(* Title: Pure/proofterm.ML
Author: Stefan Berghofer, TU Muenchen
LF style proof terms.
*)
infix 8 % %% %>;
signature BASIC_PROOFTERM =
sig
val proofs: int Unsynchronized.ref
datatype proof =
MinProof
| 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
| PAxm of string * term * typ list option
| OfClass of typ * class
| Oracle of string * term * typ list option
| Promise of serial * term * typ list
| PThm of serial * ((string * term * typ list option) * proof_body future)
and proof_body = PBody of
{oracles: (string * term) Ord_List.T,
thms: (serial * (string * term * proof_body future)) Ord_List.T,
proof: proof}
val %> : proof * term -> proof
end;
signature PROOFTERM =
sig
include BASIC_PROOFTERM
type oracle = string * term
type pthm = serial * (string * term * proof_body future)
val proof_of: proof_body -> proof
val join_proof: proof_body future -> proof
val fold_proof_atoms: bool -> (proof -> 'a -> 'a) -> proof list -> 'a -> 'a
val fold_body_thms: (string * term * proof_body -> 'a -> 'a) -> proof_body list -> 'a -> 'a
val join_bodies: proof_body list -> unit
val status_of: proof_body list -> {failed: bool, oracle: bool, unfinished: bool}
val oracle_ord: oracle * oracle -> order
val thm_ord: pthm * pthm -> order
val merge_oracles: oracle Ord_List.T -> oracle Ord_List.T -> oracle Ord_List.T
val merge_thms: pthm Ord_List.T -> pthm Ord_List.T -> pthm Ord_List.T
val all_oracles_of: proof_body -> oracle Ord_List.T
val approximate_proof_body: proof -> proof_body
(** primitive operations **)
val proofs_enabled: unit -> bool
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_body -> proof_body
val map_proof_same: term Same.operation -> typ Same.operation
-> (typ * class -> proof) -> proof Same.operation
val map_proof_terms_same: term Same.operation -> typ Same.operation -> proof Same.operation
val map_proof_types_same: typ Same.operation -> proof Same.operation
val map_proof_terms: (term -> term) -> (typ -> typ) -> proof -> proof
val map_proof_types: (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)
(** proof terms for specific inference rules **)
val implies_intr_proof: term -> proof -> proof
val implies_intr_proof': term -> proof -> proof
val forall_intr_proof: term -> string -> proof -> proof
val forall_intr_proof': term -> proof -> proof
val varify_proof: term -> (string * sort) list -> proof -> proof
val legacy_freezeT: term -> proof -> proof
val rotate_proof: term list -> term -> int -> proof -> proof
val permute_prems_proof: 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 incr_indexes: int -> 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 strip_shyps_proof: Sorts.algebra -> (typ * sort) list -> (typ * sort) list ->
sort list -> proof -> proof
val classrel_proof: theory -> class * class -> proof
val arity_proof: theory -> string * sort list * class -> proof
val of_sort_proof: theory -> (typ * class -> proof) -> typ * sort -> proof list
val install_axclass_proofs:
{classrel_proof: theory -> class * class -> proof,
arity_proof: theory -> string * sort list * class -> proof} -> unit
val axm_proof: string -> term -> proof
val oracle_proof: string -> term -> oracle * proof
(** rewriting on proof terms **)
val add_prf_rrule: proof * proof -> theory -> theory
val add_prf_rproc: (typ list -> term option list -> proof -> (proof * proof) option) -> theory -> theory
val no_skel: proof
val normal_skel: proof
val rewrite_proof: theory -> (proof * proof) list *
(typ list -> term option list -> proof -> (proof * proof) option) list -> proof -> proof
val rewrite_proof_notypes: (proof * proof) list *
(typ list -> term option list -> proof -> (proof * proof) option) list -> proof -> proof
val rew_proof: theory -> proof -> proof
val promise_proof: theory -> serial -> term -> proof
val fulfill_norm_proof: theory -> (serial * proof_body) list -> proof_body -> proof_body
val thm_proof: theory -> string -> sort list -> term list -> term ->
(serial * proof_body future) list -> proof_body -> pthm * proof
val unconstrain_thm_proof: theory -> sort list -> term ->
(serial * proof_body future) list -> proof_body -> pthm * proof
val get_name: sort list -> term list -> term -> proof -> string
val guess_name: proof -> string
end
structure Proofterm : PROOFTERM =
struct
(***** datatype proof *****)
datatype proof =
MinProof
