_ixns (i ins is, T)

  | add_term_ixns (is, Free (_, T)) = add_typ_ixns (is, T)

  | add_term_ixns (is, Const (_, T)) = add_typ_ixns (is, T)

  | add_term_ixns (is, t1 $ t2) = add_term_ixns (add_term_ixns (is, t1), t2)

  | add_term_ixns (is, Abs (_, T, t)) = add_term_ixns (add_typ_ixns (is, T), t)

  | add_term_ixns (is, _) = is;

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

fun upd_constrs env cs =

  let

    val Envir.Envir {asol, iTs, ...} = env;

    val dom = Vartab.foldl (uncurry (cons o fst) o Library.swap)

      (Vartab.foldl (uncurry (cons o fst) o Library.swap) ([], asol), iTs); 

    val vran = Vartab.foldl (add_typ_ixns o apsnd (snd o snd))

      (Vartab.foldl (add_term_ixns o apsnd (snd o snd)) ([], asol), iTs);

    fun check_cs [] = []

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

          let val vs' = vs \\ dom;

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

             else (true, p, vs' union vran)::check_cs ps

          end

  in check_cs cs end;

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

fun solve _ [] bigenv = bigenv

  | solve sg cs bigenv =

      let

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

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

                Display.pretty_flexpair (Sign.pp sg) (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 (sg, env, [(tn1, tn2)]), ps) handle Pattern.Unif =>

                       cantunify sg (tn1, tn2))

                  else

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

                    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 sg (upd_constrs env cs') (merge_envs bigenv env)

      end;

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

fun reconstruct_proof sg 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 sg

      (Envir.empty (maxidx_of_proof cprf)) cprf';

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

      (pairself (map (fst o dest_Var) o 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 sg cs' env

  in

    thawf (norm_proof env' prf)

  end;

fun prop_of_atom prop Ts =

  let val (prop', fmap) = Type.varify (prop, []);

  in subst_TVars (map fst (term_tvars prop) @ map snd fmap ~~ Ts)

    (forall_intr_vfs prop')

  end;

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

fun prop_of0 Hs (PBound i) = List.nth (Hs, i)

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

      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 (Oracle (_, prop, SOME Ts)) = prop_of_atom prop Ts

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

val prop_of' = Pattern.eta_contract oo (Envir.beta_norm oo prop_of0);

val prop_of = prop_of' [];

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

fun expand_proof sg 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, _), cprf, prop, SOME Ts)) =

          if not (exists

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

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

          then (maxidx, prfs, prf) else

          let

            fun inc i =

              map_proof_terms (Logic.incr_indexes ([], i)) (incr_tvar i);

            val (maxidx', prf, prfs') = (case assoc (prfs, (a, prop)) of

                NONE =>

                  let

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

                    val _ = message (Sign.string_of_term sg prop);

                    val prf' = forall_intr_vfs_prf prop

                      (reconstruct_proof sg prop cprf);

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

                      (maxidx_of_proof prf') prfs prf'

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

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

                  end

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

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

            val tfrees = term_tfrees prop;

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

              (term_tvars prop) @ map (rpair ~1 o fst) tfrees ~~ Ts;

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

              if p mem tfrees then TVar ((a, ~1), S) else TFree p)

          in

            (maxidx', prfs', map_proof_terms (subst_TVars tye o

               map_term_types varify) (typ_subst_TVars tye o varify) prf)

          end

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

  in #3 (expand (maxidx_of_proof prf) [] prf) end;

end;