Merge.
(* Title: Pure/Proof/reconstruct.ML
Author: Stefan Berghofer, TU Muenchen
Reconstruction of partial proof terms.
*)
signature RECONSTRUCT =
sig
val quiet_mode : bool 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 = 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_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 = fold_rev forall_intr_prf
(vars_of prop @ frees_of 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 =) (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));
val mk_abs = fold (fn T => fn u => Abs ("", T, u));
fun unifyT thy env T U =
let
val Envir.Envir {asol, iTs, maxidx} = env;
val (iTs', maxidx') = Sign.typ_unify thy (T, U) (iTs, maxidx)
in Envir.Envir {asol=asol, iTs=iTs', maxidx=maxidx'} end
handle Type.TUNIFY => error ("Non-unifiable types:\n" ^
Syntax.string_of_typ_global thy T ^ "\n\n" ^ Syntax.string_of_typ_global thy 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 thy (env as Envir.Envir {maxidx, asol, iTs}) 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, asol = asol, iTs = iTs})
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 (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 thy env Ts vTs (Var _) = error "reconstruct_proof: internal error"
| infer_type thy 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 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 (env3, V) = 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 (env, p as (t, u)) =
let fun rigrig (a, T) (b, U) uT ts us = if a <> b then cantunify thy p
else apsnd flat (Library.foldl_map (decompose thy 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 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) (unifyT thy env T U, (t, u))
| ((Abs (_, T, t), []), _) =>
decompose thy (T::Ts) (env, (t, incr_boundvars 1 u $ Bound 0))
| (_, (Abs (_, T, u), [])) =>
decompose thy (T::Ts) (env, (incr_boundvars 1 t $ Bound 0, u))
| _ => (env, [(mk_abs Ts t, mk_abs Ts u)])
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 (env', cs') = decompose thy [] (env, (t', u'))
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 = OldTerm.term_tvars prop;
val tfrees = OldTerm.term_tfrees prop;
val (env', Ts) = (case opTs of
NONE => Library.foldl_map mk_tvar (env, map snd tvars @ map snd tfrees)
| SOME Ts => (env, Ts));
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 (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 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 (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 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 (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;
(**** update list of free variables of constraints ****)
fun upd_constrs env cs =
let
val Envir.Envir {asol, iTs, ...} = env;
val dom = []
|> Vartab.fold (cons o #1) asol
|> Vartab.fold (cons o #1) iTs;
val vran = []
|> Vartab.fold (Term.add_var_names o #2 o #2) asol
|> Vartab.fold (Term.add_tvar_namesT o #2 o #2) iTs;
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, _) =>
Display.pretty_flexpair (Syntax.pp_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 (env', cs') = decompose thy [] (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 thy (upd_constrs env cs') (merge_envs 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, op union
(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 (OldTerm.term_tvars prop) @ map TFree (OldTerm.term_tfrees prop) ~~ 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 (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
fun inc i =
map_proof_terms (Logic.incr_indexes ([], i)) (Logic.incr_tvar i);
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, inc (maxidx + 1) prf,
((a, prop), (maxidx', prf)) :: prfs')
end
| SOME (maxidx', prf) => (maxidx' + maxidx + 1,
inc (maxidx + 1) prf, prfs));
val tfrees = OldTerm.term_tfrees prop;
val tye = map (fn ((s, j), _) => (s, maxidx + 1 + j))
(OldTerm.term_tvars prop) @ 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_terms (subst_TVars tye o
map_types varify) (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;