| 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
| PAxm of string * term * typ list option
| OfClass of typ * class
| Oracle of string * term * typ list option
| Promise of serial * term * typ list
| PThm of serial * ((string * term * typ list option) * proof_body future)
and proof_body = PBody of
{oracles: (string * term) Ord_List.T,
thms: (serial * (string * term * proof_body future)) Ord_List.T,
proof: proof};
type oracle = string * term;
type pthm = serial * (string * term * proof_body future);
fun proof_of (PBody {proof, ...}) = proof;
val join_proof = Future.join #> proof_of;
(***** proof atoms *****)
fun fold_proof_atoms all f =
let
fun app (Abst (_, _, prf)) = app prf
| app (AbsP (_, _, prf)) = app prf
| app (prf % _) = app prf
| app (prf1 %% prf2) = app prf1 #> app prf2
| app (prf as PThm (i, (_, body))) = (fn (x, seen) =>
if Inttab.defined seen i then (x, seen)
else
let val (x', seen') =
(if all then app (join_proof body) else I) (x, Inttab.update (i, ()) seen)
in (f prf x', seen') end)
| app prf = (fn (x, seen) => (f prf x, seen));
in fn prfs => fn x => #1 (fold app prfs (x, Inttab.empty)) end;
fun fold_body_thms f =
let
fun app (PBody {thms, ...}) =
(Future.join_results (map (#3 o #2) thms);
thms |> fold (fn (i, (name, prop, body)) => fn (x, seen) =>
if Inttab.defined seen i then (x, seen)
else
let
val body' = Future.join body;
val (x', seen') = app body' (x, Inttab.update (i, ()) seen);
in (f (name, prop, body') x', seen') end));
in fn bodies => fn x => #1 (fold app bodies (x, Inttab.empty)) end;
fun join_bodies bodies = fold_body_thms (fn _ => fn () => ()) bodies ();
fun status_of bodies =
let
fun status (PBody {oracles, thms, ...}) x =
let
val ((oracle, unfinished, failed), seen) =
(thms, x) |-> fold (fn (i, (_, _, body)) => fn (st, seen) =>
if Inttab.defined seen i then (st, seen)
else
let val seen' = Inttab.update (i, ()) seen in
(case Future.peek body of
SOME (Exn.Res body') => status body' (st, seen')
| SOME (Exn.Exn _) =>
let val (oracle, unfinished, _) = st
in ((oracle, unfinished, true), seen') end
| NONE =>
let val (oracle, _, failed) = st
in ((oracle, true, failed), seen') end)
end);
in ((oracle orelse not (null oracles), unfinished, failed), seen) end;
val (oracle, unfinished, failed) =
#1 (fold status bodies ((false, false, false), Inttab.empty));
in {oracle = oracle, unfinished = unfinished, failed = failed} end;
(* proof body *)
val oracle_ord = prod_ord fast_string_ord Term_Ord.fast_term_ord;
fun thm_ord ((i, _): pthm, (j, _)) = int_ord (j, i);
val merge_oracles = Ord_List.union oracle_ord;
val merge_thms = Ord_List.union thm_ord;
val all_oracles_of =
let
fun collect (PBody {oracles, thms, ...}) =
(Future.join_results (map (#3 o #2) thms);
thms |> fold (fn (i, (_, _, body)) => fn (x, seen) =>
if Inttab.defined seen i then (x, seen)
else
let
val body' = Future.join body;
val (x', seen') = collect body' (x, Inttab.update (i, ()) seen);
in (merge_oracles oracles x', seen') end));
in fn body => #1 (collect body ([], Inttab.empty)) end;
fun approximate_proof_body prf =
let
val (oracles, thms) = fold_proof_atoms false
(fn Oracle (s, prop, _) => apfst (cons (s, prop))
| PThm (i, ((name, prop, _), body)) => apsnd (cons (i, (name, prop, body)))
| _ => I) [prf] ([], []);
in
PBody
{oracles = Ord_List.make oracle_ord oracles,
thms = Ord_List.make thm_ord thms,
proof = prf}
end;
(***** proof objects with different levels of detail *****)
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 (body as PBody {proof, ...}) =
(case strip_combt (fst (strip_combP proof)) of
(PThm (_, (_, body')), _) => Future.join body'
| _ => body);
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 map_proof_same term typ ofclass =
let
val typs = Same.map typ;
fun proof (Abst (s, T, prf)) =
(Abst (s, Same.map_option typ T, Same.commit proof prf)
handle Same.SAME => Abst (s, T, proof prf))
| proof (AbsP (s, t, prf)) =
(AbsP (s, Same.map_option term t, Same.commit proof prf)
handle Same.SAME => AbsP (s, t, proof prf))
| proof (prf % t) =
(proof prf % Same.commit (Same.map_option term) t
handle Same.SAME => prf % Same.map_option term t)
| proof (prf1 %% prf2) =
(proof prf1 %% Same.commit proof prf2
handle Same.SAME => prf1 %% proof prf2)
| proof (PAxm (a, prop, SOME Ts)) = PAxm (a, prop, SOME (typs Ts))
| proof (OfClass T_c) = ofclass T_c
