|
(* Title: Pure/proofterm.ML
Author: Stefan Berghofer, TU Muenchen
LF style proof terms.
*)
infix 8 % %% %>;
signature PROOFTERM =
sig
type thm_header =
{serial: serial, pos: Position.T list, theory_name: string, name: string,
prop: term, types: typ list option}
type thm_body
type thm_node
datatype proof =
MinProof
| 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
| PAxm of string * term * typ list option
| PClass of typ * class
| Oracle of string * term * typ list option
| PThm of thm_header * thm_body
and proof_body = PBody of
{oracles: ((string * Position.T) * term option) Ord_List.T,
thms: (serial * thm_node) Ord_List.T,
proof: proof}
type oracle = (string * Position.T) * term option
type thm = serial * thm_node
exception MIN_PROOF of unit
val proof_of: proof_body -> proof
val join_proof: proof_body future -> proof
val map_proof_of: (proof -> proof) -> proof_body -> proof_body
val thm_header: serial -> Position.T list -> string -> string -> term -> typ list option ->
thm_header
val thm_body: proof_body -> thm_body
val thm_body_proof_raw: thm_body -> proof
val thm_body_proof_open: thm_body -> proof
val thm_node_theory_name: thm_node -> string
val thm_node_name: thm_node -> string
val thm_node_prop: thm_node -> term
val thm_node_body: thm_node -> proof_body future
val thm_node_thms: thm_node -> thm list
val join_thms: thm list -> proof_body list
val make_thm: thm_header -> thm_body -> thm
val fold_proof_atoms: bool -> (proof -> 'a -> 'a) -> proof list -> 'a -> 'a
val fold_body_thms:
({serial: serial, name: string, prop: term, body: proof_body} -> 'a -> 'a) ->
proof_body list -> 'a -> 'a
val oracle_ord: oracle ord
val thm_ord: thm ord
val unions_oracles: oracle Ord_List.T list -> oracle Ord_List.T
val unions_thms: thm Ord_List.T list -> thm Ord_List.T
val no_proof_body: proof -> proof_body
val no_thm_names: proof -> proof
val no_thm_proofs: proof -> proof
val no_body_proofs: proof -> proof
val encode: Consts.T -> proof XML.Encode.T
val encode_body: Consts.T -> proof_body XML.Encode.T
val encode_standard_term: Consts.T -> term XML.Encode.T
val encode_standard_proof: Consts.T -> proof XML.Encode.T
val decode: Consts.T -> proof XML.Decode.T
val decode_body: Consts.T -> proof_body XML.Decode.T
val %> : proof * term -> proof
(*primitive operations*)
val proofs: int Unsynchronized.ref
val proofs_enabled: unit -> bool
val atomic_proof: proof -> bool
val compact_proof: proof -> 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_body: 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) -> proof -> 'a -> 'a
val fold_proof_terms_types: (term -> 'a -> 'a) -> (typ -> 'a -> 'a) -> proof -> 'a -> 'a
val maxidx_proof: proof -> int -> int
val size_of_proof: proof -> int
val change_types: 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 trivial_proof: proof
val implies_intr_proof: term -> proof -> proof
val implies_intr_proof': term -> proof -> proof
val forall_intr_proof: string * term -> typ option -> 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 -> (string * typ) list -> term list -> int -> proof -> proof
val permute_prems_proof: term list -> int -> int -> proof -> proof
val generalize_proof: string list * string list -> int -> term -> 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 option -> term list ->
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: proof
val symmetric_proof: proof -> proof
val transitive_proof: typ -> term -> proof -> proof -> proof
val equal_intr_proof: term -> term -> proof -> proof -> proof
val equal_elim_proof: term -> term -> proof -> proof -> proof
val abstract_rule_proof: string * term -> proof -> proof
