(* Title: Pure/thm.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Makarius
The very core of Isabelle's Meta Logic: certified types and terms,
derivations, theorems, inference rules (including lifting and
resolution), oracles.
*)
signature BASIC_THM =
sig
type ctyp
type cterm
exception CTERM of string * cterm list
type thm
type conv = cterm -> thm
exception THM of string * int * thm list
end;
signature THM =
sig
include BASIC_THM
(*certified types*)
val typ_of: ctyp -> typ
val global_ctyp_of: theory -> typ -> ctyp
val ctyp_of: Proof.context -> typ -> ctyp
val dest_ctyp: ctyp -> ctyp list
(*certified terms*)
val term_of: cterm -> term
val typ_of_cterm: cterm -> typ
val ctyp_of_cterm: cterm -> ctyp
val maxidx_of_cterm: cterm -> int
val global_cterm_of: theory -> term -> cterm
val cterm_of: Proof.context -> term -> cterm
val renamed_term: term -> cterm -> cterm
val dest_comb: cterm -> cterm * cterm
val dest_fun: cterm -> cterm
val dest_arg: cterm -> cterm
val dest_fun2: cterm -> cterm
val dest_arg1: cterm -> cterm
val dest_abs: string option -> cterm -> cterm * cterm
val rename_tvar: indexname -> ctyp -> ctyp
val var: indexname * ctyp -> cterm
val apply: cterm -> cterm -> cterm
val lambda_name: string * cterm -> cterm -> cterm
val lambda: cterm -> cterm -> cterm
val adjust_maxidx_cterm: int -> cterm -> cterm
val incr_indexes_cterm: int -> cterm -> cterm
val match: cterm * cterm ->
((indexname * sort) * ctyp) list * ((indexname * typ) * cterm) list
val first_order_match: cterm * cterm ->
((indexname * sort) * ctyp) list * ((indexname * typ) * cterm) list
(*theorems*)
val fold_terms: (term -> 'a -> 'a) -> thm -> 'a -> 'a
val fold_atomic_ctyps: (ctyp -> 'a -> 'a) -> thm -> 'a -> 'a
val fold_atomic_cterms: (cterm -> 'a -> 'a) -> thm -> 'a -> 'a
val terms_of_tpairs: (term * term) list -> term list
val full_prop_of: thm -> term
val theory_id_of_thm: thm -> Context.theory_id
val theory_name_of_thm: thm -> string
val maxidx_of: thm -> int
val maxidx_thm: thm -> int -> int
val shyps_of: thm -> sort Ord_List.T
val hyps_of: thm -> term list
val prop_of: thm -> term
val tpairs_of: thm -> (term * term) list
val concl_of: thm -> term
val prems_of: thm -> term list
val nprems_of: thm -> int
val no_prems: thm -> bool
val major_prem_of: thm -> term
val cprop_of: thm -> cterm
val cprem_of: thm -> int -> cterm
val chyps_of: thm -> cterm list
exception CONTEXT of string * ctyp list * cterm list * thm list * Context.generic option
val theory_of_cterm: cterm -> theory
val theory_of_thm: thm -> theory
val trim_context_cterm: cterm -> cterm
val trim_context: thm -> thm
val transfer_cterm: theory -> cterm -> cterm
val transfer: theory -> thm -> thm
val renamed_prop: term -> thm -> thm
val weaken: cterm -> thm -> thm
val weaken_sorts: sort list -> cterm -> cterm
val extra_shyps: thm -> sort list
val proof_bodies_of: thm list -> proof_body list
val proof_body_of: thm -> proof_body
val proof_of: thm -> proof
val join_proofs: thm list -> unit
val peek_status: thm -> {oracle: bool, unfinished: bool, failed: bool}
val future: thm future -> cterm -> thm
val derivation_name: thm -> string
val name_derivation: string -> thm -> thm
val axiom: theory -> string -> thm
val axioms_of: theory -> (string * thm) list
val get_tags: thm -> Properties.T
val map_tags: (Properties.T -> Properties.T) -> thm -> thm
val norm_proof: thm -> thm
val adjust_maxidx_thm: int -> thm -> thm
(*inference rules*)
val assume: cterm -> thm
val implies_intr: cterm -> thm -> thm
val implies_elim: thm -> thm -> thm
val forall_intr: cterm -> thm -> thm
val forall_elim: cterm -> thm -> thm
val reflexive: cterm -> thm
val symmetric: thm -> thm
val transitive: thm -> thm -> thm
val beta_conversion: bool -> conv