| proof (Oracle (a, prop, SOME Ts)) = Oracle (a, prop, SOME (typs Ts))
| proof (Promise (i, prop, Ts)) = Promise (i, prop, typs Ts)
| proof (PThm (i, ((a, prop, SOME Ts), body))) =
PThm (i, ((a, prop, SOME (typs Ts)), body))
| proof _ = raise Same.SAME;
in proof end;
fun map_proof_terms_same term typ = map_proof_same term typ (fn (T, c) => OfClass (typ T, c));
fun map_proof_types_same typ = map_proof_terms_same (Term_Subst.map_types_same typ) typ;
fun same eq f x =
let val x' = f x
in if eq (x, x') then raise Same.SAME else x' end;
fun map_proof_terms f g = Same.commit (map_proof_terms_same (same (op =) f) (same (op =) g));
fun map_proof_types f = Same.commit (map_proof_types_same (same (op =) f));
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 (PAxm (_, _, SOME Ts)) = fold g Ts
| fold_proof_terms _ g (OfClass (T, _)) = g T
| fold_proof_terms _ g (Oracle (_, _, SOME Ts)) = fold g Ts
| fold_proof_terms _ g (Promise (_, _, Ts)) = fold g Ts
| fold_proof_terms _ g (PThm (_, ((_, _, 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 (prf % _) = 1 + size_of_proof prf
| size_of_proof (prf1 %% prf2) = size_of_proof prf1 + size_of_proof prf2
| size_of_proof _ = 1;
fun change_type opTs (PAxm (name, prop, _)) = PAxm (name, prop, opTs)
| change_type (SOME [T]) (OfClass (_, c)) = OfClass (T, c)
| change_type opTs (Oracle (name, prop, _)) = Oracle (name, prop, opTs)
| change_type opTs (Promise _) = raise Fail "change_type: unexpected promise"
| change_type opTs (PThm (i, ((name, prop, _), body))) =
PThm (i, ((name, prop, opTs), body))
| 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.SAME => f $ abst' lev t)
| _ => raise Same.SAME)
and absth' lev t = (abst' lev t handle Same.SAME => t);
fun abst lev (AbsP (a, t, prf)) =
(AbsP (a, Same.map_option (abst' lev) t, absth lev prf)
handle Same.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.SAME => prf1 %% abst lev prf2)
| abst lev (prf % t) = (abst lev prf % Option.map (absth' lev) t
handle Same.SAME => prf % Same.map_option (abst' lev) t)
| abst _ _ = raise Same.SAME
and absth lev prf = (abst lev prf handle Same.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.SAME
| prf_incr_bv' incP inct Plev tlev (AbsP (a, t, body)) =
(AbsP (a, Same.map_option (same (op =) (incr_bv' inct tlev)) t,
prf_incr_bv incP inct (Plev+1) tlev body) handle Same.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.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.SAME => prf % Same.map_option (same (op =) (incr_bv' inct tlev)) t)
| prf_incr_bv' _ _ _ _ _ = raise Same.SAME
and prf_incr_bv incP inct Plev tlev prf =
(prf_incr_bv' incP inct Plev tlev prf handle Same.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 = Term_Subst.instantiateT
(map_filter (fn ixnS as (_, S) =>
(Type.lookup envT ixnS; NONE) handle TYPE _ =>
SOME (ixnS, TFree ("'dummy", S))) (OldTerm.typ_tvars T)) T;
fun del_conflicting_vars env t = Term_Subst.instantiate
(map_filter (fn ixnS as (_, S) =>
(Type.lookup (Envir.type_env env) ixnS; NONE) handle TYPE _ =>
SOME (ixnS, TFree ("'dummy", S))) (OldTerm.term_tvars t),
map_filter (fn Var (ixnT as (_, T)) =>
(Envir.lookup (env, ixnT); NONE) handle TYPE _ =>
SOME (ixnT, Free ("dummy", T))) (OldTerm.term_vars t)) t;
fun norm_proof env =
let
val envT = Envir.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, Same.map_option (htypeT Envir.norm_type_same) T, Same.commit norm prf)
handle Same.SAME => Abst (s, T, norm prf))
| norm (AbsP (s, t, prf)) =
(AbsP (s, Same.map_option (htype Envir.norm_term_same) t, Same.commit norm prf)
handle Same.SAME => AbsP (s, t, norm prf))
| norm (prf % t) =
(norm prf % Option.map (htype Envir.norm_term) t
handle Same.SAME => prf % Same.map_option (htype Envir.norm_term_same) t)
| norm (prf1 %% prf2) =
(norm prf1 %% Same.commit norm prf2
handle Same.SAME => prf1 %% norm prf2)
| norm (PAxm (s, prop, Ts)) =
PAxm (s, prop, Same.map_option (htypeTs Envir.norm_types_same) Ts)
| norm (OfClass (T, c)) =
OfClass (htypeT Envir.norm_type_same T, c)
| norm (Oracle (s, prop, Ts)) =
Oracle (s, prop, Same.map_option (htypeTs Envir.norm_types_same) Ts)
| norm (Promise (i, prop, Ts)) =
Promise (i, prop, htypeTs Envir.norm_types_same Ts)
| norm (PThm (i, ((s, t, Ts), body))) =
PThm (i, ((s, t, Same.map_option (htypeTs Envir.norm_types_same) Ts), body))
| norm _ = raise Same.SAME;
