(* Title: Pure/proofterm.ML
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
LF style proof terms.
*)
infix 8 % %% %>;
signature BASIC_PROOFTERM =
sig
val proofs: int ref
datatype proof =
PBound of int
| Abst of string * typ option * proof
| AbsP of string * term option * proof
| % of proof * term option
| %% of proof * proof
| Hyp of term
| PThm of string * proof * term * typ list option
| PAxm of string * term * typ list option
| Oracle of string * term * typ list option
| MinProof of ((string * term) * proof) list * (string * term) list * (string * term) list;
val %> : proof * term -> proof
end;
signature PROOFTERM =
sig
include BASIC_PROOFTERM
val infer_derivs : (proof -> proof -> proof) -> bool * proof -> bool * proof -> bool * proof
val infer_derivs' : (proof -> proof) -> (bool * proof -> bool * proof)
(** primitive operations **)
val proof_combt : proof * term list -> proof
val proof_combt' : proof * term option list -> proof
val proof_combP : proof * proof list -> proof
val strip_combt : proof -> proof * term option list
val strip_combP : proof -> proof * proof list
val strip_thm : proof -> proof
val map_proof_terms_option : (term -> term option) -> (typ -> typ option) -> proof -> proof
val map_proof_terms : (term -> term) -> (typ -> typ) -> proof -> proof
val fold_proof_terms : (term -> 'a -> 'a) -> (typ -> 'a -> 'a) -> proof -> 'a -> 'a
val maxidx_proof : proof -> int -> int
val size_of_proof : proof -> int
val change_type : typ list option -> proof -> proof
val prf_abstract_over : term -> proof -> proof
val prf_incr_bv : int -> int -> int -> int -> proof -> proof
val incr_pboundvars : int -> int -> proof -> proof
val prf_loose_bvar1 : proof -> int -> bool
val prf_loose_Pbvar1 : proof -> int -> bool
val prf_add_loose_bnos : int -> int -> proof ->
int list * int list -> int list * int list
val norm_proof : Envir.env -> proof -> proof
val norm_proof' : Envir.env -> proof -> proof
val prf_subst_bounds : term list -> proof -> proof
val prf_subst_pbounds : proof list -> proof -> proof
val freeze_thaw_prf : proof -> proof * (proof -> proof)
val proof_of_min_axm : string * term -> proof
val proof_of_min_thm : (string * term) * proof -> proof
val thms_of_proof : proof -> (term * proof) list Symtab.table ->
(term * proof) list Symtab.table
val thms_of_proof' : proof -> (term * proof) list Symtab.table ->
(term * proof) list Symtab.table
val axms_of_proof : proof -> proof Symtab.table -> proof Symtab.table
val oracles_of_proof : (string * term) list -> proof -> (string * term) list
(** proof terms for specific inference rules **)
val implies_intr_proof : term -> proof -> proof
val forall_intr_proof : term -> string -> proof -> proof
val varify_proof : term -> (string * sort) list -> proof -> proof
val freezeT : term -> proof -> proof
val rotate_proof : term list -> term -> int -> proof -> proof
val permute_prems_prf : term list -> int -> int -> proof -> proof
val generalize: string list * string list -> int -> proof -> proof
val instantiate : ((indexname * sort) * typ) list * ((indexname * typ) * term) list
-> proof -> proof
val lift_proof : term -> int -> term -> proof -> proof
val assumption_proof : term list -> term -> int -> proof -> proof
val bicompose_proof : bool -> term list -> term list -> term list -> term option ->
int -> int -> proof -> proof -> proof
val equality_axms : (string * term) list
val reflexive_axm : proof
val symmetric_axm : proof
val transitive_axm : proof
val equal_intr_axm : proof
val equal_elim_axm : proof
val abstract_rule_axm : proof
val combination_axm : proof
val reflexive : proof
val symmetric : proof -> proof
val transitive : term -> typ -> proof -> proof -> proof
val abstract_rule : term -> string -> proof -> proof
val combination : term -> term -> term -> term -> typ -> proof -> proof -> proof
val equal_intr : term -> term -> proof -> proof -> proof
val equal_elim : term -> term -> proof -> proof -> proof
val axm_proof : string -> term -> proof
val oracle_proof : string -> term -> proof
val thm_proof : theory -> string -> term list -> term -> proof -> proof
val get_name : term list -> term -> proof -> string
(** rewriting on proof terms **)
val add_prf_rrule : proof * proof -> theory -> theory
val add_prf_rproc : string * (Term.typ list -> proof -> proof option) ->
theory -> theory
val rewrite_proof : theory -> (proof * proof) list *
(string * (typ list -> proof -> proof option)) list -> proof -> proof
val rewrite_proof_notypes : (proof * proof) list *
(string * (typ list -> proof -> proof option)) list -> proof -> proof
val rew_proof : theory -> proof -> proof
end
structure Proofterm : PROOFTERM =
struct
open Envir;
datatype proof =
PBound of int
| Abst of string * typ option * proof
| AbsP of string * term option * proof
| op % of proof * term option
| op %% of proof * proof
| Hyp of term
| PThm of string * proof * term * typ list option
| PAxm of string * term * typ list option
| Oracle of string * term * typ list option
| MinProof of ((string * term) * proof) list * (string * term) list * (string * term) list;
fun proof_of_min_axm (s, prop) = PAxm (s, prop, NONE);
fun proof_of_min_thm ((s, prop), prf) = PThm (s, prf, prop, NONE);
val string_term_ord = prod_ord fast_string_ord Term.fast_term_ord;
fun oracles_of_proof oras prf =
let
fun oras_of (Abst (_, _, prf)) = oras_of prf
| oras_of (AbsP (_, _, prf)) = oras_of prf
| oras_of (prf % _) = oras_of prf
| oras_of (prf1 %% prf2) = oras_of prf1 #> oras_of prf2
| oras_of (PThm (name, prf, prop, _)) = (fn tabs as (thms, oras) =>
case Symtab.lookup thms name of
NONE => oras_of prf (Symtab.update (name, [prop]) thms, oras)
| SOME ps => if member (op =) ps prop then tabs else
oras_of prf (Symtab.update (name, prop::ps) thms, oras))
| oras_of (Oracle (s, prop, _)) =
apsnd (OrdList.insert string_term_ord (s, prop))
| oras_of (MinProof (thms, _, oras)) =
apsnd (OrdList.union string_term_ord oras) #>
fold (oras_of o proof_of_min_thm) thms
| oras_of _ = I
in
snd (oras_of prf (Symtab.empty, oras))
end;
fun thms_of_proof (Abst (_, _, prf)) = thms_of_proof prf
| thms_of_proof (AbsP (_, _, prf)) = thms_of_proof prf
| thms_of_proof (prf1 %% prf2) = thms_of_proof prf1 #> thms_of_proof prf2
| thms_of_proof (prf % _) = thms_of_proof prf
| thms_of_proof (prf' as PThm (s, prf, prop, _)) = (fn tab =>
case Symtab.lookup tab s of