val eta_conversion: conv
val eta_long_conversion: conv
val abstract_rule: string -> cterm -> thm -> thm
val combination: thm -> thm -> thm
val equal_intr: thm -> thm -> thm
val equal_elim: thm -> thm -> thm
val flexflex_rule: Proof.context option -> thm -> thm Seq.seq
val generalize: string list * string list -> int -> thm -> thm
val instantiate: ((indexname * sort) * ctyp) list * ((indexname * typ) * cterm) list ->
thm -> thm
val instantiate_cterm: ((indexname * sort) * ctyp) list * ((indexname * typ) * cterm) list ->
cterm -> cterm
val trivial: cterm -> thm
val of_class: ctyp * class -> thm
val strip_shyps: thm -> thm
val unconstrainT: thm -> thm
val varifyT_global': (string * sort) list -> thm -> ((string * sort) * indexname) list * thm
val varifyT_global: thm -> thm
val legacy_freezeT: thm -> thm
val dest_state: thm * int -> (term * term) list * term list * term * term
val lift_rule: cterm -> thm -> thm
val incr_indexes: int -> thm -> thm
val assumption: Proof.context option -> int -> thm -> thm Seq.seq
val eq_assumption: int -> thm -> thm
val rotate_rule: int -> int -> thm -> thm
val permute_prems: int -> int -> thm -> thm
val bicompose: Proof.context option -> {flatten: bool, match: bool, incremented: bool} ->
bool * thm * int -> int -> thm -> thm Seq.seq
val biresolution: Proof.context option -> bool -> (bool * thm) list -> int -> thm -> thm Seq.seq
(*oracles*)
val extern_oracles: bool -> Proof.context -> (Markup.T * xstring) list
val add_oracle: binding * ('a -> cterm) -> theory -> (string * ('a -> thm)) * theory
end;
structure Thm: THM =
struct
(*** Certified terms and types ***)
(** certified types **)
abstype ctyp = Ctyp of {cert: Context.certificate, T: typ, maxidx: int, sorts: sort Ord_List.T}
with
fun typ_of (Ctyp {T, ...}) = T;
fun global_ctyp_of thy raw_T =
let
val T = Sign.certify_typ thy raw_T;
val maxidx = Term.maxidx_of_typ T;
val sorts = Sorts.insert_typ T [];
in Ctyp {cert = Context.Certificate thy, T = T, maxidx = maxidx, sorts = sorts} end;
val ctyp_of = global_ctyp_of o Proof_Context.theory_of;
fun dest_ctyp (Ctyp {cert, T = Type (_, Ts), maxidx, sorts}) =
map (fn T => Ctyp {cert = cert, T = T, maxidx = maxidx, sorts = sorts}) Ts
| dest_ctyp cT = raise TYPE ("dest_ctyp", [typ_of cT], []);
(** certified terms **)
(*certified terms with checked typ, maxidx, and sorts*)
abstype cterm =
Cterm of {cert: Context.certificate, t: term, T: typ, maxidx: int, sorts: sort Ord_List.T}
with
exception CTERM of string * cterm list;
fun term_of (Cterm {t, ...}) = t;
fun typ_of_cterm (Cterm {T, ...}) = T;
fun ctyp_of_cterm (Cterm {cert, T, maxidx, sorts, ...}) =
Ctyp {cert = cert, T = T, maxidx = maxidx, sorts = sorts};
fun maxidx_of_cterm (Cterm {maxidx, ...}) = maxidx;
fun global_cterm_of thy tm =
let
val (t, T, maxidx) = Sign.certify_term thy tm;
val sorts = Sorts.insert_term t [];
in Cterm {cert = Context.Certificate thy, t = t, T = T, maxidx = maxidx, sorts = sorts} end;
val cterm_of = global_cterm_of o Proof_Context.theory_of;
fun join_certificate0 (Cterm {cert = cert1, ...}, Cterm {cert = cert2, ...}) =
Context.join_certificate (cert1, cert2);
fun renamed_term t' (Cterm {cert, t, T, maxidx, sorts}) =
if t aconv t' then Cterm {cert = cert, t = t', T = T, maxidx = maxidx, sorts = sorts}
else raise TERM ("renamed_term: terms disagree", [t, t']);
(* destructors *)
fun dest_comb (Cterm {t = c $ a, T, cert, maxidx, sorts}) =
let val A = Term.argument_type_of c 0 in
(Cterm {t = c, T = A --> T, cert = cert, maxidx = maxidx, sorts = sorts},
Cterm {t = a, T = A, cert = cert, maxidx = maxidx, sorts = sorts})
end
| dest_comb ct = raise CTERM ("dest_comb", [ct]);
fun dest_fun (Cterm {t = c $ _, T, cert, maxidx, sorts}) =
let val A = Term.argument_type_of c 0