in Same.commit norm 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 {maxidx, tenv, tyenv}) =
Envir.Envir {maxidx = maxidx, tenv = Vartab.map (K (apsnd remove_types)) tenv, tyenv = tyenv};
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.SAME (*var is locally bound*)
else incr_boundvars lev (nth args (i-lev))
handle General.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.SAME => f $ subst' lev t)
| subst' _ _ = raise Same.SAME
and substh' lev t = (subst' lev t handle Same.SAME => t);
fun subst lev (AbsP (a, t, body)) =
(AbsP (a, Same.map_option (subst' lev) t, substh lev body)
handle Same.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.SAME => prf %% subst lev prf')
| subst lev (prf % t) = (subst lev prf % Option.map (substh' lev) t
handle Same.SAME => prf % Same.map_option (subst' lev) t)
| subst _ _ = raise Same.SAME
and substh lev prf = (subst lev prf handle Same.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.SAME (*var is locally bound*)
else incr_pboundvars Plev tlev (nth args (i-Plev))
handle General.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.SAME => prf %% subst prf' Plev tlev)
| subst (prf % t) Plev tlev = subst prf Plev tlev % t
| subst prf _ _ = raise Same.SAME
and substh prf Plev tlev = (subst prf Plev tlev handle Same.SAME => prf)
in case args of [] => prf | _ => substh prf 0 0 end;
(**** Freezing and thawing of variables in proof terms ****)
local
fun frzT names =
map_type_tvar (fn (ixn, S) => TFree (the (AList.lookup (op =) names ixn), S));
fun thawT names =
map_type_tfree (fn (a, S) =>
(case AList.lookup (op =) names a of
NONE => TFree (a, S)
| SOME ixn => TVar (ixn, S)));
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 (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;
in
fun freeze_thaw_prf prf =
let
val (fs, Tfs, vs, Tvs) = fold_proof_terms
(fn t => fn (fs, Tfs, vs, Tvs) =>
(Term.add_free_names t fs, Term.add_tfree_names t Tfs,
Term.add_var_names t vs, Term.add_tvar_names t Tvs))
(fn T => fn (fs, Tfs, vs, Tvs) =>
(fs, Term.add_tfree_namesT T Tfs,
vs, Term.add_tvar_namesT T Tvs))
prf ([], [], [], []);
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;
end;
(***** implication introduction *****)
fun gen_implies_intr_proof f h prf =
let
fun abshyp i (Hyp t) = if h aconv t then PBound i else raise Same.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.SAME => prf1 %% abshyp i prf2)
| abshyp _ _ = raise Same.SAME
and abshyph i prf = (abshyp i prf handle Same.SAME => prf);
in
AbsP ("H", f h, abshyph 0 prf)
end;
val implies_intr_proof = gen_implies_intr_proof (K NONE);
val implies_intr_proof' = gen_implies_intr_proof SOME;
(***** forall introduction *****)
fun forall_intr_proof x a prf = Abst (a, NONE, prf_abstract_over x prf);
fun forall_intr_proof' 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;
(***** 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 used = Name.context
|> fold_types (fold_atyps (fn TVar ((a, _), _) => Name.declare a | _ => I)) t;
val fmap = fs ~~ #1 (fold_map Name.variant (map fst fs) used);
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 = singleton (Name.variant_list 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 legacy_freezeT t prf =
let
val used = OldTerm.it_term_types OldTerm.add_typ_tfree_names (t, [])
and tvars = map #1 (OldTerm.it_term_types OldTerm.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_proof 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 =
Same.commit (map_proof_terms_same
(Term_Subst.generalize_same (tfrees, frees) idx)
(Term_Subst.generalizeT_same tfrees idx));
(***** instantiation *****)
fun instantiate (instT, inst) =
Same.commit (map_proof_terms_same
(Term_Subst.instantiate_same (instT, map (apsnd remove_types) inst))
(Term_Subst.instantiateT_same 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, Same.map_option (Logic.incr_tvar_same inc) T, lifth' Us (dummyT::Ts) prf)
handle Same.SAME => Abst (s, T, lift' Us (dummyT::Ts) prf))
| lift' Us Ts (AbsP (s, t, prf)) =
(AbsP (s, Same.map_option (same (op =) (lift'' Us Ts)) t, lifth' Us Ts prf)
handle Same.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.SAME => prf % Same.map_option (same (op =) (lift'' Us Ts)) t)
| lift' Us Ts (prf1 %% prf2) = (lift' Us Ts prf1 %% lifth' Us Ts prf2
handle Same.SAME => prf1 %% lift' Us Ts prf2)
| lift' _ _ (PAxm (s, prop, Ts)) =