NONE => thms_of_proof prf (Symtab.update (s, [(prop, prf')]) tab)
| SOME ps => if exists (fn (p, _) => p = prop) ps then tab else
thms_of_proof prf (Symtab.update (s, (prop, prf')::ps) tab))
| thms_of_proof (MinProof (prfs, _, _)) = fold (thms_of_proof o proof_of_min_thm) prfs
| thms_of_proof _ = I;
(* this version does not recursively descend into proofs of (named) theorems *)
fun thms_of_proof' (Abst (_, _, prf)) = thms_of_proof' prf
| thms_of_proof' (AbsP (_, _, prf)) = thms_of_proof' prf
| thms_of_proof' (prf1 %% prf2) = thms_of_proof' prf1 #> thms_of_proof' prf2
| thms_of_proof' (prf % _) = thms_of_proof' prf
| thms_of_proof' (PThm ("", prf, prop, _)) = thms_of_proof' prf
| thms_of_proof' (prf' as PThm (s, _, prop, _)) = (fn tab =>
case Symtab.lookup tab s of
NONE => Symtab.update (s, [(prop, prf')]) tab
| SOME ps => if exists (fn (p, _) => p = prop) ps then tab else
Symtab.update (s, (prop, prf')::ps) tab)
| thms_of_proof' (MinProof (prfs, _, _)) = fold (thms_of_proof' o proof_of_min_thm) prfs
| thms_of_proof' _ = I;
fun axms_of_proof (Abst (_, _, prf)) = axms_of_proof prf
| axms_of_proof (AbsP (_, _, prf)) = axms_of_proof prf
| axms_of_proof (prf1 %% prf2) = axms_of_proof prf1 #> axms_of_proof prf2
| axms_of_proof (prf % _) = axms_of_proof prf
| axms_of_proof (prf as PAxm (s, _, _)) = Symtab.update (s, prf)
| axms_of_proof (MinProof (_, prfs, _)) = fold (axms_of_proof o proof_of_min_axm) prfs
| axms_of_proof _ = I;
(** collect all theorems, axioms and oracles **)
fun map3 f g h (thms, axms, oras) = (f thms, g axms, h oras);
fun mk_min_proof (Abst (_, _, prf)) = mk_min_proof prf
| mk_min_proof (AbsP (_, _, prf)) = mk_min_proof prf
| mk_min_proof (prf % _) = mk_min_proof prf
| mk_min_proof (prf1 %% prf2) = mk_min_proof prf1 #> mk_min_proof prf2
| mk_min_proof (PThm (s, prf, prop, _)) =
map3 (OrdList.insert (string_term_ord o pairself fst) ((s, prop), prf)) I I
| mk_min_proof (PAxm (s, prop, _)) =
map3 I (OrdList.insert string_term_ord (s, prop)) I
| mk_min_proof (Oracle (s, prop, _)) =
map3 I I (OrdList.insert string_term_ord (s, prop))
| mk_min_proof (MinProof (thms, axms, oras)) =
map3 (OrdList.union (string_term_ord o pairself fst) thms)
(OrdList.union string_term_ord axms) (OrdList.union string_term_ord oras)
| mk_min_proof _ = I;
(** proof objects with different levels of detail **)
val proofs = ref 2;
fun err_illegal_level i =
error ("Illegal level of detail for proof objects: " ^ string_of_int i);
fun if_ora b = if b then oracles_of_proof else K;
val min_proof = MinProof ([], [], []);
fun infer_derivs f (ora1, prf1) (ora2, prf2) =
let val ora = ora1 orelse ora2 in
(ora,
case !proofs of
2 => f prf1 prf2
| 1 => MinProof (([], [], []) |> mk_min_proof prf1 |> mk_min_proof prf2)
| 0 => if ora then MinProof ([], [], if_ora ora2 (if_ora ora1 [] prf1) prf2) else min_proof
| i => err_illegal_level i)
end;
fun infer_derivs' f = infer_derivs (K f) (false, min_proof);
fun (prf %> t) = prf % SOME t;
val proof_combt = Library.foldl (op %>);
val proof_combt' = Library.foldl (op %);
val proof_combP = Library.foldl (op %%);
fun strip_combt prf =
let fun stripc (prf % t, ts) = stripc (prf, t::ts)
| stripc x = x
in stripc (prf, []) end;
fun strip_combP prf =
let fun stripc (prf %% prf', prfs) = stripc (prf, prf'::prfs)
| stripc x = x
in stripc (prf, []) end;
fun strip_thm prf = (case strip_combt (fst (strip_combP prf)) of
(PThm (_, prf', _, _), _) => prf'
| _ => prf);
val mk_Abst = fold_rev (fn (s, T:typ) => fn prf => Abst (s, NONE, prf));
fun mk_AbsP (i, prf) = funpow i (fn prf => AbsP ("H", NONE, prf)) prf;
fun apsome f NONE = raise SAME
| apsome f (SOME x) = (case f x of NONE => raise SAME | some => some);
fun apsome' f NONE = raise SAME
| apsome' f (SOME x) = SOME (f x);
fun map_proof_terms_option f g =
let
fun map_typs (T :: Ts) =
(case g T of
NONE => T :: map_typs Ts
| SOME T' => T' :: (map_typs Ts handle SAME => Ts))
| map_typs [] = raise SAME;
fun mapp (Abst (s, T, prf)) = (Abst (s, apsome g T, mapph prf)
handle SAME => Abst (s, T, mapp prf))
| mapp (AbsP (s, t, prf)) = (AbsP (s, apsome f t, mapph prf)
handle SAME => AbsP (s, t, mapp prf))
| mapp (prf % t) = (mapp prf % (apsome f t handle SAME => t)
handle SAME => prf % apsome f t)
| mapp (prf1 %% prf2) = (mapp prf1 %% mapph prf2
handle SAME => prf1 %% mapp prf2)
| mapp (PThm (a, prf, prop, SOME Ts)) =
PThm (a, prf, prop, SOME (map_typs Ts))
| mapp (PAxm (a, prop, SOME Ts)) =
PAxm (a, prop, SOME (map_typs Ts))
| mapp _ = raise SAME
and mapph prf = (mapp prf handle SAME => prf)
in mapph end;
fun same eq f x =
let val x' = f x
in if eq (x, x') then raise SAME else x' end;
fun map_proof_terms f g =
map_proof_terms_option
(fn t => SOME (same (op =) f t) handle SAME => NONE)
(fn T => SOME (same (op =) g T) handle SAME => NONE);
fun fold_proof_terms f g (Abst (_, SOME T, prf)) = g T #> fold_proof_terms f g prf
| fold_proof_terms f g (Abst (_, NONE, prf)) = fold_proof_terms f g prf
| fold_proof_terms f g (AbsP (_, SOME t, prf)) = f t #> fold_proof_terms f g prf
| fold_proof_terms f g (AbsP (_, NONE, prf)) = fold_proof_terms f g prf
| fold_proof_terms f g (prf % SOME t) = fold_proof_terms f g prf #> f t
| fold_proof_terms f g (prf % NONE) = fold_proof_terms f g prf
| fold_proof_terms f g (prf1 %% prf2) =
fold_proof_terms f g prf1 #> fold_proof_terms f g prf2
| fold_proof_terms _ g (PThm (_, _, _, SOME Ts)) = fold g Ts
| fold_proof_terms _ g (PAxm (_, _, SOME Ts)) = fold g Ts
| fold_proof_terms _ _ _ = I;
fun maxidx_proof prf = fold_proof_terms Term.maxidx_term Term.maxidx_typ prf;
fun size_of_proof (Abst (_, _, prf)) = 1 + size_of_proof prf
| size_of_proof (AbsP (_, t, prf)) = 1 + size_of_proof prf
| size_of_proof (prf1 %% prf2) = size_of_proof prf1 + size_of_proof prf2
| size_of_proof (prf % _) = 1 + size_of_proof prf
| size_of_proof _ = 1;
fun change_type opTs (PThm (name, prf, prop, _)) = PThm (name, prf, prop, opTs)
| change_type opTs (PAxm (name, prop, _)) = PAxm (name, prop, opTs)
| change_type opTs (Oracle (name, prop, _)) = Oracle (name, prop, opTs)
| change_type _ prf = prf;
(***** utilities *****)
fun strip_abs (_::Ts) (Abs (_, _, t)) = strip_abs Ts t
| strip_abs _ t = t;
fun mk_abs Ts t = Library.foldl (fn (t', T) => Abs ("", T, t')) (t, Ts);
(*Abstraction of a proof term over its occurrences of v,
which must contain no loose bound variables.