in Cterm {t = c, T = A --> T, cert = cert, maxidx = maxidx, sorts = sorts} end
| dest_fun ct = raise CTERM ("dest_fun", [ct]);
fun dest_arg (Cterm {t = c $ a, T = _, cert, maxidx, sorts}) =
let val A = Term.argument_type_of c 0
in Cterm {t = a, T = A, cert = cert, maxidx = maxidx, sorts = sorts} end
| dest_arg ct = raise CTERM ("dest_arg", [ct]);
fun dest_fun2 (Cterm {t = c $ _ $ _, T, cert, maxidx, sorts}) =
let
val A = Term.argument_type_of c 0;
val B = Term.argument_type_of c 1;
in Cterm {t = c, T = A --> B --> T, cert = cert, maxidx = maxidx, sorts = sorts} end
| dest_fun2 ct = raise CTERM ("dest_fun2", [ct]);
fun dest_arg1 (Cterm {t = c $ a $ _, T = _, cert, maxidx, sorts}) =
let val A = Term.argument_type_of c 0
in Cterm {t = a, T = A, cert = cert, maxidx = maxidx, sorts = sorts} end
| dest_arg1 ct = raise CTERM ("dest_arg1", [ct]);
fun dest_abs a (Cterm {t = Abs (x, T, t), T = Type ("fun", [_, U]), cert, maxidx, sorts}) =
let val (y', t') = Term.dest_abs (the_default x a, T, t) in
(Cterm {t = Free (y', T), T = T, cert = cert, maxidx = maxidx, sorts = sorts},
Cterm {t = t', T = U, cert = cert, maxidx = maxidx, sorts = sorts})
end
| dest_abs _ ct = raise CTERM ("dest_abs", [ct]);
(* constructors *)
fun rename_tvar (a, i) (Ctyp {cert, T, maxidx, sorts}) =
let
val S =
(case T of
TFree (_, S) => S
| TVar (_, S) => S
| _ => raise TYPE ("rename_tvar: no variable", [T], []));
val _ = if i < 0 then raise TYPE ("rename_tvar: bad index", [TVar ((a, i), S)], []) else ();
in Ctyp {cert = cert, T = TVar ((a, i), S), maxidx = Int.max (i, maxidx), sorts = sorts} end;
fun var ((x, i), Ctyp {cert, T, maxidx, sorts}) =
if i < 0 then raise TERM ("var: bad index", [Var ((x, i), T)])
else Cterm {cert = cert, t = Var ((x, i), T), T = T, maxidx = Int.max (i, maxidx), sorts = sorts};
fun apply
(cf as Cterm {t = f, T = Type ("fun", [dty, rty]), maxidx = maxidx1, sorts = sorts1, ...})
(cx as Cterm {t = x, T, maxidx = maxidx2, sorts = sorts2, ...}) =
if T = dty then
Cterm {cert = join_certificate0 (cf, cx),
t = f $ x,
T = rty,
maxidx = Int.max (maxidx1, maxidx2),
sorts = Sorts.union sorts1 sorts2}
else raise CTERM ("apply: types don't agree", [cf, cx])
| apply cf cx = raise CTERM ("apply: first arg is not a function", [cf, cx]);
fun lambda_name
(x, ct1 as Cterm {t = t1, T = T1, maxidx = maxidx1, sorts = sorts1, ...})
(ct2 as Cterm {t = t2, T = T2, maxidx = maxidx2, sorts = sorts2, ...}) =
let val t = Term.lambda_name (x, t1) t2 in
Cterm {cert = join_certificate0 (ct1, ct2),
t = t, T = T1 --> T2,
maxidx = Int.max (maxidx1, maxidx2),
sorts = Sorts.union sorts1 sorts2}
end;
fun lambda t u = lambda_name ("", t) u;
(* indexes *)
fun adjust_maxidx_cterm i (ct as Cterm {cert, t, T, maxidx, sorts}) =
if maxidx = i then ct
else if maxidx < i then
Cterm {maxidx = i, cert = cert, t = t, T = T, sorts = sorts}
else
Cterm {maxidx = Int.max (maxidx_of_term t, i), cert = cert, t = t, T = T, sorts = sorts};
fun incr_indexes_cterm i (ct as Cterm {cert, t, T, maxidx, sorts}) =
if i < 0 then raise CTERM ("negative increment", [ct])
else if i = 0 then ct
else Cterm {cert = cert, t = Logic.incr_indexes ([], [], i) t,
T = Logic.incr_tvar i T, maxidx = maxidx + i, sorts = sorts};
(*** Derivations and Theorems ***)
abstype thm = Thm of
deriv * (*derivation*)
{cert: Context.certificate, (*background theory certificate*)
tags: Properties.T, (*additional annotations/comments*)
maxidx: int, (*maximum index of any Var or TVar*)
shyps: sort Ord_List.T, (*sort hypotheses*)
hyps: term Ord_List.T, (*hypotheses*)
tpairs: (term * term) list, (*flex-flex pairs*)
prop: term} (*conclusion*)
and deriv = Deriv of
{promises: (serial * thm future) Ord_List.T,
body: Proofterm.proof_body}
with
type conv = cterm -> thm;