PAxm (s, prop, (Same.map_option o Same.map) (Logic.incr_tvar_same inc) Ts)
| lift' _ _ (OfClass (T, c)) =
OfClass (Logic.incr_tvar_same inc T, c)
| lift' _ _ (Oracle (s, prop, Ts)) =
Oracle (s, prop, (Same.map_option o Same.map) (Logic.incr_tvar_same inc) Ts)
| lift' _ _ (Promise (i, prop, Ts)) =
Promise (i, prop, Same.map (Logic.incr_tvar_same inc) Ts)
| lift' _ _ (PThm (i, ((s, prop, Ts), body))) =
PThm (i, ((s, prop, (Same.map_option o Same.map) (Logic.incr_tvar inc) Ts), body))
| lift' _ _ _ = raise Same.SAME
and lifth' Us Ts prf = (lift' Us Ts prf handle Same.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;
fun incr_indexes i =
Same.commit (map_proof_terms_same
(Logic.incr_indexes_same ([], i)) (Logic.incr_tvar_same i));
(***** 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);
val equality_axms =
[("reflexive", Logic.mk_equals (x, x)),
("symmetric", Logic.mk_implies (Logic.mk_equals (x, y), Logic.mk_equals (y, x))),
("transitive",
Logic.list_implies ([Logic.mk_equals (x, y), Logic.mk_equals (y, z)], Logic.mk_equals (x, z))),
("equal_intr",
Logic.list_implies ([Logic.mk_implies (A, B), Logic.mk_implies (B, A)], Logic.mk_equals (A, B))),
("equal_elim", Logic.list_implies ([Logic.mk_equals (A, B), A], B)),
("abstract_rule",
Logic.mk_implies
(Logic.all x
(Logic.mk_equals (f $ x, g $ x)), Logic.mk_equals (lambda x (f $ x), lambda x (g $ x)))),
("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 ("Pure." ^ s, Logic.varify_global t, NONE)) equality_axms;
val reflexive = reflexive_axm % NONE;
fun symmetric (prf as PAxm ("Pure.reflexive", _, _) % _) = prf
| symmetric prf = symmetric_axm % NONE % NONE %% prf;
fun transitive _ _ (PAxm ("Pure.reflexive", _, _) % _) prf2 = prf2
| transitive _ _ prf1 (PAxm ("Pure.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 ("Pure.combination", _, _) % f % g % _ % _ %% prf %% _) =
is_some f orelse check_comb prf
| check_comb (PAxm ("Pure.transitive", _, _) % _ % _ % _ %% prf1 %% prf2) =
check_comb prf1 andalso check_comb prf2
| check_comb (PAxm ("Pure.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 ("Pure.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;
(**** type classes ****)
fun strip_shyps_proof algebra present witnessed extra_sorts prf =
let
fun get S2 (T, S1) = if Sorts.sort_le algebra (S1, S2) then SOME T else NONE;
val extra = map (fn S => (TFree ("'dummy", S), S)) extra_sorts;
val replacements = present @ extra @ witnessed;
fun replace T =
if exists (fn (T', _) => T' = T) present then raise Same.SAME
else
(case get_first (get (Type.sort_of_atyp T)) replacements of
SOME T' => T'
| NONE => raise Fail "strip_shyps_proof: bad type variable in proof term");
in Same.commit (map_proof_types_same (Term_Subst.map_atypsT_same replace)) prf end;
local
type axclass_proofs =
{classrel_proof: theory -> class * class -> proof,
arity_proof: theory -> string * sort list * class -> proof};
val axclass_proofs: axclass_proofs Single_Assignment.var =
Single_Assignment.var "Proofterm.axclass_proofs";
fun axclass_proof which thy x =
(case Single_Assignment.peek axclass_proofs of
NONE => raise Fail "Axclass proof operations not installed"
| SOME prfs => which prfs thy x);
in
val classrel_proof = axclass_proof #classrel_proof;
val arity_proof = axclass_proof #arity_proof;
fun install_axclass_proofs prfs = Single_Assignment.assign axclass_proofs prfs;
end;
local
fun canonical_instance typs =
let
val names = Name.invent Name.context Name.aT (length typs);
val instT = map2 (fn a => fn T => (((a, 0), []), Type.strip_sorts T)) names typs;
in instantiate (instT, []) end;
in
fun of_sort_proof thy hyps =
Sorts.of_sort_derivation (Sign.classes_of thy)
{class_relation = fn typ => fn (prf, c1) => fn c2 =>
if c1 = c2 then prf
else canonical_instance [typ] (classrel_proof thy (c1, c2)) %% prf,
type_constructor = fn (a, typs) => fn dom => fn c =>
let val Ss = map (map snd) dom and prfs = maps (map fst) dom
in proof_combP (canonical_instance typs (arity_proof thy (a, Ss, c)), prfs) end,
type_variable = fn typ => map (fn c => (hyps (typ, c), c)) (Type.sort_of_atyp typ)};
end;
(***** axioms and theorems *****)
val proofs = Unsynchronized.ref 2;
fun proofs_enabled () = ! proofs >= 2;
fun vars_of t = map Var (rev (Term.add_vars t []));
fun frees_of t = map Free (rev (Term.add_frees 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