The resulting proof term is ready to become the body of an Abst.*)
fun prf_abstract_over v =
let
fun abst' lev u = if v aconv u then Bound lev else
(case u of
Abs (a, T, t) => Abs (a, T, abst' (lev + 1) t)
| f $ t => (abst' lev f $ absth' lev t handle SAME => f $ abst' lev t)
| _ => raise SAME)
and absth' lev t = (abst' lev t handle SAME => t);
fun abst lev (AbsP (a, t, prf)) =
(AbsP (a, apsome' (abst' lev) t, absth lev prf)
handle SAME => AbsP (a, t, abst lev prf))
| abst lev (Abst (a, T, prf)) = Abst (a, T, abst (lev + 1) prf)
| abst lev (prf1 %% prf2) = (abst lev prf1 %% absth lev prf2
handle SAME => prf1 %% abst lev prf2)
| abst lev (prf % t) = (abst lev prf % Option.map (absth' lev) t
handle SAME => prf % apsome' (abst' lev) t)
| abst _ _ = raise SAME
and absth lev prf = (abst lev prf handle SAME => prf)
in absth 0 end;
(*increments a proof term's non-local bound variables
required when moving a proof term within abstractions
inc is increment for bound variables
lev is level at which a bound variable is considered 'loose'*)
fun incr_bv' inct tlev t = incr_bv (inct, tlev, t);
fun prf_incr_bv' incP inct Plev tlev (PBound i) =
if i >= Plev then PBound (i+incP) else raise SAME
| prf_incr_bv' incP inct Plev tlev (AbsP (a, t, body)) =
(AbsP (a, apsome' (same (op =) (incr_bv' inct tlev)) t,
prf_incr_bv incP inct (Plev+1) tlev body) handle SAME =>
AbsP (a, t, prf_incr_bv' incP inct (Plev+1) tlev body))
| prf_incr_bv' incP inct Plev tlev (Abst (a, T, body)) =
Abst (a, T, prf_incr_bv' incP inct Plev (tlev+1) body)
| prf_incr_bv' incP inct Plev tlev (prf %% prf') =
(prf_incr_bv' incP inct Plev tlev prf %% prf_incr_bv incP inct Plev tlev prf'
handle SAME => prf %% prf_incr_bv' incP inct Plev tlev prf')
| prf_incr_bv' incP inct Plev tlev (prf % t) =
(prf_incr_bv' incP inct Plev tlev prf % Option.map (incr_bv' inct tlev) t
handle SAME => prf % apsome' (same (op =) (incr_bv' inct tlev)) t)
| prf_incr_bv' _ _ _ _ _ = raise SAME
and prf_incr_bv incP inct Plev tlev prf =
(prf_incr_bv' incP inct Plev tlev prf handle SAME => prf);
fun incr_pboundvars 0 0 prf = prf
| incr_pboundvars incP inct prf = prf_incr_bv incP inct 0 0 prf;
fun prf_loose_bvar1 (prf1 %% prf2) k = prf_loose_bvar1 prf1 k orelse prf_loose_bvar1 prf2 k
| prf_loose_bvar1 (prf % SOME t) k = prf_loose_bvar1 prf k orelse loose_bvar1 (t, k)
| prf_loose_bvar1 (_ % NONE) _ = true
| prf_loose_bvar1 (AbsP (_, SOME t, prf)) k = loose_bvar1 (t, k) orelse prf_loose_bvar1 prf k
| prf_loose_bvar1 (AbsP (_, NONE, _)) k = true
| prf_loose_bvar1 (Abst (_, _, prf)) k = prf_loose_bvar1 prf (k+1)
| prf_loose_bvar1 _ _ = false;
fun prf_loose_Pbvar1 (PBound i) k = i = k
| prf_loose_Pbvar1 (prf1 %% prf2) k = prf_loose_Pbvar1 prf1 k orelse prf_loose_Pbvar1 prf2 k
| prf_loose_Pbvar1 (prf % _) k = prf_loose_Pbvar1 prf k
| prf_loose_Pbvar1 (AbsP (_, _, prf)) k = prf_loose_Pbvar1 prf (k+1)
| prf_loose_Pbvar1 (Abst (_, _, prf)) k = prf_loose_Pbvar1 prf k
| prf_loose_Pbvar1 _ _ = false;
fun prf_add_loose_bnos plev tlev (PBound i) (is, js) =
if i < plev then (is, js) else (insert (op =) (i-plev) is, js)
| prf_add_loose_bnos plev tlev (prf1 %% prf2) p =
prf_add_loose_bnos plev tlev prf2
(prf_add_loose_bnos plev tlev prf1 p)
| prf_add_loose_bnos plev tlev (prf % opt) (is, js) =
prf_add_loose_bnos plev tlev prf (case opt of
NONE => (is, insert (op =) ~1 js)
| SOME t => (is, add_loose_bnos (t, tlev, js)))
| prf_add_loose_bnos plev tlev (AbsP (_, opt, prf)) (is, js) =
prf_add_loose_bnos (plev+1) tlev prf (case opt of
NONE => (is, insert (op =) ~1 js)
| SOME t => (is, add_loose_bnos (t, tlev, js)))
| prf_add_loose_bnos plev tlev (Abst (_, _, prf)) p =
prf_add_loose_bnos plev (tlev+1) prf p
| prf_add_loose_bnos _ _ _ _ = ([], []);
(**** substitutions ****)
fun del_conflicting_tvars envT T = TermSubst.instantiateT
(map_filter (fn ixnS as (_, S) =>
(Type.lookup (envT, ixnS); NONE) handle TYPE _ =>
SOME (ixnS, TFree ("'dummy", S))) (typ_tvars T)) T;
fun del_conflicting_vars env t = TermSubst.instantiate
(map_filter (fn ixnS as (_, S) =>
(Type.lookup (type_env env, ixnS); NONE) handle TYPE _ =>
SOME (ixnS, TFree ("'dummy", S))) (term_tvars t),
map_filter (fn Var (ixnT as (_, T)) =>
(Envir.lookup (env, ixnT); NONE) handle TYPE _ =>
SOME (ixnT, Free ("dummy", T))) (term_vars t)) t;
fun norm_proof env =
let
val envT = type_env env;
fun msg s = warning ("type conflict in norm_proof:\n" ^ s);
fun htype f t = f env t handle TYPE (s, _, _) =>
(msg s; f env (del_conflicting_vars env t));
fun htypeT f T = f envT T handle TYPE (s, _, _) =>
(msg s; f envT (del_conflicting_tvars envT T));
fun htypeTs f Ts = f envT Ts handle TYPE (s, _, _) =>
(msg s; f envT (map (del_conflicting_tvars envT) Ts));
fun norm (Abst (s, T, prf)) = (Abst (s, apsome' (htypeT norm_type_same) T, normh prf)
handle SAME => Abst (s, T, norm prf))
| norm (AbsP (s, t, prf)) = (AbsP (s, apsome' (htype norm_term_same) t, normh prf)
handle SAME => AbsP (s, t, norm prf))
| norm (prf % t) = (norm prf % Option.map (htype norm_term) t
handle SAME => prf % apsome' (htype norm_term_same) t)
| norm (prf1 %% prf2) = (norm prf1 %% normh prf2
handle SAME => prf1 %% norm prf2)
| norm (PThm (s, prf, t, Ts)) = PThm (s, prf, t, apsome' (htypeTs norm_types_same) Ts)
| norm (PAxm (s, prop, Ts)) = PAxm (s, prop, apsome' (htypeTs norm_types_same) Ts)
| norm _ = raise SAME
and normh prf = (norm prf handle SAME => prf);
in normh end;
