(* Title: Pure/Proof/reconstruct.ML
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
License: GPL (GNU GENERAL PUBLIC LICENSE)
Reconstruction of partial proof terms.
*)
signature RECONSTRUCT =
sig
val quiet_mode : bool ref
val reconstruct_proof : Sign.sg -> term -> Proofterm.proof -> Proofterm.proof
val expand_proof : Sign.sg -> string 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
(vars_of prop @ sort (make_ord atless) (term_frees prop), 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
(vars_of prop @ sort (make_ord atless) (term_frees prop), prf);
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 aconv) (asol1, asol2),
iTs=Vartab.merge (op =) (iTs1, iTs2),
maxidx=Int.max (maxidx1, maxidx2)};
fun strip_abs (_::Ts) (Abs (_, _, t)) = strip_abs Ts t
| strip_abs _ t = t;
(********************************************************************************
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 = foldl (fn (u, T) => Abs ("", T, u)) (t, Ts);
fun make_Tconstraints_cprf maxidx cprf =
let
fun mk_Tcnstrts maxidx Ts (Abst (s, Some T, cprf)) =
let val (cs, cprf', maxidx') = mk_Tcnstrts maxidx (T::Ts) cprf;
in (cs, Abst (s, Some T, cprf'), maxidx') end
| mk_Tcnstrts maxidx Ts (Abst (s, None, cprf)) =
let
val T' = TVar (("'t", maxidx+1), ["logic"]);
val (cs, cprf', maxidx') = mk_Tcnstrts (maxidx+1) (T'::Ts) cprf;
in (cs, Abst (s, Some T', cprf'), maxidx') end
| mk_Tcnstrts maxidx Ts (AbsP (s, Some t, cprf)) =
let val (cs, cprf', maxidx') = mk_Tcnstrts maxidx Ts cprf;
in ((mk_abs Ts t, rev Ts ---> propT)::cs, AbsP (s, Some t, cprf'), maxidx') end
| mk_Tcnstrts maxidx Ts (AbsP (s, None, cprf)) =
let val (cs, cprf', maxidx') = mk_Tcnstrts maxidx Ts cprf;
in (cs, AbsP (s, None, cprf'), maxidx') end
| mk_Tcnstrts maxidx Ts (cprf1 %% cprf2) =
let
val (cs, cprf1', maxidx') = mk_Tcnstrts maxidx Ts cprf1;
val (cs', cprf2', maxidx'') = mk_Tcnstrts maxidx' Ts cprf2;
in (cs' @ cs, cprf1' %% cprf2', maxidx'') end
| mk_Tcnstrts maxidx Ts (cprf % Some t) =
let val (cs, cprf', maxidx') = mk_Tcnstrts maxidx Ts cprf;
in ((mk_abs Ts t, rev Ts ---> TypeInfer.logicT)::cs,
cprf' % Some t, maxidx')
end
| mk_Tcnstrts maxidx Ts (cprf % None) =
let val (cs, cprf', maxidx') = mk_Tcnstrts maxidx Ts cprf;
in (cs, cprf % None, maxidx') end
| mk_Tcnstrts maxidx _ cprf = ([], cprf, maxidx);
in mk_Tcnstrts maxidx [] cprf end;
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 decompose sg env Ts t u = case (Envir.head_norm env t, Envir.head_norm env u) of
(Const ("all", _) $ t, Const ("all", _) $ u) => decompose sg env Ts t u
| (Const ("==>", _) $ t1 $ t2, Const ("==>", _) $ u1 $ u2) =>
let val (env', cs) = decompose sg env Ts t1 u1
in apsnd (curry op @ cs) (decompose sg env' Ts t2 u2) end
| (Abs (_, T, t), Abs (_, U, u)) =>
decompose sg (unifyT sg env T U) (T::Ts) t u
| (t, u) => (env, [(mk_abs Ts t, mk_abs Ts u)]);
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 make_constraints_cprf sg env ts cprf =
let
fun add_cnstrt Ts prop prf cs env ts (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')]), ts)
handle Pattern.Pattern =>
let val (env', cs') = decompose sg env [] t' u'
in (prop, prf, cs @ cs', env', ts) end
| Pattern.Unif =>
cantunify sg (Envir.norm_term env t') (Envir.norm_term env u')
end;
fun mk_cnstrts_atom env ts prop opTs mk_prf =
let
val tvars = term_tvars prop;
val (env', Ts) = if_none (apsome (pair env) opTs)
(foldl_map (mk_tvar o apsnd snd) (env, tvars));
val prop' = subst_TVars (map fst tvars ~~ Ts) (forall_intr_vfs prop);
in (prop', mk_prf (Some Ts), [], env', ts) end;
fun mk_cnstrts env _ Hs ts (PBound i) = (nth_elem (i, Hs), PBound i, [], env, ts)
| mk_cnstrts env Ts Hs ts (Abst (s, Some T, cprf)) =
let val (t, prf, cnstrts, env', ts') =
mk_cnstrts env (T::Ts) (map (incr_boundvars 1) Hs) ts cprf;
in (Const ("all", (T --> propT) --> propT) $ Abs (s, T, t), Abst (s, Some T, prf),
cnstrts, env', ts')
end
| mk_cnstrts env Ts Hs (t::ts) (AbsP (s, Some _, cprf)) =
let
val (u, prf, cnstrts, env', ts') = mk_cnstrts env Ts (t::Hs) ts cprf;
val t' = strip_abs Ts t;
in (Logic.mk_implies (t', u), AbsP (s, Some t', prf), cnstrts, env', ts')
end
| mk_cnstrts env Ts Hs ts (AbsP (s, None, cprf)) =
let
val (env', t) = mk_var env Ts propT;
val (u, prf, cnstrts, env'', ts') = mk_cnstrts env' Ts (t::Hs) ts cprf;