union (op =) vs (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) = union (op =) (prop_vars P) (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 =
union (op =) (Library.foldl (uncurry (union (op =)))
([], map (uncurry (insert (op =))) (add_npvars true true [] ([], prop))))
(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 (frees_of 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 ((name, dummy), Oracle (name, dummy, NONE))
else ((name, prop), gen_axm_proof Oracle name prop);
fun shrink_proof thy =
let
val (typ, term) = Term_Sharing.init 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 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 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 (union (op =) is 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 term t' else prf) end
| shrink' ls lev ts prfs (prf as PBound i) =
(if exists (fn SOME (Bound j) => lev-j <= 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 (term t))
| shrink' ls lev ts prfs (prf as MinProof) = ([], false, map (pair false) ts, prf)
| shrink' ls lev ts prfs (prf as OfClass _) = ([], false, map (pair false) ts, prf)
| shrink' ls lev ts prfs prf =
let
val prop =
(case prf of
PAxm (_, prop, _) => prop
| Oracle (_, prop, _) => prop
| Promise (_, prop, _) => prop
| PThm (_, ((_, prop, _), _)) => prop
| _ => raise Fail "shrink: proof not in normal form");
val vs = vars_of prop;
val (ts', ts'') = chop (length vs) ts;
val insts = 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 union (op =) ixns ixns' else ixns'
| _ => union (op =) ixns 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) =
union (op =) (if b then map (fst o dest_Var) (vars_of t) else []) (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) = filter (fn n => n < i);
fun fomatch Ts tymatch j instsp p =
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 instsp (pairself Envir.beta_eta_contract p) 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 (PAxm (s1, _, opTs), PAxm (s2, _, opUs)) =
if s1 = s2 then optmatch matchTs inst (opTs, opUs)
else raise PMatch
| mtch Ts i j inst (OfClass (T1, c1), OfClass (T2, c2)) =
if c1 = c2 then matchT inst (T1, T2)
else raise PMatch
| 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 _ _ _ 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.subst_type_same tyinsts;
val substTs = Same.map substT;
fun subst' lev (Var (xi, _)) =
(case AList.lookup (op =) insts xi of
NONE => raise Same.SAME
| SOME u => incr_boundvars lev u)
| subst' _ (Const (s, T)) = Const (s, substT T)
| subst' _ (Free (s, T)) = Free (s, substT T)
| subst' lev (Abs (a, T, body)) =
(Abs (a, substT T, Same.commit (subst' (lev + 1)) body)
handle Same.SAME => Abs (a, T, subst' (lev + 1) body))
| subst' lev (f $ t) =
(subst' lev f $ Same.commit (subst' lev) t
handle Same.SAME => f $ subst' lev t)
| subst' _ _ = raise Same.SAME;
fun subst plev tlev (AbsP (a, t, body)) =
(AbsP (a, Same.map_option (subst' tlev) t, Same.commit (subst (plev + 1) tlev) body)
handle Same.SAME => AbsP (a, t, subst (plev + 1) tlev body))
| subst plev tlev (Abst (a, T, body)) =
(Abst (a, Same.map_option substT T, Same.commit (subst plev (tlev + 1)) body)
handle Same.SAME => Abst (a, T, subst plev (tlev + 1) body))
| subst plev tlev (prf %% prf') =
(subst plev tlev prf %% Same.commit (subst plev tlev) prf'
handle Same.SAME => prf %% subst plev tlev prf')
| subst plev tlev (prf % t) =
(subst plev tlev prf % Same.commit (Same.map_option (subst' tlev)) t
handle Same.SAME => prf % Same.map_option (subst' tlev) t)
| subst plev tlev (Hyp (Var (xi, _))) =
(case AList.lookup (op =) pinst xi of
NONE => raise Same.SAME
| SOME prf' => incr_pboundvars plev tlev prf')
| subst _ _ (PAxm (id, prop, Ts)) = PAxm (id, prop, Same.map_option substTs Ts)
| subst _ _ (OfClass (T, c)) = OfClass (substT T, c)
| subst _ _ (Oracle (id, prop, Ts)) = Oracle (id, prop, Same.map_option substTs Ts)
| subst _ _ (Promise (i, prop, Ts)) = Promise (i, prop, substTs Ts)
| subst _ _ (PThm (i, ((id, prop, Ts), body))) =
PThm (i, ((id, prop, Same.map_option substTs Ts), body))
| subst _ _ _ = raise Same.SAME;
in fn t => subst 0 0 t handle Same.SAME => t 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
| (PAxm (a, _, _), PAxm (b, _, _)) => a = b andalso matchrands prf1 prf2