(***** Remove some types in proof term (to save space) *****)
fun remove_types (Abs (s, _, t)) = Abs (s, dummyT, remove_types t)
| remove_types (t $ u) = remove_types t $ remove_types u
| remove_types (Const (s, _)) = Const (s, dummyT)
| remove_types t = t;
fun remove_types_env (Envir.Envir {iTs, asol, maxidx}) =
Envir.Envir {iTs = iTs, asol = Vartab.map (apsnd remove_types) asol,
maxidx = maxidx};
fun norm_proof' env prf = norm_proof (remove_types_env env) prf;
(**** substitution of bound variables ****)
fun prf_subst_bounds args prf =
let
val n = length args;
fun subst' lev (Bound i) =
(if i<lev then raise SAME (*var is locally bound*)
else incr_boundvars lev (List.nth (args, i-lev))
handle Subscript => Bound (i-n) (*loose: change it*))
| subst' lev (Abs (a, T, body)) = Abs (a, T, subst' (lev+1) body)
| subst' lev (f $ t) = (subst' lev f $ substh' lev t
handle SAME => f $ subst' lev t)
| subst' _ _ = raise SAME
and substh' lev t = (subst' lev t handle SAME => t);
fun subst lev (AbsP (a, t, body)) = (AbsP (a, apsome' (subst' lev) t, substh lev body)
handle SAME => AbsP (a, t, subst lev body))
| subst lev (Abst (a, T, body)) = Abst (a, T, subst (lev+1) body)
| subst lev (prf %% prf') = (subst lev prf %% substh lev prf'
handle SAME => prf %% subst lev prf')
| subst lev (prf % t) = (subst lev prf % Option.map (substh' lev) t
handle SAME => prf % apsome' (subst' lev) t)
| subst _ _ = raise SAME
and substh lev prf = (subst lev prf handle SAME => prf)
in case args of [] => prf | _ => substh 0 prf end;
fun prf_subst_pbounds args prf =
let
val n = length args;
fun subst (PBound i) Plev tlev =
(if i < Plev then raise SAME (*var is locally bound*)
else incr_pboundvars Plev tlev (List.nth (args, i-Plev))
handle Subscript => PBound (i-n) (*loose: change it*))
| subst (AbsP (a, t, body)) Plev tlev = AbsP (a, t, subst body (Plev+1) tlev)
| subst (Abst (a, T, body)) Plev tlev = Abst (a, T, subst body Plev (tlev+1))
| subst (prf %% prf') Plev tlev = (subst prf Plev tlev %% substh prf' Plev tlev
handle SAME => prf %% subst prf' Plev tlev)
| subst (prf % t) Plev tlev = subst prf Plev tlev % t
| subst prf _ _ = raise SAME
and substh prf Plev tlev = (subst prf Plev tlev handle SAME => prf)
in case args of [] => prf | _ => substh prf 0 0 end;
(**** Freezing and thawing of variables in proof terms ****)
fun frzT names =
map_type_tvar (fn (ixn, xs) => TFree ((the o AList.lookup (op =) names) ixn, xs));
fun thawT names =
map_type_tfree (fn (s, xs) => case AList.lookup (op =) names s of
NONE => TFree (s, xs)
| SOME ixn => TVar (ixn, xs));
fun freeze names names' (t $ u) =
freeze names names' t $ freeze names names' u
| freeze names names' (Abs (s, T, t)) =
Abs (s, frzT names' T, freeze names names' t)
| freeze names names' (Const (s, T)) = Const (s, frzT names' T)
| freeze names names' (Free (s, T)) = Free (s, frzT names' T)
| freeze names names' (Var (ixn, T)) =
Free ((the o AList.lookup (op =) names) ixn, frzT names' T)
| freeze names names' t = t;
fun thaw names names' (t $ u) =
thaw names names' t $ thaw names names' u
| thaw names names' (Abs (s, T, t)) =
Abs (s, thawT names' T, thaw names names' t)
| thaw names names' (Const (s, T)) = Const (s, thawT names' T)
| thaw names names' (Free (s, T)) =
let val T' = thawT names' T
in case AList.lookup (op =) names s of
NONE => Free (s, T')
| SOME ixn => Var (ixn, T')
end
| thaw names names' (Var (ixn, T)) = Var (ixn, thawT names' T)
| thaw names names' t = t;
fun freeze_thaw_prf prf =
let
val (fs, Tfs, vs, Tvs) = fold_proof_terms
(fn t => fn (fs, Tfs, vs, Tvs) =>
(add_term_frees (t, fs), add_term_tfree_names (t, Tfs),
add_term_vars (t, vs), add_term_tvar_ixns (t, Tvs)))
(fn T => fn (fs, Tfs, vs, Tvs) =>
(fs, add_typ_tfree_names (T, Tfs),
vs, add_typ_ixns (Tvs, T)))
prf ([], [], [], []);
val fs' = map (fst o dest_Free) fs;
val vs' = map (fst o dest_Var) vs;
val names = vs' ~~ Name.variant_list fs' (map fst vs');
val names' = Tvs ~~ Name.variant_list Tfs (map fst Tvs);
val rnames = map swap names;
val rnames' = map swap names';
in
(map_proof_terms (freeze names names') (frzT names') prf,
map_proof_terms (thaw rnames rnames') (thawT rnames'))
end;
(***** implication introduction *****)
fun implies_intr_proof h prf =
let
fun abshyp i (Hyp t) = if h aconv t then PBound i else raise SAME
| abshyp i (Abst (s, T, prf)) = Abst (s, T, abshyp i prf)
| abshyp i (AbsP (s, t, prf)) = AbsP (s, t, abshyp (i+1) prf)
| abshyp i (prf % t) = abshyp i prf % t
| abshyp i (prf1 %% prf2) = (abshyp i prf1 %% abshyph i prf2
handle SAME => prf1 %% abshyp i prf2)
| abshyp _ _ = raise SAME
and abshyph i prf = (abshyp i prf handle SAME => prf)
in
AbsP ("H", NONE (*h*), abshyph 0 prf)
end;
(***** forall introduction *****)
fun forall_intr_proof x a prf = Abst (a, NONE, prf_abstract_over x prf);
(***** varify *****)
fun varify_proof t fixed prf =
let
val fs = Term.fold_types (Term.fold_atyps
(fn TFree v => if member (op =) fixed v then I else insert (op =) v | _ => I)) t [];
val ixns = add_term_tvar_ixns (t, []);
val fmap = fs ~~ Name.variant_list (map #1 ixns) (map fst fs)
fun thaw (f as (a, S)) =
(case AList.lookup (op =) fmap f of
NONE => TFree f
| SOME b => TVar ((b, 0), S));
in map_proof_terms (map_types (map_type_tfree thaw)) (map_type_tfree thaw) prf
end;
local
fun new_name (ix, (pairs,used)) =
let val v = Name.variant used (string_of_indexname ix)
in ((ix, v) :: pairs, v :: used) end;
fun freeze_one alist (ix, sort) = (case AList.lookup (op =) alist ix of