(*errors involving theorems*)
exception THM of string * int * thm list;
fun rep_thm (Thm (_, args)) = args;
fun fold_terms f (Thm (_, {tpairs, prop, hyps, ...})) =
fold (fn (t, u) => f t #> f u) tpairs #> f prop #> fold f hyps;
fun fold_atomic_ctyps f (th as Thm (_, {cert, maxidx, shyps, ...})) =
let fun ctyp T = Ctyp {cert = cert, T = T, maxidx = maxidx, sorts = shyps}
in (fold_terms o fold_types o fold_atyps) (f o ctyp) th end;
fun fold_atomic_cterms f (th as Thm (_, {cert, maxidx, shyps, ...})) =
let fun cterm t T = Cterm {cert = cert, t = t, T = T, maxidx = maxidx, sorts = shyps} in
(fold_terms o fold_aterms)
(fn t as Const (_, T) => f (cterm t T)
| t as Free (_, T) => f (cterm t T)
| t as Var (_, T) => f (cterm t T)
| _ => I) th
end;
fun terms_of_tpairs tpairs = fold_rev (fn (t, u) => cons t o cons u) tpairs [];
fun eq_tpairs ((t, u), (t', u')) = t aconv t' andalso u aconv u';
fun union_tpairs ts us = Library.merge eq_tpairs (ts, us);
val maxidx_tpairs = fold (fn (t, u) => Term.maxidx_term t #> Term.maxidx_term u);
fun attach_tpairs tpairs prop =
Logic.list_implies (map Logic.mk_equals tpairs, prop);
fun full_prop_of (Thm (_, {tpairs, prop, ...})) = attach_tpairs tpairs prop;
val union_hyps = Ord_List.union Term_Ord.fast_term_ord;
val insert_hyps = Ord_List.insert Term_Ord.fast_term_ord;
val remove_hyps = Ord_List.remove Term_Ord.fast_term_ord;
fun join_certificate1 (Cterm {cert = cert1, ...}, Thm (_, {cert = cert2, ...})) =
Context.join_certificate (cert1, cert2);
fun join_certificate2 (Thm (_, {cert = cert1, ...}), Thm (_, {cert = cert2, ...})) =
Context.join_certificate (cert1, cert2);
(* basic components *)
val cert_of = #cert o rep_thm;
val theory_id_of_thm = Context.certificate_theory_id o cert_of;
val theory_name_of_thm = Context.theory_id_name o theory_id_of_thm;
val maxidx_of = #maxidx o rep_thm;
fun maxidx_thm th i = Int.max (maxidx_of th, i);
val shyps_of = #shyps o rep_thm;
val hyps_of = #hyps o rep_thm;
val prop_of = #prop o rep_thm;
val tpairs_of = #tpairs o rep_thm;
val concl_of = Logic.strip_imp_concl o prop_of;
val prems_of = Logic.strip_imp_prems o prop_of;
val nprems_of = Logic.count_prems o prop_of;
fun no_prems th = nprems_of th = 0;
fun major_prem_of th =
(case prems_of th of
prem :: _ => Logic.strip_assums_concl prem
| [] => raise THM ("major_prem_of: rule with no premises", 0, [th]));
fun cprop_of (Thm (_, {cert, maxidx, shyps, prop, ...})) =
Cterm {cert = cert, maxidx = maxidx, T = propT, t = prop, sorts = shyps};
fun cprem_of (th as Thm (_, {cert, maxidx, shyps, prop, ...})) i =
Cterm {cert = cert, maxidx = maxidx, T = propT, sorts = shyps,
t = Logic.nth_prem (i, prop) handle TERM _ => raise THM ("cprem_of", i, [th])};
fun chyps_of (Thm (_, {cert, shyps, hyps, ...})) =
map (fn t => Cterm {cert = cert, maxidx = ~1, T = propT, sorts = shyps, t = t}) hyps;
(* implicit theory context *)
exception CONTEXT of string * ctyp list * cterm list * thm list * Context.generic option;
fun theory_of_cterm (ct as Cterm {cert, ...}) =
Context.certificate_theory cert
handle ERROR msg => raise CONTEXT (msg, [], [ct], [], NONE);
fun theory_of_thm th =
Context.certificate_theory (cert_of th)
handle ERROR msg => raise CONTEXT (msg, [], [], [th], NONE);
fun trim_context_cterm ct =
(case ct of
Cterm {cert = Context.Certificate_Id _, ...} => ct
| Cterm {cert = Context.Certificate thy, t, T, maxidx, sorts} =>
Cterm {cert = Context.Certificate_Id (Context.theory_id thy),
t = t, T = T, maxidx = maxidx, sorts = sorts});
fun trim_context th =
(case th of
Thm (_, {cert = Context.Certificate_Id _, ...}) => th
| Thm (der, {cert = Context.Certificate thy, tags, maxidx, shyps, hyps, tpairs, prop}) =>
Thm (der,
{cert = Context.Certificate_Id (Context.theory_id thy),