val combination_proof: term -> term -> term -> term -> proof -> proof -> proof
val strip_shyps_proof: Sorts.algebra -> (typ * sort) list -> (typ * sort) list ->
sort list -> proof -> proof
val of_sort_proof: Sorts.algebra ->
(class * class -> proof) ->
(string * class list list * class -> proof) ->
(typ * class -> proof) -> typ * sort -> proof list
val axm_proof: string -> term -> proof
val oracle_proof: string -> term -> proof
val shrink_proof: proof -> 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 set_preproc: (theory -> proof -> proof) -> theory -> theory
val apply_preproc: theory -> proof -> proof
val forall_intr_variables_term: term -> term
val forall_intr_variables: term -> proof -> proof
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 reconstruct_proof: theory -> term -> proof -> proof
val prop_of': term list -> proof -> term
val prop_of: proof -> term
val expand_name_empty: thm_header -> string option
val expand_proof: theory -> (thm_header -> string option) -> proof -> proof
val standard_vars: Name.context -> term * proof option -> term * proof option
val standard_vars_term: Name.context -> term -> term
val add_standard_vars: proof -> (string * typ) list -> (string * typ) list
val add_standard_vars_term: term -> (string * typ) list -> (string * typ) list
val export_enabled: unit -> bool
val export_standard_enabled: unit -> bool
val export_proof_boxes_required: theory -> bool
val export_proof_boxes: proof_body list -> unit
val fulfill_norm_proof: theory -> (serial * proof_body) list -> proof_body -> proof_body
val thm_proof: theory -> (class * class -> proof) ->
(string * class list list * class -> proof) -> string * Position.T -> sort list ->
term list -> term -> (serial * proof_body future) list -> proof_body -> thm * proof
val unconstrain_thm_proof: theory -> (class * class -> proof) ->
(string * class list list * class -> proof) -> sort list -> term ->
(serial * proof_body future) list -> proof_body -> thm * proof
val get_identity: sort list -> term list -> term -> proof ->
{serial: serial, theory_name: string, name: string} option
val get_approximative_name: sort list -> term list -> term -> proof -> string
type thm_id = {serial: serial, theory_name: string}
val make_thm_id: serial * string -> thm_id
val thm_header_id: thm_header -> thm_id
val thm_id: thm -> thm_id
val get_id: sort list -> term list -> term -> proof -> thm_id option
val this_id: thm_id option -> thm_id -> bool
val proof_boxes: {excluded: thm_id -> bool, included: thm_id -> bool} ->
proof list -> (thm_header * proof) list (*exception MIN_PROOF*)
end
structure Proofterm : PROOFTERM =
struct
(** datatype proof **)
type thm_header =
{serial: serial, pos: Position.T list, theory_name: string, name: string,
prop: term, types: typ list option};
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
| PClass of typ * class
| Oracle of string * term * typ list option
| PThm of thm_header * thm_body
and proof_body = PBody of
{oracles: ((string * Position.T) * term option) Ord_List.T,
thms: (serial * thm_node) Ord_List.T,
proof: proof}
and thm_body =
Thm_Body of {open_proof: proof -> proof, body: proof_body future}
and thm_node =
Thm_Node of {theory_name: string, name: string, prop: term,
body: proof_body future, export: unit lazy, consolidate: unit lazy};
type oracle = (string * Position.T) * term option;
val oracle_ord: oracle ord =
prod_ord (prod_ord fast_string_ord Position.ord) (option_ord Term_Ord.fast_term_ord);
type thm = serial * thm_node;
val thm_ord: thm ord = fn ((i, _), (j, _)) => int_ord (j, i);
exception MIN_PROOF of unit;
fun proof_of (PBody {proof, ...}) = proof;
val join_proof = Future.join #> proof_of;