NONE => TVar (ix, sort)
| SOME name => TFree (name, sort));
in
fun freezeT t prf =
let
val used = it_term_types add_typ_tfree_names (t, [])
and tvars = map #1 (it_term_types add_typ_tvars (t, []));
val (alist, _) = List.foldr new_name ([], used) tvars;
in
(case alist of
[] => prf (*nothing to do!*)
| _ =>
let val frzT = map_type_tvar (freeze_one alist)
in map_proof_terms (map_types frzT) frzT prf end)
end;
end;
(***** rotate assumptions *****)
fun rotate_proof Bs Bi m prf =
let
val params = Term.strip_all_vars Bi;
val asms = Logic.strip_imp_prems (Term.strip_all_body Bi);
val i = length asms;
val j = length Bs;
in
mk_AbsP (j+1, proof_combP (prf, map PBound
(j downto 1) @ [mk_Abst params (mk_AbsP (i,
proof_combP (proof_combt (PBound i, map Bound ((length params - 1) downto 0)),
map PBound (((i-m-1) downto 0) @ ((i-1) downto (i-m))))))]))
end;
(***** permute premises *****)
fun permute_prems_prf prems j k prf =
let val n = length prems
in mk_AbsP (n, proof_combP (prf,
map PBound ((n-1 downto n-j) @ (k-1 downto 0) @ (n-j-1 downto k))))
end;
(***** generalization *****)
fun generalize (tfrees, frees) idx =
map_proof_terms_option
(TermSubst.generalize_option (tfrees, frees) idx)
(TermSubst.generalizeT_option tfrees idx);
(***** instantiation *****)
fun instantiate (instT, inst) =
map_proof_terms_option
(TermSubst.instantiate_option (instT, map (apsnd remove_types) inst))
(TermSubst.instantiateT_option instT);
(***** lifting *****)
fun lift_proof Bi inc prop prf =
let
fun lift'' Us Ts t = strip_abs Ts (Logic.incr_indexes (Us, inc) (mk_abs Ts t));
fun lift' Us Ts (Abst (s, T, prf)) =
(Abst (s, apsome' (same (op =) (Logic.incr_tvar inc)) T, lifth' Us (dummyT::Ts) prf)
handle SAME => Abst (s, T, lift' Us (dummyT::Ts) prf))
| lift' Us Ts (AbsP (s, t, prf)) =
(AbsP (s, apsome' (same (op =) (lift'' Us Ts)) t, lifth' Us Ts prf)
handle SAME => AbsP (s, t, lift' Us Ts prf))
| lift' Us Ts (prf % t) = (lift' Us Ts prf % Option.map (lift'' Us Ts) t
handle SAME => prf % apsome' (same (op =) (lift'' Us Ts)) t)
| lift' Us Ts (prf1 %% prf2) = (lift' Us Ts prf1 %% lifth' Us Ts prf2
handle SAME => prf1 %% lift' Us Ts prf2)
| lift' _ _ (PThm (s, prf, prop, Ts)) =
PThm (s, prf, prop, apsome' (same (op =) (map (Logic.incr_tvar inc))) Ts)
| lift' _ _ (PAxm (s, prop, Ts)) =
PAxm (s, prop, apsome' (same (op =) (map (Logic.incr_tvar inc))) Ts)
| lift' _ _ _ = raise SAME
and lifth' Us Ts prf = (lift' Us Ts prf handle SAME => prf);
val ps = map (Logic.lift_all inc Bi) (Logic.strip_imp_prems prop);
val k = length ps;
fun mk_app b (i, j, prf) =
if b then (i-1, j, prf %% PBound i) else (i, j-1, prf %> Bound j);
fun lift Us bs i j (Const ("==>", _) $ A $ B) =
AbsP ("H", NONE (*A*), lift Us (true::bs) (i+1) j B)
| lift Us bs i j (Const ("all", _) $ Abs (a, T, t)) =
Abst (a, NONE (*T*), lift (T::Us) (false::bs) i (j+1) t)
| lift Us bs i j _ = proof_combP (lifth' (rev Us) [] prf,
map (fn k => (#3 (fold_rev mk_app bs (i-1, j-1, PBound k))))
(i + k - 1 downto i));
in
mk_AbsP (k, lift [] [] 0 0 Bi)
end;
(***** proof by assumption *****)
fun mk_asm_prf t i m =
let
fun imp_prf _ i 0 = PBound i
| imp_prf (Const ("==>", _) $ A $ B) i m = AbsP ("H", NONE (*A*), imp_prf B (i+1) (m-1))
| imp_prf _ i _ = PBound i;
fun all_prf (Const ("all", _) $ Abs (a, T, t)) = Abst (a, NONE (*T*), all_prf t)
| all_prf t = imp_prf t (~i) m
in all_prf t end;
fun assumption_proof Bs Bi n prf =
mk_AbsP (length Bs, proof_combP (prf,
map PBound (length Bs - 1 downto 0) @ [mk_asm_prf Bi n ~1]));
(***** Composition of object rule with proof state *****)
fun flatten_params_proof i j n (Const ("==>", _) $ A $ B, k) =
AbsP ("H", NONE (*A*), flatten_params_proof (i+1) j n (B, k))
| flatten_params_proof i j n (Const ("all", _) $ Abs (a, T, t), k) =
Abst (a, NONE (*T*), flatten_params_proof i (j+1) n (t, k))
| flatten_params_proof i j n (_, k) = proof_combP (proof_combt (PBound (k+i),
map Bound (j-1 downto 0)), map PBound (remove (op =) (i-n) (i-1 downto 0)));
fun bicompose_proof flatten Bs oldAs newAs A n m rprf sprf =
let
val la = length newAs;
val lb = length Bs;
in
mk_AbsP (lb+la, proof_combP (sprf,
map PBound (lb + la - 1 downto la)) %%
proof_combP (rprf, (if n>0 then [mk_asm_prf (the A) n m] else []) @
map (if flatten then flatten_params_proof 0 0 n else PBound o snd)
(oldAs ~~ (la - 1 downto 0))))
end;
(***** axioms for equality *****)
val aT = TFree ("'a", []);
val bT = TFree ("'b", []);
val x = Free ("x", aT);
val y = Free ("y", aT);
val z = Free ("z", aT);
val A = Free ("A", propT);
val B = Free ("B", propT);
val f = Free ("f", aT --> bT);
val g = Free ("g", aT --> bT);
local open Logic in
val equality_axms =
[("reflexive", mk_equals (x, x)),
("symmetric", mk_implies (mk_equals (x, y), mk_equals (y, x))),
("transitive", list_implies ([mk_equals (x, y), mk_equals (y, z)], mk_equals (x, z))),
("equal_intr", list_implies ([mk_implies (A, B), mk_implies (B, A)], mk_equals (A, B))),
("equal_elim", list_implies ([mk_equals (A, B), A], B)),
("abstract_rule", Logic.mk_implies
(all aT $ Abs ("x", aT, equals bT $ (f $ Bound 0) $ (g $ Bound 0)),
equals (aT --> bT) $
Abs ("x", aT, f $ Bound 0) $ Abs ("x", aT, g $ Bound 0))),
("combination", Logic.list_implies
([Logic.mk_equals (f, g), Logic.mk_equals (x, y)],
Logic.mk_equals (f $ x, g $ y)))];
val [reflexive_axm, symmetric_axm, transitive_axm, equal_intr_axm,
equal_elim_axm, abstract_rule_axm, combination_axm] =