tags = tags, maxidx = maxidx, shyps = shyps, hyps = hyps, tpairs = tpairs, prop = prop}));
fun transfer_cterm thy' ct =
let
val Cterm {cert, t, T, maxidx, sorts} = ct;
val _ =
Context.subthy_id (Context.certificate_theory_id cert, Context.theory_id thy') orelse
raise CONTEXT ("Cannot transfer: not a super theory", [], [ct], [],
SOME (Context.Theory thy'));
val cert' = Context.join_certificate (Context.Certificate thy', cert);
in
if Context.eq_certificate (cert, cert') then ct
else Cterm {cert = cert', t = t, T = T, maxidx = maxidx, sorts = sorts}
end;
fun transfer thy' th =
let
val Thm (der, {cert, tags, maxidx, shyps, hyps, tpairs, prop}) = th;
val _ =
Context.subthy_id (Context.certificate_theory_id cert, Context.theory_id thy') orelse
raise CONTEXT ("Cannot transfer: not a super theory", [], [], [th],
SOME (Context.Theory thy'));
val cert' = Context.join_certificate (Context.Certificate thy', cert);
in
if Context.eq_certificate (cert, cert') then th
else
Thm (der,
{cert = cert',
tags = tags,
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
tpairs = tpairs,
prop = prop})
end;
(* matching *)
local
fun gen_match match
(ct1 as Cterm {t = t1, sorts = sorts1, ...},
ct2 as Cterm {t = t2, sorts = sorts2, maxidx = maxidx2, ...}) =
let
val cert = join_certificate0 (ct1, ct2);
val thy = Context.certificate_theory cert
handle ERROR msg => raise CONTEXT (msg, [], [ct1, ct2], [], NONE);
val (Tinsts, tinsts) = match thy (t1, t2) (Vartab.empty, Vartab.empty);
val sorts = Sorts.union sorts1 sorts2;
fun mk_cTinst ((a, i), (S, T)) =
(((a, i), S), Ctyp {T = T, cert = cert, maxidx = maxidx2, sorts = sorts});
fun mk_ctinst ((x, i), (U, t)) =
let val T = Envir.subst_type Tinsts U in
(((x, i), T), Cterm {t = t, T = T, cert = cert, maxidx = maxidx2, sorts = sorts})
end;
in (Vartab.fold (cons o mk_cTinst) Tinsts [], Vartab.fold (cons o mk_ctinst) tinsts []) end;
in
val match = gen_match Pattern.match;
val first_order_match = gen_match Pattern.first_order_match;
end;
(*implicit alpha-conversion*)
fun renamed_prop prop' (Thm (der, {cert, tags, maxidx, shyps, hyps, tpairs, prop})) =
if prop aconv prop' then
Thm (der, {cert = cert, tags = tags, maxidx = maxidx, shyps = shyps,
hyps = hyps, tpairs = tpairs, prop = prop'})
else raise TERM ("renamed_prop: props disagree", [prop, prop']);
fun make_context ths NONE cert =
(Context.Theory (Context.certificate_theory cert)
handle ERROR msg => raise CONTEXT (msg, [], [], ths, NONE))
| make_context ths (SOME ctxt) cert =
let
val thy_id = Context.certificate_theory_id cert;
val thy_id' = Context.theory_id (Proof_Context.theory_of ctxt);
in
if Context.subthy_id (thy_id, thy_id') then Context.Proof ctxt
else raise CONTEXT ("Bad context", [], [], ths, SOME (Context.Proof ctxt))
end;
(*explicit weakening: maps |- B to A |- B*)
fun weaken raw_ct th =
let
val ct as Cterm {t = A, T, sorts, maxidx = maxidxA, ...} = adjust_maxidx_cterm ~1 raw_ct;
val Thm (der, {tags, maxidx, shyps, hyps, tpairs, prop, ...}) = th;
in
if T <> propT then
raise THM ("weaken: assumptions must have type prop", 0, [])
else if maxidxA <> ~1 then
raise THM ("weaken: assumptions may not contain schematic variables", maxidxA, [])
else
Thm (der,
{cert = join_certificate1 (ct, th),
tags = tags,
maxidx = maxidx,
shyps = Sorts.union sorts shyps,
hyps = insert_hyps A hyps,
tpairs = tpairs,
prop = prop})
end;
fun weaken_sorts raw_sorts ct =
let
val Cterm {cert, t, T, maxidx, sorts} = ct;
val thy = theory_of_cterm ct;
val more_sorts = Sorts.make (map (Sign.certify_sort thy) raw_sorts);
val sorts' = Sorts.union sorts more_sorts;
in Cterm {cert = cert, t = t, T = T, maxidx = maxidx, sorts = sorts'} end;
(*dangling sort constraints of a thm*)