fun map_proof_of f (PBody {oracles, thms, proof}) =
PBody {oracles = oracles, thms = thms, proof = f proof};
fun thm_header serial pos theory_name name prop types : thm_header =
{serial = serial, pos = pos, theory_name = theory_name, name = name, prop = prop, types = types};
fun thm_body body = Thm_Body {open_proof = I, body = Future.value body};
fun thm_body_proof_raw (Thm_Body {body, ...}) = join_proof body;
fun thm_body_proof_open (Thm_Body {open_proof, body, ...}) = open_proof (join_proof body);
fun rep_thm_node (Thm_Node args) = args;
val thm_node_theory_name = #theory_name o rep_thm_node;
val thm_node_name = #name o rep_thm_node;
val thm_node_prop = #prop o rep_thm_node;
val thm_node_body = #body o rep_thm_node;
val thm_node_thms = thm_node_body #> Future.join #> (fn PBody {thms, ...} => thms);
val thm_node_export = #export o rep_thm_node;
val thm_node_consolidate = #consolidate o rep_thm_node;
fun join_thms (thms: thm list) =
Future.joins (map (thm_node_body o #2) thms);
val consolidate_bodies =
maps (fn PBody {thms, ...} => map (thm_node_consolidate o #2) thms)
#> Lazy.consolidate #> map Lazy.force #> ignore;
fun make_thm_node theory_name name prop body export =
let
val consolidate =
Lazy.lazy_name "Proofterm.make_thm_node" (fn () =>
let val PBody {thms, ...} = Future.join body
in consolidate_bodies (join_thms thms) end);
in
Thm_Node {theory_name = theory_name, name = name, prop = prop, body = body,
export = export, consolidate = consolidate}
end;
val no_export = Lazy.value ();
fun make_thm ({serial, theory_name, name, prop, ...}: thm_header) (Thm_Body {body, ...}) =
(serial, make_thm_node theory_name name prop body no_export);
(* 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 ({serial = i, ...}, Thm_Body {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, ...}) =
tap join_thms thms |> fold (fn (i, thm_node) => fn (x, seen) =>
if Inttab.defined seen i then (x, seen)
else
let
val name = thm_node_name thm_node;
val prop = thm_node_prop thm_node;
val body = Future.join (thm_node_body thm_node);
val (x', seen') = app body (x, Inttab.update (i, ()) seen);
in (f {serial = i, name = name, prop = prop, body = body} x', seen') end);
in fn bodies => fn x => #1 (fold app bodies (x, Inttab.empty)) end;
(* proof body *)
val unions_oracles = Ord_List.unions oracle_ord;
val unions_thms = Ord_List.unions thm_ord;
fun no_proof_body proof = PBody {oracles = [], thms = [], proof = proof};
val no_thm_body = thm_body (no_proof_body MinProof);
fun no_thm_names (Abst (x, T, prf)) = Abst (x, T, no_thm_names prf)
| no_thm_names (AbsP (x, t, prf)) = AbsP (x, t, no_thm_names prf)
| no_thm_names (prf % t) = no_thm_names prf % t
| no_thm_names (prf1 %% prf2) = no_thm_names prf1 %% no_thm_names prf2
| no_thm_names (PThm ({serial, pos, theory_name, name = _, prop, types}, thm_body)) =
PThm (thm_header serial pos theory_name "" prop types, thm_body)
| no_thm_names a = a;
fun no_thm_proofs (Abst (x, T, prf)) = Abst (x, T, no_thm_proofs prf)
| no_thm_proofs (AbsP (x, t, prf)) = AbsP (x, t, no_thm_proofs prf)
| no_thm_proofs (prf % t) = no_thm_proofs prf % t
| no_thm_proofs (prf1 %% prf2) = no_thm_proofs prf1 %% no_thm_proofs prf2
| no_thm_proofs (PThm (header, _)) = PThm (header, no_thm_body)
| no_thm_proofs a = a;
fun no_body_proofs (Abst (x, T, prf)) = Abst (x, T, no_body_proofs prf)
| no_body_proofs (AbsP (x, t, prf)) = AbsP (x, t, no_body_proofs prf)
| no_body_proofs (prf % t) = no_body_proofs prf % t
| no_body_proofs (prf1 %% prf2) = no_body_proofs prf1 %% no_body_proofs prf2
| no_body_proofs (PThm (header, Thm_Body {open_proof, body})) =
let
val body' = Future.value (no_proof_body (join_proof body));