map (fn (s, t) => PAxm ("ProtoPure." ^ s, varify t, NONE)) equality_axms;
end;
val reflexive = reflexive_axm % NONE;
fun symmetric (prf as PAxm ("ProtoPure.reflexive", _, _) % _) = prf
| symmetric prf = symmetric_axm % NONE % NONE %% prf;
fun transitive _ _ (PAxm ("ProtoPure.reflexive", _, _) % _) prf2 = prf2
| transitive _ _ prf1 (PAxm ("ProtoPure.reflexive", _, _) % _) = prf1
| transitive u (Type ("prop", [])) prf1 prf2 =
transitive_axm % NONE % SOME (remove_types u) % NONE %% prf1 %% prf2
| transitive u T prf1 prf2 =
transitive_axm % NONE % NONE % NONE %% prf1 %% prf2;
fun abstract_rule x a prf =
abstract_rule_axm % NONE % NONE %% forall_intr_proof x a prf;
fun check_comb (PAxm ("ProtoPure.combination", _, _) % f % g % _ % _ %% prf %% _) =
is_some f orelse check_comb prf
| check_comb (PAxm ("ProtoPure.transitive", _, _) % _ % _ % _ %% prf1 %% prf2) =
check_comb prf1 andalso check_comb prf2
| check_comb (PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% prf) = check_comb prf
| check_comb _ = false;
fun combination f g t u (Type (_, [T, U])) prf1 prf2 =
let
val f = Envir.beta_norm f;
val g = Envir.beta_norm g;
val prf = if check_comb prf1 then
combination_axm % NONE % NONE
else (case prf1 of
PAxm ("ProtoPure.reflexive", _, _) % _ =>
combination_axm %> remove_types f % NONE
| _ => combination_axm %> remove_types f %> remove_types g)
in
(case T of
Type ("fun", _) => prf %
(case head_of f of
Abs _ => SOME (remove_types t)
| Var _ => SOME (remove_types t)
| _ => NONE) %
(case head_of g of
Abs _ => SOME (remove_types u)
| Var _ => SOME (remove_types u)
| _ => NONE) %% prf1 %% prf2
| _ => prf % NONE % NONE %% prf1 %% prf2)
end;
fun equal_intr A B prf1 prf2 =
equal_intr_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
fun equal_elim A B prf1 prf2 =
equal_elim_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
(***** axioms and theorems *****)
fun vars_of t = rev (fold_aterms (fn v as Var _ => insert (op =) v | _ => I) t []);
fun test_args _ [] = true
| test_args is (Bound i :: ts) =
not (member (op =) is i) andalso test_args (i :: is) ts
| test_args _ _ = false;
fun is_fun (Type ("fun", _)) = true
| is_fun (TVar _) = true
| is_fun _ = false;
fun add_funvars Ts (vs, t) =
if is_fun (fastype_of1 (Ts, t)) then
vs union map_filter (fn Var (ixn, T) =>
if is_fun T then SOME ixn else NONE | _ => NONE) (vars_of t)
else vs;
fun add_npvars q p Ts (vs, Const ("==>", _) $ t $ u) =
add_npvars q p Ts (add_npvars q (not p) Ts (vs, t), u)
| add_npvars q p Ts (vs, Const ("all", Type (_, [Type (_, [T, _]), _])) $ t) =
add_npvars q p Ts (vs, if p andalso q then betapply (t, Var (("",0), T)) else t)
| add_npvars q p Ts (vs, Abs (_, T, t)) = add_npvars q p (T::Ts) (vs, t)
| add_npvars _ _ Ts (vs, t) = add_npvars' Ts (vs, t)
and add_npvars' Ts (vs, t) = (case strip_comb t of
(Var (ixn, _), ts) => if test_args [] ts then vs
else Library.foldl (add_npvars' Ts)
(AList.update (op =) (ixn,
Library.foldl (add_funvars Ts) ((these ooo AList.lookup) (op =) vs ixn, ts)) vs, ts)
| (Abs (_, T, u), ts) => Library.foldl (add_npvars' (T::Ts)) (vs, u :: ts)
| (_, ts) => Library.foldl (add_npvars' Ts) (vs, ts));
fun prop_vars (Const ("==>", _) $ P $ Q) = prop_vars P union prop_vars Q
| prop_vars (Const ("all", _) $ Abs (_, _, t)) = prop_vars t
| prop_vars t = (case strip_comb t of
(Var (ixn, _), _) => [ixn] | _ => []);
fun is_proj t =
let
fun is_p i t = (case strip_comb t of
(Bound j, []) => false
| (Bound j, ts) => j >= i orelse exists (is_p i) ts
| (Abs (_, _, u), _) => is_p (i+1) u
| (_, ts) => exists (is_p i) ts)
in (case strip_abs_body t of
Bound _ => true
| t' => is_p 0 t')
end;
fun needed_vars prop =
Library.foldl (op union)
([], map (uncurry (insert (op =))) (add_npvars true true [] ([], prop))) union
prop_vars prop;
fun gen_axm_proof c name prop =
let
val nvs = needed_vars prop;
val args = map (fn (v as Var (ixn, _)) =>
if member (op =) nvs ixn then SOME v else NONE) (vars_of prop) @
map SOME (sort Term.term_ord (term_frees prop));
in
proof_combt' (c (name, prop, NONE), args)
end;
val axm_proof = gen_axm_proof PAxm;
val dummy = Const (Term.dummy_patternN, dummyT);
fun oracle_proof name prop =
if !proofs = 0 then Oracle (name, dummy, NONE)
else gen_axm_proof Oracle name prop;
fun shrink_proof thy =
let
val compress_typ = Compress.typ thy;
val compress_term = Compress.term thy;
fun shrink ls lev (prf as Abst (a, T, body)) =
let val (b, is, ch, body') = shrink ls (lev+1) body
in (b, is, ch, if ch then Abst (a, Option.map compress_typ T, body') else prf) end
| shrink ls lev (prf as AbsP (a, t, body)) =
let val (b, is, ch, body') = shrink (lev::ls) lev body
in (b orelse member (op =) is 0, map_filter (fn 0 => NONE | i => SOME (i-1)) is,
ch, if ch then AbsP (a, Option.map compress_term t, body') else prf)
end
| shrink ls lev prf =
let val (is, ch, _, prf') = shrink' ls lev [] [] prf
in (false, is, ch, prf') end
and shrink' ls lev ts prfs (prf as prf1 %% prf2) =
let
val p as (_, is', ch', prf') = shrink ls lev prf2;
val (is, ch, ts', prf'') = shrink' ls lev ts (p::prfs) prf1
in (is union is', ch orelse ch', ts',
if ch orelse ch' then prf'' %% prf' else prf)
end
| shrink' ls lev ts prfs (prf as prf1 % t) =
let val (is, ch, (ch', t')::ts', prf') = shrink' ls lev (t::ts) prfs prf1
in (is, ch orelse ch', ts',
if ch orelse ch' then prf' % Option.map compress_term t' else prf) end
| shrink' ls lev ts prfs (prf as PBound i) =