fun extra_shyps (th as Thm (_, {shyps, ...})) =
Sorts.subtract (fold_terms Sorts.insert_term th []) shyps;
(** derivations and promised proofs **)
fun make_deriv promises oracles thms proof =
Deriv {promises = promises, body = PBody {oracles = oracles, thms = thms, proof = proof}};
val empty_deriv = make_deriv [] [] [] Proofterm.MinProof;
(* inference rules *)
fun promise_ord ((i, _), (j, _)) = int_ord (j, i);
fun deriv_rule2 f
(Deriv {promises = ps1, body = PBody {oracles = oras1, thms = thms1, proof = prf1}})
(Deriv {promises = ps2, body = PBody {oracles = oras2, thms = thms2, proof = prf2}}) =
let
val ps = Ord_List.union promise_ord ps1 ps2;
val oras = Proofterm.unions_oracles [oras1, oras2];
val thms = Proofterm.unions_thms [thms1, thms2];
val prf =
(case ! Proofterm.proofs of
2 => f prf1 prf2
| 1 => MinProof
| 0 => MinProof
| i => error ("Illegal level of detail for proof objects: " ^ string_of_int i));
in make_deriv ps oras thms prf end;
fun deriv_rule1 f = deriv_rule2 (K f) empty_deriv;
fun deriv_rule0 prf = deriv_rule1 I (make_deriv [] [] [] prf);
fun deriv_rule_unconditional f (Deriv {promises, body = PBody {oracles, thms, proof}}) =
make_deriv promises oracles thms (f proof);
(* fulfilled proofs *)
fun raw_body_of (Thm (Deriv {body, ...}, _)) = body;
fun raw_promises_of (Thm (Deriv {promises, ...}, _)) = promises;
fun join_promises [] = ()
| join_promises promises = join_promises_of (Future.joins (map snd promises))
and join_promises_of thms = join_promises (Ord_List.make promise_ord (maps raw_promises_of thms));
fun fulfill_body (th as Thm (Deriv {promises, body}, _)) =
Proofterm.fulfill_norm_proof (theory_of_thm th) (fulfill_promises promises) body
and fulfill_promises promises =
map fst promises ~~ map fulfill_body (Future.joins (map snd promises));
fun proof_bodies_of thms =
let
val _ = join_promises_of thms;
val bodies = map fulfill_body thms;
val _ = Proofterm.join_bodies bodies;
in bodies end;
val proof_body_of = singleton proof_bodies_of;
val proof_of = Proofterm.proof_of o proof_body_of;
val join_proofs = ignore o proof_bodies_of;
(* derivation status *)
fun peek_status (Thm (Deriv {promises, body}, _)) =
let
val ps = map (Future.peek o snd) promises;
val bodies = body ::
map_filter (fn SOME (Exn.Res th) => SOME (raw_body_of th) | _ => NONE) ps;
val {oracle, unfinished, failed} = Proofterm.peek_status bodies;
in
{oracle = oracle,
unfinished = unfinished orelse exists is_none ps,
failed = failed orelse exists (fn SOME (Exn.Exn _) => true | _ => false) ps}
end;
(* future rule *)
fun future_result i orig_cert orig_shyps orig_prop thm =
let
fun err msg = raise THM ("future_result: " ^ msg, 0, [thm]);
val Thm (Deriv {promises, ...}, {cert, shyps, hyps, tpairs, prop, ...}) = thm;
val _ = Context.eq_certificate (cert, orig_cert) orelse err "bad theory";
val _ = prop aconv orig_prop orelse err "bad prop";
val _ = null tpairs orelse err "bad tpairs";
val _ = null hyps orelse err "bad hyps";
val _ = Sorts.subset (shyps, orig_shyps) orelse err "bad shyps";
val _ = forall (fn (j, _) => i <> j) promises orelse err "bad dependencies";
val _ = join_promises promises;
in thm end;
fun future future_thm ct =
let
val Cterm {cert = cert, t = prop, T, maxidx, sorts} = ct;
val _ = T <> propT andalso raise CTERM ("future: prop expected", [ct]);
val thy = theory_of_cterm ct;
val i = serial ();
val future = future_thm |> Future.map (future_result i cert sorts prop);
in
Thm (make_deriv [(i, future)] [] [] (Proofterm.promise_proof thy i prop),
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = prop})
end;
(* closed derivations with official name *)
(*non-deterministic, depends on unknown promises*)
fun derivation_name (Thm (Deriv {body, ...}, {shyps, hyps, prop, ...})) =