val thm_body' = Thm_Body {open_proof = open_proof, body = body'};
in PThm (header, thm_body') end
| no_body_proofs a = a;
(** XML data representation **)
(* encode *)
local
open XML.Encode Term_XML.Encode;
fun proof consts prf = prf |> variant
[fn MinProof => ([], []),
fn PBound a => ([], int a),
fn Abst (a, b, c) => ([a], pair (option typ) (proof consts) (b, c)),
fn AbsP (a, b, c) => ([a], pair (option (term consts)) (proof consts) (b, c)),
fn a % b => ([], pair (proof consts) (option (term consts)) (a, b)),
fn a %% b => ([], pair (proof consts) (proof consts) (a, b)),
fn Hyp a => ([], term consts a),
fn PAxm (a, b, c) => ([a], pair (term consts) (option (list typ)) (b, c)),
fn PClass (a, b) => ([b], typ a),
fn Oracle (a, b, c) => ([a], pair (term consts) (option (list typ)) (b, c)),
fn PThm ({serial, pos, theory_name, name, prop, types}, Thm_Body {open_proof, body, ...}) =>
([int_atom serial, theory_name, name],
pair (list properties) (pair (term consts) (pair (option (list typ)) (proof_body consts)))
(map Position.properties_of pos, (prop, (types, map_proof_of open_proof (Future.join body)))))]
and proof_body consts (PBody {oracles, thms, proof = prf}) =
triple (list (pair (pair string (properties o Position.properties_of))
(option (term consts)))) (list (thm consts)) (proof consts) (oracles, thms, prf)
and thm consts (a, thm_node) =
pair int (pair string (pair string (pair (term consts) (proof_body consts))))
(a, (thm_node_theory_name thm_node, (thm_node_name thm_node, (thm_node_prop thm_node,
(Future.join (thm_node_body thm_node))))));
fun standard_term consts t = t |> variant
[fn Const (a, b) => ([a], list typ (Consts.typargs consts (a, b))),
fn Free (a, _) => ([a], []),
fn Var (a, _) => (indexname a, []),
fn Bound a => ([], int a),
fn Abs (a, b, c) => ([a], pair typ (standard_term consts) (b, c)),
fn op $ a => ([], pair (standard_term consts) (standard_term consts) a)];
fun standard_proof consts prf = prf |> variant
[fn MinProof => ([], []),
fn PBound a => ([], int a),
fn Abst (a, SOME b, c) => ([a], pair typ (standard_proof consts) (b, c)),
fn AbsP (a, SOME b, c) => ([a], pair (standard_term consts) (standard_proof consts) (b, c)),
fn a % SOME b => ([], pair (standard_proof consts) (standard_term consts) (a, b)),
fn a %% b => ([], pair (standard_proof consts) (standard_proof consts) (a, b)),
fn Hyp a => ([], standard_term consts a),
fn PAxm (name, _, SOME Ts) => ([name], list typ Ts),
fn PClass (T, c) => ([c], typ T),
fn Oracle (name, prop, SOME Ts) => ([name], pair (standard_term consts) (list typ) (prop, Ts)),
fn PThm ({serial, theory_name, name, types = SOME Ts, ...}, _) =>
([int_atom serial, theory_name, name], list typ Ts)];
in
val encode = proof;
val encode_body = proof_body;
val encode_standard_term = standard_term;
val encode_standard_proof = standard_proof;
end;
(* decode *)
local
open XML.Decode Term_XML.Decode;
fun proof consts prf = prf |> variant
[fn ([], []) => MinProof,
fn ([], a) => PBound (int a),
fn ([a], b) => let val (c, d) = pair (option typ) (proof consts) b in Abst (a, c, d) end,
fn ([a], b) => let val (c, d) = pair (option (term consts)) (proof consts) b in AbsP (a, c, d) end,
fn ([], a) => op % (pair (proof consts) (option (term consts)) a),
fn ([], a) => op %% (pair (proof consts) (proof consts) a),
fn ([], a) => Hyp (term consts a),
fn ([a], b) => let val (c, d) = pair (term consts) (option (list typ)) b in PAxm (a, c, d) end,
fn ([b], a) => PClass (typ a, b),
fn ([a], b) => let val (c, d) = pair (term consts) (option (list typ)) b in Oracle (a, c, d) end,
fn ([a, b, c], d) =>
let
val ((e, (f, (g, h)))) =