(if exists (fn SOME (Bound j) => lev-j <= List.nth (ls, i) | _ => true) ts
orelse has_duplicates (op =)
(Library.foldl (fn (js, SOME (Bound j)) => j :: js | (js, _) => js) ([], ts))
orelse exists #1 prfs then [i] else [], false, map (pair false) ts, prf)
| shrink' ls lev ts prfs (Hyp t) = ([], false, map (pair false) ts, Hyp (compress_term t))
| shrink' ls lev ts prfs (prf as MinProof _) =
([], false, map (pair false) ts, prf)
| shrink' ls lev ts prfs prf =
let
val prop = (case prf of PThm (_, _, prop, _) => prop | PAxm (_, prop, _) => prop
| Oracle (_, prop, _) => prop | _ => error "shrink: proof not in normal form");
val vs = vars_of prop;
val (ts', ts'') = chop (length vs) ts;
val insts = Library.take (length ts', map (fst o dest_Var) vs) ~~ ts';
val nvs = Library.foldl (fn (ixns', (ixn, ixns)) =>
insert (op =) ixn (case AList.lookup (op =) insts ixn of
SOME (SOME t) => if is_proj t then ixns union ixns' else ixns'
| _ => ixns union ixns'))
(needed prop ts'' prfs, add_npvars false true [] ([], prop));
val insts' = map
(fn (ixn, x as SOME _) => if member (op =) nvs ixn then (false, x) else (true, NONE)
| (_, x) => (false, x)) insts
in ([], false, insts' @ map (pair false) ts'', prf) end
and needed (Const ("==>", _) $ t $ u) ts ((b, _, _, _)::prfs) =
(if b then map (fst o dest_Var) (vars_of t) else []) union needed u ts prfs
| needed (Var (ixn, _)) (_::_) _ = [ixn]
| needed _ _ _ = [];
in shrink end;
(**** Simple first order matching functions for terms and proofs ****)
exception PMatch;
(** see pattern.ML **)
fun flt (i: int) = List.filter (fn n => n < i);
fun fomatch Ts tymatch j =
let
fun mtch (instsp as (tyinsts, insts)) = fn
(Var (ixn, T), t) =>
if j>0 andalso not (null (flt j (loose_bnos t)))
then raise PMatch
else (tymatch (tyinsts, fn () => (T, fastype_of1 (Ts, t))),
(ixn, t) :: insts)
| (Free (a, T), Free (b, U)) =>
if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
| (Const (a, T), Const (b, U)) =>
if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
| (f $ t, g $ u) => mtch (mtch instsp (f, g)) (t, u)
| (Bound i, Bound j) => if i=j then instsp else raise PMatch
| _ => raise PMatch
in mtch end;
fun match_proof Ts tymatch =
let
fun optmatch _ inst (NONE, _) = inst
| optmatch _ _ (SOME _, NONE) = raise PMatch
| optmatch mtch inst (SOME x, SOME y) = mtch inst (x, y)
fun matcht Ts j (pinst, tinst) (t, u) =
(pinst, fomatch Ts tymatch j tinst (t, Envir.beta_norm u));
fun matchT (pinst, (tyinsts, insts)) p =
(pinst, (tymatch (tyinsts, K p), insts));
fun matchTs inst (Ts, Us) = Library.foldl (uncurry matchT) (inst, Ts ~~ Us);
fun mtch Ts i j (pinst, tinst) (Hyp (Var (ixn, _)), prf) =
if i = 0 andalso j = 0 then ((ixn, prf) :: pinst, tinst)
else (case apfst (flt i) (apsnd (flt j)
(prf_add_loose_bnos 0 0 prf ([], []))) of
([], []) => ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
| ([], _) => if j = 0 then
((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
else raise PMatch
| _ => raise PMatch)
| mtch Ts i j inst (prf1 % opt1, prf2 % opt2) =
optmatch (matcht Ts j) (mtch Ts i j inst (prf1, prf2)) (opt1, opt2)
| mtch Ts i j inst (prf1 %% prf2, prf1' %% prf2') =
mtch Ts i j (mtch Ts i j inst (prf1, prf1')) (prf2, prf2')
| mtch Ts i j inst (Abst (_, opT, prf1), Abst (_, opU, prf2)) =
mtch (the_default dummyT opU :: Ts) i (j+1)
(optmatch matchT inst (opT, opU)) (prf1, prf2)
| mtch Ts i j inst (prf1, Abst (_, opU, prf2)) =
mtch (the_default dummyT opU :: Ts) i (j+1) inst
(incr_pboundvars 0 1 prf1 %> Bound 0, prf2)
| mtch Ts i j inst (AbsP (_, opt, prf1), AbsP (_, opu, prf2)) =
mtch Ts (i+1) j (optmatch (matcht Ts j) inst (opt, opu)) (prf1, prf2)
| mtch Ts i j inst (prf1, AbsP (_, _, prf2)) =
mtch Ts (i+1) j inst (incr_pboundvars 1 0 prf1 %% PBound 0, prf2)
| mtch Ts i j inst (PThm (name1, _, prop1, opTs), PThm (name2, _, prop2, opUs)) =
if name1=name2 andalso prop1=prop2 then
optmatch matchTs inst (opTs, opUs)
else raise PMatch
| mtch Ts i j inst (PAxm (s1, _, opTs), PAxm (s2, _, opUs)) =
if s1=s2 then optmatch matchTs inst (opTs, opUs)
else raise PMatch
| mtch _ _ _ inst (PBound i, PBound j) = if i = j then inst else raise PMatch
| mtch _ _ _ _ _ = raise PMatch
in mtch Ts 0 0 end;
fun prf_subst (pinst, (tyinsts, insts)) =
let
val substT = Envir.typ_subst_TVars tyinsts;
fun subst' lev (t as Var (ixn, _)) = (case AList.lookup (op =) insts ixn of
NONE => t
| SOME u => incr_boundvars lev u)
| subst' lev (Const (s, T)) = Const (s, substT T)
| subst' lev (Free (s, T)) = Free (s, substT T)
| subst' lev (Abs (a, T, body)) = Abs (a, substT T, subst' (lev+1) body)
| subst' lev (f $ t) = subst' lev f $ subst' lev t
| subst' _ t = t;
fun subst plev tlev (AbsP (a, t, body)) =
AbsP (a, Option.map (subst' tlev) t, subst (plev+1) tlev body)
| subst plev tlev (Abst (a, T, body)) =
Abst (a, Option.map substT T, subst plev (tlev+1) body)
| subst plev tlev (prf %% prf') = subst plev tlev prf %% subst plev tlev prf'
| subst plev tlev (prf % t) = subst plev tlev prf % Option.map (subst' tlev) t
| subst plev tlev (prf as Hyp (Var (ixn, _))) = (case AList.lookup (op =) pinst ixn of
NONE => prf
| SOME prf' => incr_pboundvars plev tlev prf')
| subst _ _ (PThm (id, prf, prop, Ts)) =
PThm (id, prf, prop, Option.map (map substT) Ts)
| subst _ _ (PAxm (id, prop, Ts)) =
PAxm (id, prop, Option.map (map substT) Ts)
| subst _ _ t = t
in subst 0 0 end;
(*A fast unification filter: true unless the two terms cannot be unified.