Proofterm.get_name shyps hyps prop (Proofterm.proof_of body);
fun name_derivation name (thm as Thm (der, args)) =
let
val Deriv {promises, body} = der;
val {shyps, hyps, prop, tpairs, ...} = args;
val _ = null tpairs orelse raise THM ("put_name: unsolved flex-flex constraints", 0, [thm]);
val thy = theory_of_thm thm;
val ps = map (apsnd (Future.map fulfill_body)) promises;
val (pthm, proof) = Proofterm.thm_proof thy name shyps hyps prop ps body;
val der' = make_deriv [] [] [pthm] proof;
in Thm (der', args) end;
(** Axioms **)
fun axiom thy0 name =
let
fun get_ax thy =
Name_Space.lookup (Theory.axiom_table thy) name
|> Option.map (fn prop =>
let
val der = deriv_rule0 (Proofterm.axm_proof name prop);
val cert = Context.Certificate thy;
val maxidx = maxidx_of_term prop;
val shyps = Sorts.insert_term prop [];
in
Thm (der, {cert = cert, tags = [],
maxidx = maxidx, shyps = shyps, hyps = [], tpairs = [], prop = prop})
end);
in
(case get_first get_ax (Theory.nodes_of thy0) of
SOME thm => thm
| NONE => raise THEORY ("No axiom " ^ quote name, [thy0]))
end;
(*return additional axioms of this theory node*)
fun axioms_of thy =
map (fn (name, _) => (name, axiom thy name)) (Theory.axioms_of thy);
(* tags *)
val get_tags = #tags o rep_thm;
fun map_tags f (Thm (der, {cert, tags, maxidx, shyps, hyps, tpairs, prop})) =
Thm (der, {cert = cert, tags = f tags, maxidx = maxidx,
shyps = shyps, hyps = hyps, tpairs = tpairs, prop = prop});
(* technical adjustments *)
fun norm_proof (th as Thm (der, args)) =
Thm (deriv_rule1 (Proofterm.rew_proof (theory_of_thm th)) der, args);
fun adjust_maxidx_thm i (th as Thm (der, {cert, tags, maxidx, shyps, hyps, tpairs, prop})) =
if maxidx = i then th
else if maxidx < i then
Thm (der, {maxidx = i, cert = cert, tags = tags, shyps = shyps,
hyps = hyps, tpairs = tpairs, prop = prop})
else
Thm (der, {maxidx = Int.max (maxidx_tpairs tpairs (maxidx_of_term prop), i),
cert = cert, tags = tags, shyps = shyps, hyps = hyps, tpairs = tpairs, prop = prop});
(*** Meta rules ***)
(** primitive rules **)
(*The assumption rule A |- A*)
fun assume raw_ct =
let val Cterm {cert, t = prop, T, maxidx, sorts} = adjust_maxidx_cterm ~1 raw_ct in
if T <> propT then
raise THM ("assume: prop", 0, [])
else if maxidx <> ~1 then
raise THM ("assume: variables", maxidx, [])
else Thm (deriv_rule0 (Proofterm.Hyp prop),
{cert = cert,
tags = [],
maxidx = ~1,
shyps = sorts,
hyps = [prop],
tpairs = [],
prop = prop})
end;
(*Implication introduction
[A]
:
B
-------
A ==> B
*)
fun implies_intr
(ct as Cterm {t = A, T, maxidx = maxidxA, sorts, ...})
(th as Thm (der, {maxidx, hyps, shyps, tpairs, prop, ...})) =
if T <> propT then
raise THM ("implies_intr: assumptions must have type prop", 0, [th])
else
Thm (deriv_rule1 (Proofterm.implies_intr_proof A) der,
{cert = join_certificate1 (ct, th),
tags = [],
maxidx = Int.max (maxidxA, maxidx),
shyps = Sorts.union sorts shyps,
hyps = remove_hyps A hyps,
tpairs = tpairs,
prop = Logic.mk_implies (A, prop)});
(*Implication elimination
A ==> B A
------------
B
*)
fun implies_elim thAB thA =
let
val Thm (derA, {maxidx = maxA, hyps = hypsA, shyps = shypsA, tpairs = tpairsA,
prop = propA, ...}) = thA
and Thm (der, {maxidx, hyps, shyps, tpairs, prop, ...}) = thAB;
fun err () = raise THM ("implies_elim: major premise", 0, [thAB, thA]);
in
case prop of
Const ("Pure.imp", _) $ A $ B =>
if A aconv propA then
Thm (deriv_rule2 (curry Proofterm.%%) der derA,
{cert = join_certificate2 (thAB, thA),
tags = [],
maxidx = Int.max (maxA, maxidx),
shyps = Sorts.union shypsA shyps,
hyps = union_hyps hypsA hyps,
tpairs = union_tpairs tpairsA tpairs,
prop = B})
else err ()
| _ => err ()
end;
(*Forall introduction. The Free or Var x must not be free in the hypotheses.