in (Logic.mk_implies (t, u), AbsP (s, Some t, prf), cnstrts, env'', ts')
end
| mk_cnstrts env Ts Hs ts (cprf1 %% cprf2) =
let val (u, prf2, cnstrts, env', ts') = mk_cnstrts env Ts Hs ts cprf2
in (case mk_cnstrts env' Ts Hs ts' cprf1 of
(Const ("==>", _) $ u' $ t', prf1, cnstrts', env'', ts'') =>
add_cnstrt Ts t' (prf1 %% prf2) (cnstrts' @ cnstrts)
env'' ts'' (u, u')
| (t, prf1, cnstrts', env'', ts'') =>
let val (env''', v) = mk_var env'' Ts propT
in add_cnstrt Ts v (prf1 %% prf2) (cnstrts' @ cnstrts)
env''' ts'' (t, Logic.mk_implies (u, v))
end)
end
| mk_cnstrts env Ts Hs (t::ts) (cprf % Some _) =
let val t' = strip_abs Ts t
in (case mk_cnstrts env Ts Hs ts cprf of
(Const ("all", Type ("fun", [Type ("fun", [T, _]), _])) $ f,
prf, cnstrts, env', ts') =>
let val env'' = unifyT sg env' T (Envir.fastype env' Ts t')
in (betapply (f, t'), prf % Some t', cnstrts, env'', ts')
end
| (u, prf, cnstrts, env', ts') =>
let
val T = Envir.fastype env' Ts t';
val (env'', v) = mk_var env' Ts (T --> propT);
in
add_cnstrt Ts (v $ t') (prf % Some t') cnstrts env'' ts'
(u, Const ("all", (T --> propT) --> propT) $ v)
end)
end
| mk_cnstrts env Ts Hs ts (cprf % None) =
(case mk_cnstrts env Ts Hs ts cprf of
(Const ("all", Type ("fun", [Type ("fun", [T, _]), _])) $ f,
prf, cnstrts, env', ts') =>
let val (env'', t) = mk_var env' Ts T
in (betapply (f, t), prf % Some t, cnstrts, env'', ts')
end
| (u, prf, cnstrts, env', ts') =>
let
val (env1, T) = mk_tvar (env', ["logic"]);
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 ts'
(u, Const ("all", (T --> propT) --> propT) $ v)
end)
| mk_cnstrts env _ _ ts (PThm (name, prf, prop, opTs)) =
mk_cnstrts_atom env ts prop opTs (fn x => PThm (name, prf, prop, x))
| mk_cnstrts env _ _ ts (PAxm (name, prop, opTs)) =
mk_cnstrts_atom env ts prop opTs (fn x => PAxm (name, prop, x))
| mk_cnstrts env _ _ ts (Oracle (name, prop, opTs)) =
mk_cnstrts_atom env ts prop opTs (fn x => Oracle (name, prop, x))
| mk_cnstrts env _ _ ts (Hyp t) = (t, Hyp t, [], env, ts)
| mk_cnstrts _ _ _ _ _ = error "reconstruct_proof: minimal proof object"
in mk_cnstrts env [] [] ts 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)
(Vartab.foldl (add_term_ixns o apsnd 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, _) =>
Sign.pretty_term sg (Logic.mk_flexpair (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 type constraints...";
val (Tcs, cprf'', maxidx) = make_Tconstraints_cprf 0 cprf';
val (ts, Ts) = ListPair.unzip Tcs;
val tsig = Sign.tsig_of sg;
val {classrel, arities, ...} = Type.rep_tsig tsig;
val _ = message "Solving type constraints...";
val (ts', _, unifier) = TypeInfer.infer_types (Sign.pretty_term sg) (Sign.pretty_typ sg)
(Sign.const_type sg) classrel arities [] false (K true) ts Ts;
val env = Envir.Envir {asol = Vartab.empty, iTs = Vartab.make unifier, maxidx = maxidx};
val _ = message "Collecting term constraints...";
val (t, prf, cs, env, _) = make_constraints_cprf sg env ts' 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 full_prf_of thm =
let val {prop, der = (_, prf), sign, ...} = rep_thm thm
in reconstruct_proof sign prop prf end;
(********************************************************************************
expand and reconstruct subproofs
*********************************************************************************)
fun expand_proof sg names prf =
let
fun expand prfs (AbsP (s, t, prf)) =
let val (prfs', prf') = expand prfs prf
in (prfs', AbsP (s, t, prf')) end
| expand prfs (Abst (s, T, prf)) =
let val (prfs', prf') = expand prfs prf
in (prfs', Abst (s, T, prf')) end
| expand prfs (prf1 %% prf2) =
let
val (prfs', prf1') = expand prfs prf1;
val (prfs'', prf2') = expand prfs' prf2;
in (prfs'', prf1' %% prf2') end
| expand prfs (prf % t) =
let val (prfs', prf') = expand prfs prf
in (prfs', prf' % t) end
| expand prfs (prf as PThm ((a, _), cprf, prop, Some Ts)) =
if not (a mem names) then (prfs, prf) else
let
val (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 (prfs', prf) = expand prfs (forall_intr_vfs_prf prop
(reconstruct_proof sg prop cprf))
in (prf, ((a, prop), prf) :: prfs') end
| Some prf => (prf, prfs));
val tye = map fst (term_tvars prop) ~~ Ts
in
(prfs', map_proof_terms (subst_TVars tye) (typ_subst_TVars tye) prf)
end
| expand prfs prf = (prfs, prf);
in snd (expand [] prf) end;
end;