pair (list properties) (pair (term consts) (pair (option (list typ)) (proof_body consts))) d;
val header = thm_header (int_atom a) (map Position.of_properties e) b c f g;
in PThm (header, thm_body h) end]
and proof_body consts x =
let
val (a, b, c) =
triple (list (pair (pair string (Position.of_properties o properties))
(option (term consts)))) (list (thm consts)) (proof consts) x;
in PBody {oracles = a, thms = b, proof = c} end
and thm consts x =
let
val (a, (b, (c, (d, e)))) =
pair int (pair string (pair string (pair (term consts) (proof_body consts)))) x
in (a, make_thm_node b c d (Future.value e) no_export) end;
in
val decode = proof;
val decode_body = proof_body;
end;
(** proof objects with different levels of detail **)
val proofs = Unsynchronized.ref 2;
fun proofs_enabled () = ! proofs >= 2;
fun atomic_proof prf =
(case prf of
Abst _ => false
| AbsP _ => false
| op % _ => false
| op %% _ => false
| MinProof => false
| _ => true);
fun compact_proof (prf % _) = compact_proof prf
| compact_proof (prf1 %% prf2) = atomic_proof prf2 andalso compact_proof prf1
| compact_proof prf = atomic_proof prf;
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 (body as PBody {proof, ...}) =
(case fst (strip_combt (fst (strip_combP proof))) of
PThm (_, Thm_Body {body = body', ...}) => Future.join body'
| _ => body);
val mk_Abst = fold_rev (fn (x, _: typ) => fn prf => Abst (x, NONE, prf));
val mk_AbsP = fold_rev (fn _: term => fn prf => AbsP ("H", NONE, 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 (PClass T_c) = ofclass T_c
| proof (Oracle (a, prop, SOME Ts)) = Oracle (a, prop, SOME (typs Ts))
| proof (PThm ({serial, pos, theory_name, name, prop, types = SOME Ts}, thm_body)) =
PThm (thm_header serial pos theory_name name prop (SOME (typs Ts)), thm_body)
| proof _ = raise Same.SAME;
in proof end;
fun map_proof_terms_same term typ = map_proof_same term typ (fn (T, c) => PClass (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 (Abst (_, _, prf)) = fold_proof_terms f prf
| fold_proof_terms f (AbsP (_, SOME t, prf)) = f t #> fold_proof_terms f prf
| fold_proof_terms f (AbsP (_, NONE, prf)) = fold_proof_terms f prf
| fold_proof_terms f (prf % SOME t) = fold_proof_terms f prf #> f t
| fold_proof_terms f (prf % NONE) = fold_proof_terms f prf
| fold_proof_terms f (prf1 %% prf2) = fold_proof_terms f prf1 #> fold_proof_terms f prf2
| fold_proof_terms _ _ = I;
fun fold_proof_terms_types f g (Abst (_, SOME T, prf)) = g T #> fold_proof_terms_types f g prf
| fold_proof_terms_types f g (Abst (_, NONE, prf)) = fold_proof_terms_types f g prf
| fold_proof_terms_types f g (AbsP (_, SOME t, prf)) = f t #> fold_proof_terms_types f g prf
| fold_proof_terms_types f g (AbsP (_, NONE, prf)) = fold_proof_terms_types f g prf
| fold_proof_terms_types f g (prf % SOME t) = fold_proof_terms_types f g prf #> f t
| fold_proof_terms_types f g (prf % NONE) = fold_proof_terms_types f g prf
| fold_proof_terms_types f g (prf1 %% prf2) =
fold_proof_terms_types f g prf1 #> fold_proof_terms_types f g prf2
| fold_proof_terms_types _ g (PAxm (_, _, SOME Ts)) = fold g Ts
| fold_proof_terms_types _ g (PClass (T, _)) = g T
| fold_proof_terms_types _ g (Oracle (_, _, SOME Ts)) = fold g Ts
| fold_proof_terms_types _ g (PThm ({types = SOME Ts, ...}, _)) = fold g Ts
| fold_proof_terms_types _ _ _ = I;
fun maxidx_proof prf = fold_proof_terms_types Term.maxidx_term Term.maxidx_typ prf;
fun size_of_proof (Abst (_, _, prf)) = 1 + size_of_proof prf
| size_of_proof (AbsP (_, _, 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_types types (PAxm (name, prop, _)) = PAxm (name, prop, types)
| change_types (SOME [T]) (PClass (_, c)) = PClass (T, c)