Terms must be NORMAL. Treats all Vars as distinct. *)
fun could_unify prf1 prf2 =
let
fun matchrands (prf1 %% prf2) (prf1' %% prf2') =
could_unify prf2 prf2' andalso matchrands prf1 prf1'
| matchrands (prf % SOME t) (prf' % SOME t') =
Term.could_unify (t, t') andalso matchrands prf prf'
| matchrands (prf % _) (prf' % _) = matchrands prf prf'
| matchrands _ _ = true
fun head_of (prf %% _) = head_of prf
| head_of (prf % _) = head_of prf
| head_of prf = prf
in case (head_of prf1, head_of prf2) of
(_, Hyp (Var _)) => true
| (Hyp (Var _), _) => true
| (PThm (a, _, propa, _), PThm (b, _, propb, _)) =>
a = b andalso propa = propb andalso matchrands prf1 prf2
| (PAxm (a, _, _), PAxm (b, _, _)) => a = b andalso matchrands prf1 prf2
| (PBound i, PBound j) => i = j andalso matchrands prf1 prf2
| (AbsP _, _) => true (*because of possible eta equality*)
| (Abst _, _) => true
| (_, AbsP _) => true
| (_, Abst _) => true
| _ => false
end;
(**** rewriting on proof terms ****)
val skel0 = PBound 0;
fun rewrite_prf tymatch (rules, procs) prf =
let
fun rew _ (Abst (_, _, body) % SOME t) = SOME (prf_subst_bounds [t] body, skel0)
| rew _ (AbsP (_, _, body) %% prf) = SOME (prf_subst_pbounds [prf] body, skel0)
| rew Ts prf = (case get_first (fn (_, r) => r Ts prf) procs of
SOME prf' => SOME (prf', skel0)
| NONE => get_first (fn (prf1, prf2) => SOME (prf_subst
(match_proof Ts tymatch ([], (Vartab.empty, [])) (prf1, prf)) prf2, prf2)
handle PMatch => NONE) (List.filter (could_unify prf o fst) rules));
fun rew0 Ts (prf as AbsP (_, _, prf' %% PBound 0)) =
if prf_loose_Pbvar1 prf' 0 then rew Ts prf
else
let val prf'' = incr_pboundvars (~1) 0 prf'
in SOME (the_default (prf'', skel0) (rew Ts prf'')) end
| rew0 Ts (prf as Abst (_, _, prf' % SOME (Bound 0))) =
if prf_loose_bvar1 prf' 0 then rew Ts prf
else
let val prf'' = incr_pboundvars 0 (~1) prf'
in SOME (the_default (prf'', skel0) (rew Ts prf'')) end
| rew0 Ts prf = rew Ts prf;
fun rew1 _ (Hyp (Var _)) _ = NONE
| rew1 Ts skel prf = (case rew2 Ts skel prf of
SOME prf1 => (case rew0 Ts prf1 of
SOME (prf2, skel') => SOME (the_default prf2 (rew1 Ts skel' prf2))
| NONE => SOME prf1)
| NONE => (case rew0 Ts prf of
SOME (prf1, skel') => SOME (the_default prf1 (rew1 Ts skel' prf1))
| NONE => NONE))
and rew2 Ts skel (prf % SOME t) = (case prf of
Abst (_, _, body) =>
let val prf' = prf_subst_bounds [t] body
in SOME (the_default prf' (rew2 Ts skel0 prf')) end
| _ => (case rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf of
SOME prf' => SOME (prf' % SOME t)
| NONE => NONE))
| rew2 Ts skel (prf % NONE) = Option.map (fn prf' => prf' % NONE)
(rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf)
| rew2 Ts skel (prf1 %% prf2) = (case prf1 of
AbsP (_, _, body) =>
let val prf' = prf_subst_pbounds [prf2] body
in SOME (the_default prf' (rew2 Ts skel0 prf')) end
| _ =>
let val (skel1, skel2) = (case skel of
skel1 %% skel2 => (skel1, skel2)
| _ => (skel0, skel0))
in case rew1 Ts skel1 prf1 of
SOME prf1' => (case rew1 Ts skel2 prf2 of
SOME prf2' => SOME (prf1' %% prf2')
| NONE => SOME (prf1' %% prf2))
| NONE => (case rew1 Ts skel2 prf2 of
SOME prf2' => SOME (prf1 %% prf2')
| NONE => NONE)
end)
| rew2 Ts skel (Abst (s, T, prf)) = (case rew1 (the_default dummyT T :: Ts)
(case skel of Abst (_, _, skel') => skel' | _ => skel0) prf of
SOME prf' => SOME (Abst (s, T, prf'))
| NONE => NONE)
| rew2 Ts skel (AbsP (s, t, prf)) = (case rew1 Ts
(case skel of AbsP (_, _, skel') => skel' | _ => skel0) prf of
SOME prf' => SOME (AbsP (s, t, prf'))
| NONE => NONE)
| rew2 _ _ _ = NONE
in the_default prf (rew1 [] skel0 prf) end;
fun rewrite_proof thy = rewrite_prf (fn (tyenv, f) =>
Sign.typ_match thy (f ()) tyenv handle Type.TYPE_MATCH => raise PMatch);
fun rewrite_proof_notypes rews = rewrite_prf fst rews;
(**** theory data ****)
structure ProofData = TheoryDataFun
(
type T = (proof * proof) list * (string * (typ list -> proof -> proof option)) list;
val empty = ([], []);
val copy = I;
val extend = I;
fun merge _ ((rules1, procs1) : T, (rules2, procs2)) =
(Library.merge (op =) (rules1, rules2),
AList.merge (op =) (K true) (procs1, procs2));
);
fun rew_proof thy = rewrite_prf fst (ProofData.get thy);
fun add_prf_rrule r = (ProofData.map o apfst) (insert (op =) r);
fun add_prf_rproc p = (ProofData.map o apsnd) (AList.update (op =) p);
fun thm_proof thy name hyps prop prf =
let
val prop = Logic.list_implies (hyps, prop);
val nvs = needed_vars prop;
val args = map (fn (v as Var (ixn, _)) =>
if member (op =) nvs ixn then SOME v else NONE) (vars_of prop) @
map SOME (sort Term.term_ord (term_frees prop));
val opt_prf = if ! proofs = 2 then
#4 (shrink_proof thy [] 0 (rewrite_prf fst (ProofData.get thy)
(fold_rev implies_intr_proof hyps prf)))
else MinProof (mk_min_proof prf ([], [], []));
val head = (case strip_combt (fst (strip_combP prf)) of
(PThm (old_name, prf', prop', NONE), args') =>
if (old_name="" orelse old_name=name) andalso
prop = prop' andalso args = args' then
PThm (name, prf', prop, NONE)
else
PThm (name, opt_prf, prop, NONE)
| _ => PThm (name, opt_prf, prop, NONE))
in
proof_combP (proof_combt' (head, args), map Hyp hyps)
end;
fun get_name hyps prop prf =
let val prop = Logic.list_implies (hyps, prop) in
(case strip_combt (fst (strip_combP prf)) of
(PThm (name, _, prop', _), _) => if prop=prop' then name else ""
| (PAxm (name, prop', _), _) => if prop=prop' then name else ""
| _ => "")
end;
end;
structure BasicProofterm : BASIC_PROOFTERM = Proofterm;
open BasicProofterm;