| (OfClass (_, c), OfClass (_, d)) => c = d andalso matchrands prf1 prf2
| (PThm (_, ((a, propa, _), _)), PThm (_, ((b, propb, _), _))) =>
a = b andalso propa = propb 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 no_skel = PBound 0;
val normal_skel = Hyp (Var ((Name.uu, 0), propT));
fun rewrite_prf tymatch (rules, procs) prf =
let
fun rew _ _ (Abst (_, _, body) % SOME t) = SOME (prf_subst_bounds [t] body, no_skel)
| rew _ _ (AbsP (_, _, body) %% prf) = SOME (prf_subst_pbounds [prf] body, no_skel)
| rew Ts hs prf =
(case get_first (fn r => r Ts hs prf) procs of
NONE => get_first (fn (prf1, prf2) => SOME (prf_subst
(match_proof Ts tymatch ([], (Vartab.empty, [])) (prf1, prf)) prf2, prf2)
handle PMatch => NONE) (filter (could_unify prf o fst) rules)
| some => some);
fun rew0 Ts hs (prf as AbsP (_, _, prf' %% PBound 0)) =
if prf_loose_Pbvar1 prf' 0 then rew Ts hs prf
else
let val prf'' = incr_pboundvars (~1) 0 prf'
in SOME (the_default (prf'', no_skel) (rew Ts hs prf'')) end
| rew0 Ts hs (prf as Abst (_, _, prf' % SOME (Bound 0))) =
if prf_loose_bvar1 prf' 0 then rew Ts hs prf
else
let val prf'' = incr_pboundvars 0 (~1) prf'
in SOME (the_default (prf'', no_skel) (rew Ts hs prf'')) end
| rew0 Ts hs prf = rew Ts hs prf;
fun rew1 _ _ (Hyp (Var _)) _ = NONE
| rew1 Ts hs skel prf = (case rew2 Ts hs skel prf of
SOME prf1 => (case rew0 Ts hs prf1 of
SOME (prf2, skel') => SOME (the_default prf2 (rew1 Ts hs skel' prf2))
| NONE => SOME prf1)
| NONE => (case rew0 Ts hs prf of
SOME (prf1, skel') => SOME (the_default prf1 (rew1 Ts hs skel' prf1))
| NONE => NONE))
and rew2 Ts hs skel (prf % SOME t) = (case prf of
Abst (_, _, body) =>
let val prf' = prf_subst_bounds [t] body
in SOME (the_default prf' (rew2 Ts hs no_skel prf')) end
| _ => (case rew1 Ts hs (case skel of skel' % _ => skel' | _ => no_skel) prf of
SOME prf' => SOME (prf' % SOME t)
| NONE => NONE))
| rew2 Ts hs skel (prf % NONE) = Option.map (fn prf' => prf' % NONE)
(rew1 Ts hs (case skel of skel' % _ => skel' | _ => no_skel) prf)
| rew2 Ts hs skel (prf1 %% prf2) = (case prf1 of
AbsP (_, _, body) =>
let val prf' = prf_subst_pbounds [prf2] body
in SOME (the_default prf' (rew2 Ts hs no_skel prf')) end
| _ =>
let val (skel1, skel2) = (case skel of
skel1 %% skel2 => (skel1, skel2)
| _ => (no_skel, no_skel))
in case rew1 Ts hs skel1 prf1 of
SOME prf1' => (case rew1 Ts hs skel2 prf2 of
SOME prf2' => SOME (prf1' %% prf2')
| NONE => SOME (prf1' %% prf2))
| NONE => (case rew1 Ts hs skel2 prf2 of
SOME prf2' => SOME (prf1 %% prf2')
| NONE => NONE)
end)
| rew2 Ts hs skel (Abst (s, T, prf)) = (case rew1 (the_default dummyT T :: Ts) hs
(case skel of Abst (_, _, skel') => skel' | _ => no_skel) prf of
SOME prf' => SOME (Abst (s, T, prf'))
| NONE => NONE)
| rew2 Ts hs skel (AbsP (s, t, prf)) = (case rew1 Ts (t :: hs)
(case skel of AbsP (_, _, skel') => skel' | _ => no_skel) prf of
SOME prf' => SOME (AbsP (s, t, prf'))
| NONE => NONE)
| rew2 _ _ _ _ = NONE;
in the_default prf (rew1 [] [] no_skel 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 Data = Theory_Data
(
type T =
(stamp * (proof * proof)) list *
(stamp * (typ list -> term option list -> proof -> (proof * proof) option)) list;
val empty = ([], []);
val extend = I;
fun merge ((rules1, procs1), (rules2, procs2)) : T =
(AList.merge (op =) (K true) (rules1, rules2),
AList.merge (op =) (K true) (procs1, procs2));
);
fun get_data thy = let val (rules, procs) = Data.get thy in (map #2 rules, map #2 procs) end;
fun rew_proof thy = rewrite_prf fst (get_data thy);
fun add_prf_rrule r = (Data.map o apfst) (cons (stamp (), r));
fun add_prf_rproc p = (Data.map o apsnd) (cons (stamp (), p));
(***** promises *****)
fun promise_proof thy i prop =
let
val _ = prop |> Term.exists_subterm (fn t =>
(Term.is_Free t orelse Term.is_Var t) andalso
raise Fail ("promise_proof: illegal variable " ^ Syntax.string_of_term_global thy t));
val _ = prop |> Term.exists_type (Term.exists_subtype
(fn TFree (a, _) => raise Fail ("promise_proof: illegal type variable " ^ quote a)
| _ => false));
in Promise (i, prop, map TVar (Term.add_tvars prop [])) end;
fun fulfill_norm_proof thy ps body0 =
let
val PBody {oracles = oracles0, thms = thms0, proof = proof0} = body0;
val oracles = fold (fn (_, PBody {oracles, ...}) => merge_oracles oracles) ps oracles0;
val thms = fold (fn (_, PBody {thms, ...}) => merge_thms thms) ps thms0;