| change_types types (Oracle (name, prop, _)) = Oracle (name, prop, types)
| change_types types (PThm ({serial, pos, theory_name, name, prop, types = _}, thm_body)) =
PThm (thm_header serial pos theory_name name prop types, thm_body)
| change_types _ 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 _ Plev _ (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, _)) _ = 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 _ (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, Logic.dummy_tfree S)) (Term.add_tvarsT 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, Logic.dummy_tfree S)) (Term.add_tvars t []),
map_filter (fn (ixnT as (_, T)) =>
(Envir.lookup env ixnT; NONE) handle TYPE _ =>
SOME (ixnT, Free ("dummy", T))) (Term.add_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 (PClass (T, c)) =
PClass (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 (PThm ({serial = i, pos = p, theory_name, name = a, prop = t, types = Ts}, thm_body)) =
PThm (thm_header i p theory_name a t
(Same.map_option (htypeTs Envir.norm_types_same) Ts), thm_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 _ _ _ = 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' (Const (s, T)) = Const (s, frzT names' T)
| freeze _ 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 _ _ 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' (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' (Var (ixn, T)) = Var (ixn, thawT names' T)
| thaw _ _ t = t;
in
fun freeze_thaw_prf prf =
let
val (fs, Tfs, vs, Tvs) = fold_proof_terms_types
(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;
(** inference rules **)
(* trivial implication *)
val trivial_proof = AbsP ("H", NONE, PBound 0);
(* 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 (a, v) opt_T prf = Abst (a, opt_T, prf_abstract_over v prf);
fun forall_intr_proof' v prf =
let val (a, T) = (case v of Var ((a, _), T) => (a, T) | Free (a, T) => (a, T))
in forall_intr_proof (a, v) (SOME 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 (a, S) =
(case AList.lookup (op =) fmap (a, S) of
NONE => TFree (a, S)
| 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 = Term.add_tfree_names t [];
val (alist, _) = fold_rev new_name (map #1 (Term.add_tvars t [])) ([], used);
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' params asms m prf =
let
val i = length asms;
val j = length Bs;
in
mk_AbsP (Bs @ [Bi']) (proof_combP (prf, map PBound
(j downto 1) @ [mk_Abst params (mk_AbsP asms
(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 prems'
(proof_combP (prf, map PBound ((n-1 downto n-j) @ (k-1 downto 0) @ (n-j-1 downto k))))
end;
(* generalization *)
fun generalize_proof (tfrees, frees) idx prop prf =
let
val gen =
if null frees then []
else
fold_aterms (fn Free (x, T) => member (op =) frees x ? insert (op =) (x, T) | _ => I)
(Term_Subst.generalize (tfrees, []) idx prop) [];
in
prf
|> Same.commit (map_proof_terms_same
(Term_Subst.generalize_same (tfrees, []) idx)
(Term_Subst.generalizeT_same tfrees idx))
|> fold (fn (x, T) => forall_intr_proof (x, Free (x, T)) NONE) gen
|> fold_rev (fn (x, T) => fn prf' => prf' %> Var (Name.clean_index (x, idx), T)) gen
end;
(* 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' _ _ (PClass (T, c)) =
PClass (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' _ _ (PThm ({serial = i, pos = p, theory_name, name = s, prop, types = Ts}, thm_body)) =
PThm (thm_header i p theory_name s prop
((Same.map_option o Same.map) (Logic.incr_tvar inc) Ts), thm_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 ("Pure.imp", _) $ A $ B) =
AbsP ("H", NONE (*A*), lift Us (true::bs) (i+1) j B)
| lift Us bs i j (Const ("Pure.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 ps (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 ("Pure.imp", _) $ A $ B) i m = AbsP ("H", NONE (*A*), imp_prf B (i+1) (m-1))
| imp_prf _ i _ = PBound i;
fun all_prf (Const ("Pure.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 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 ("Pure.imp", _) $ A $ B, k) =
AbsP ("H", NONE (*A*), flatten_params_proof (i+1) j n (B, k))
| flatten_params_proof i j n (Const ("Pure.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 As A oldAs n m rprf sprf =
let
val lb = length Bs;
val la = length As;
in
mk_AbsP (Bs @ As) (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;
(** type classes **)
fun strip_shyps_proof algebra present witnessed extra prf =
let
val replacements = present @ witnessed @ map (`Logic.dummy_tfree) extra;
fun get_replacement S =
replacements |> get_first (fn (T', S') =>
if Sorts.sort_le algebra (S', S) then SOME T' else NONE);
fun replace T =
if exists (fn (T', _) => T' = T) present then raise Same.SAME
else
(case get_replacement (Type.sort_of_atyp T) 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;
fun of_sort_proof algebra classrel_proof arity_proof hyps =
Sorts.of_sort_derivation algebra
{class_relation = fn _ => fn _ => fn (prf, c1) => fn c2 =>
if c1 = c2 then prf else classrel_proof (c1, c2) %% prf,
type_constructor = fn (a, _) => fn dom => fn c =>
let val Ss = map (map snd) dom and prfs = maps (map fst) dom
in proof_combP (arity_proof (a, Ss, c), prfs) end,
type_variable = fn typ => map (fn c => (hyps (typ, c), c)) (Type.sort_of_atyp typ)};
(** axioms and theorems **)
val add_type_variables = (fold_types o fold_atyps) (insert (op =));
fun type_variables_of t = rev (add_type_variables t []);
val add_variables = fold_aterms (fn a => (is_Var a orelse is_Free a) ? insert (op =) a);
fun variables_of t = rev (add_variables 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 vars_of t = map Var (rev (Term.add_vars t []));
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 ("Pure.imp", _) $ t $ u) =
add_npvars q p Ts (add_npvars q (not p) Ts (vs, t), u)
| add_npvars q p Ts (vs, Const ("Pure.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 ("Pure.imp", _) $ P $ Q) = union (op =) (prop_vars P) (prop_vars Q)
| prop_vars (Const ("Pure.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 _, []) => 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 prop_args prop =
let
val needed_vars =
union (op =) (Library.foldl (uncurry (union (op =)))
([], map (uncurry (insert (op =))) (add_npvars true true [] ([], prop))))
(prop_vars prop);
in
variables_of prop |> map
(fn var as Var (ixn, _) => if member (op =) needed_vars ixn then SOME var else NONE
| free => SOME free)
end;
fun const_proof mk name prop =
let
val args = prop_args prop;
val ({outer_constraints, ...}, prop1) = Logic.unconstrainT [] prop;
val head = mk (name, prop1, NONE);
in proof_combP (proof_combt' (head, args), map PClass outer_constraints) end;
val axm_proof = const_proof PAxm;
val oracle_proof = const_proof Oracle;
val shrink_proof =
let
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, 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, 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' % 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' _ _ ts _ (Hyp t) = ([], false, map (pair false) ts, Hyp t)
| shrink' _ _ ts _ (prf as MinProof) = ([], false, map (pair false) ts, prf)
| shrink' _ _ ts _ (prf as PClass _) = ([], false, map (pair false) ts, prf)
| shrink' _ _ ts prfs prf =
let
val prop =
(case prf of
PAxm (_, prop, _) => prop