val proofs = fold (fn (i, PBody {proof, ...}) => Inttab.update (i, proof)) ps Inttab.empty;
fun fill (Promise (i, prop, Ts)) =
(case Inttab.lookup proofs i of
NONE => NONE
| SOME prf => SOME (instantiate (Term.add_tvars prop [] ~~ Ts, []) prf, normal_skel))
| fill _ = NONE;
val (rules, procs) = get_data thy;
val proof = rewrite_prf fst (rules, K (K fill) :: procs) proof0;
in PBody {oracles = oracles, thms = thms, proof = proof} end;
fun fulfill_proof_future _ [] postproc body =
if not (Multithreading.enabled ()) then Future.value (postproc (Future.join body))
else Future.map postproc body
| fulfill_proof_future thy promises postproc body =
singleton
(Future.forks {name = "Proofterm.fulfill_proof_future", group = NONE,
deps = Future.task_of body :: map (Future.task_of o snd) promises, pri = 0,
interrupts = true})
(fn () =>
postproc (fulfill_norm_proof thy (map (apsnd Future.join) promises) (Future.join body)));
(***** abstraction over sort constraints *****)
fun unconstrainT_prf thy (atyp_map, constraints) =
let
fun hyp_map hyp =
(case AList.lookup (op =) constraints hyp of
SOME t => Hyp t
| NONE => raise Fail "unconstrainT_prf: missing constraint");
val typ = Term_Subst.map_atypsT_same (Type.strip_sorts o atyp_map);
fun ofclass (ty, c) =
let val ty' = Term.map_atyps atyp_map ty;
in the_single (of_sort_proof thy hyp_map (ty', [c])) end;
in
Same.commit (map_proof_same (Term_Subst.map_types_same typ) typ ofclass)
#> fold_rev (implies_intr_proof o snd) constraints
end;
fun unconstrainT_body thy constrs (PBody {oracles, thms, proof}) =
PBody
{oracles = oracles, (* FIXME merge (!), unconstrain (!?!) *)
thms = thms, (* FIXME merge (!) *)
proof = unconstrainT_prf thy constrs proof};
(***** theorems *****)
fun prepare_thm_proof thy name shyps hyps concl promises body =
let
val PBody {oracles = oracles0, thms = thms0, proof = prf} = body;
val prop = Logic.list_implies (hyps, concl);
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 (frees_of prop);
val ((atyp_map, constraints, outer_constraints), prop1) = Logic.unconstrainT shyps prop;
val postproc = unconstrainT_body thy (atyp_map, constraints);
val args1 =
(map o Option.map o Term.map_types o Term.map_atyps)
(Type.strip_sorts o atyp_map) args;
val argsP = map OfClass outer_constraints @ map Hyp hyps;
fun full_proof0 () =
#4 (shrink_proof thy [] 0 (rew_proof thy (fold_rev implies_intr_proof hyps prf)));
fun make_body0 proof0 = PBody {oracles = oracles0, thms = thms0, proof = proof0};
val body0 =
if not (proofs_enabled ()) then Future.value (make_body0 MinProof)
else if not (Multithreading.enabled ()) then Future.value (make_body0 (full_proof0 ()))
else
singleton
(Future.forks
{name = "Proofterm.prepare_thm_proof", group = NONE, deps = [], pri = ~1,
interrupts = true})
(make_body0 o full_proof0);
fun new_prf () = (serial (), fulfill_proof_future thy promises postproc body0);
val (i, body') =
(*non-deterministic, depends on unknown promises*)
(case strip_combt (fst (strip_combP prf)) of
(PThm (i, ((old_name, prop', NONE), body')), args') =>
if (old_name = "" orelse old_name = name) andalso prop1 = prop' andalso args = args'
then (i, body')
else new_prf ()
| _ => new_prf ());
val head = PThm (i, ((name, prop1, NONE), body'));
in ((i, (name, prop1, body')), head, args, argsP, args1) end;
fun thm_proof thy name shyps hyps concl promises body =
let val (pthm, head, args, argsP, _) = prepare_thm_proof thy name shyps hyps concl promises body
in (pthm, proof_combP (proof_combt' (head, args), argsP)) end;
fun unconstrain_thm_proof thy shyps concl promises body =
let
val (pthm, head, _, _, args) = prepare_thm_proof thy "" shyps [] concl promises body
in (pthm, proof_combt' (head, args)) end;
fun get_name shyps hyps prop prf =
let val (_, prop) = Logic.unconstrainT shyps (Logic.list_implies (hyps, prop)) in
(case strip_combt (fst (strip_combP prf)) of
(PThm (_, ((name, prop', _), _)), _) => if prop = prop' then name else ""
| _ => "")
end;
fun guess_name (PThm (_, ((name, _, _), _))) = name
| guess_name (prf %% Hyp _) = guess_name prf
| guess_name (prf %% OfClass _) = guess_name prf
| guess_name (prf % NONE) = guess_name prf
| guess_name (prf % SOME (Var _)) = guess_name prf
| guess_name _ = "";
end;
structure Basic_Proofterm : BASIC_PROOFTERM = Proofterm;
open Basic_Proofterm;