| Oracle (_, 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 ("Pure.imp", _) $ 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 fn prf => #4 (shrink [] 0 prf) 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_proof = reflexive_axm % NONE;
val is_reflexive_proof = fn PAxm ("Pure.reflexive", _, _) % _ => true | _ => false;
fun symmetric_proof prf =
if is_reflexive_proof prf then prf
else symmetric_axm % NONE % NONE %% prf;
fun transitive_proof U u prf1 prf2 =
if is_reflexive_proof prf1 then prf2
else if is_reflexive_proof prf2 then prf1
else if U = propT then transitive_axm % NONE % SOME (remove_types u) % NONE %% prf1 %% prf2
else transitive_axm % NONE % NONE % NONE %% prf1 %% prf2;
fun equal_intr_proof A B prf1 prf2 =
equal_intr_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
fun equal_elim_proof A B prf1 prf2 =
equal_elim_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
fun abstract_rule_proof (a, x) prf =
abstract_rule_axm % NONE % NONE %% forall_intr_proof (a, x) NONE prf;
fun check_comb (PAxm ("Pure.combination", _, _) % f % _ % _ % _ %% 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_proof f g 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
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
end;
(** rewriting on proof terms **)
(* simple first order matching functions for terms and proofs (see pattern.ML) *)
exception PMatch;
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 (apply2 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 (PClass (T1, c1), PClass (T2, c2)) =
if c1 = c2 then matchT inst (T1, T2)
else raise PMatch
| mtch Ts i j inst
(PThm ({name = name1, prop = prop1, types = types1, ...}, _),
PThm ({name = name2, prop = prop2, types = types2, ...}, _)) =
if name1 = name2 andalso prop1 = prop2
then optmatch matchTs inst (types1, types2)
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 _ _ (PClass (T, c)) = PClass (substT T, c)
| subst _ _ (Oracle (id, prop, Ts)) = Oracle (id, prop, Same.map_option substTs Ts)
| subst _ _ (PThm ({serial = i, pos = p, theory_name, name = id, prop, types}, thm_body)) =
PThm (thm_header i p theory_name id prop (Same.map_option substTs types), thm_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
| (PClass (_, c), PClass (_, d)) => c = d andalso matchrands prf1 prf2
| (PThm ({name = a, prop = propa, ...}, _), PThm ({name = b, prop = 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;
(* rewrite proof *)
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) *
(theory -> proof -> proof) option;
val empty = (([], []), NONE);
val extend = I;
fun merge (((rules1, procs1), preproc1), ((rules2, procs2), preproc2)) : T =
((AList.merge (op =) (K true) (rules1, rules2),
AList.merge (op =) (K true) (procs1, procs2)),
merge_options (preproc1, preproc2));
);
fun get_rew_data thy =
let val (rules, procs) = #1 (Data.get thy)
in (map #2 rules, map #2 procs) end;
fun rew_proof thy = rewrite_prf fst (get_rew_data thy);
fun add_prf_rrule r = (Data.map o apfst o apfst) (cons (stamp (), r));
fun add_prf_rproc p = (Data.map o apfst o apsnd) (cons (stamp (), p));
fun set_preproc f = (Data.map o apsnd) (K (SOME f));
fun apply_preproc thy = (case #2 (Data.get thy) of NONE => I | SOME f => f thy);
(** reconstruction of partial proof terms **)
fun forall_intr_variables_term prop = fold_rev Logic.all (variables_of prop) prop;
fun forall_intr_variables prop prf = fold_rev forall_intr_proof' (variables_of prop) prf;
local
fun app_types shift prop Ts prf =
let
--> --------------------
--> maximum size reached
--> --------------------
¤ Dauer der Verarbeitung: 0.59 Sekunden
(vorverarbeitet)
¤
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