[x]
:
A
------
!!x. A
*)
fun forall_intr
(ct as Cterm {t = x, T, sorts, ...})
(th as Thm (der, {maxidx, shyps, hyps, tpairs, prop, ...})) =
let
fun result a =
Thm (deriv_rule1 (Proofterm.forall_intr_proof x a) der,
{cert = join_certificate1 (ct, th),
tags = [],
maxidx = maxidx,
shyps = Sorts.union sorts shyps,
hyps = hyps,
tpairs = tpairs,
prop = Logic.all_const T $ Abs (a, T, abstract_over (x, prop))});
fun check_occs a x ts =
if exists (fn t => Logic.occs (x, t)) ts then
raise THM ("forall_intr: variable " ^ quote a ^ " free in assumptions", 0, [th])
else ();
in
(case x of
Free (a, _) => (check_occs a x hyps; check_occs a x (terms_of_tpairs tpairs); result a)
| Var ((a, _), _) => (check_occs a x (terms_of_tpairs tpairs); result a)
| _ => raise THM ("forall_intr: not a variable", 0, [th]))
end;
(*Forall elimination
!!x. A
------
A[t/x]
*)
fun forall_elim
(ct as Cterm {t, T, maxidx = maxt, sorts, ...})
(th as Thm (der, {maxidx, shyps, hyps, tpairs, prop, ...})) =
(case prop of
Const ("Pure.all", Type ("fun", [Type ("fun", [qary, _]), _])) $ A =>
if T <> qary then
raise THM ("forall_elim: type mismatch", 0, [th])
else
Thm (deriv_rule1 (Proofterm.% o rpair (SOME t)) der,
{cert = join_certificate1 (ct, th),
tags = [],
maxidx = Int.max (maxidx, maxt),
shyps = Sorts.union sorts shyps,
hyps = hyps,
tpairs = tpairs,
prop = Term.betapply (A, t)})
| _ => raise THM ("forall_elim: not quantified", 0, [th]));
(* Equality *)
(*Reflexivity
t == t
*)
fun reflexive (Cterm {cert, t, T = _, maxidx, sorts}) =
Thm (deriv_rule0 Proofterm.reflexive,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_equals (t, t)});
(*Symmetry
t == u
------
u == t
*)
fun symmetric (th as Thm (der, {cert, maxidx, shyps, hyps, tpairs, prop, ...})) =
(case prop of
(eq as Const ("Pure.eq", _)) $ t $ u =>
Thm (deriv_rule1 Proofterm.symmetric der,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
tpairs = tpairs,
prop = eq $ u $ t})
| _ => raise THM ("symmetric", 0, [th]));
(*Transitivity
t1 == u u == t2
------------------
t1 == t2
*)
fun transitive th1 th2 =
let
val Thm (der1, {maxidx = max1, hyps = hyps1, shyps = shyps1, tpairs = tpairs1,
prop = prop1, ...}) = th1
and Thm (der2, {maxidx = max2, hyps = hyps2, shyps = shyps2, tpairs = tpairs2,
prop = prop2, ...}) = th2;
fun err msg = raise THM ("transitive: " ^ msg, 0, [th1, th2]);
in
case (prop1, prop2) of
((eq as Const ("Pure.eq", Type (_, [T, _]))) $ t1 $ u, Const ("Pure.eq", _) $ u' $ t2) =>
if not (u aconv u') then err "middle term"
else
Thm (deriv_rule2 (Proofterm.transitive u T) der1 der2,
{cert = join_certificate2 (th1, th2),
tags = [],
maxidx = Int.max (max1, max2),
shyps = Sorts.union shyps1 shyps2,
hyps = union_hyps hyps1 hyps2,
tpairs = union_tpairs tpairs1 tpairs2,
prop = eq $ t1 $ t2})
| _ => err "premises"
end;
(*Beta-conversion
(%x. t)(u) == t[u/x]
fully beta-reduces the term if full = true
*)
fun beta_conversion full (Cterm {cert, t, T = _, maxidx, sorts}) =
let val t' =
if full then Envir.beta_norm t
else
(case t of Abs (_, _, bodt) $ u => subst_bound (u, bodt)
| _ => raise THM ("beta_conversion: not a redex", 0, []));
in
Thm (deriv_rule0 Proofterm.reflexive,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_equals (t, t')})
end;
fun eta_conversion (Cterm {cert, t, T = _, maxidx, sorts}) =
Thm (deriv_rule0 Proofterm.reflexive,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_equals (t, Envir.eta_contract t)});
fun eta_long_conversion (Cterm {cert, t, T = _, maxidx, sorts}) =
Thm (deriv_rule0 Proofterm.reflexive,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_equals (t, Envir.eta_long [] t)});
(*The abstraction rule. The Free or Var x must not be free in the hypotheses.
The bound variable will be named "a" (since x will be something like x320)
t == u
--------------
%x. t == %x. u
*)
fun abstract_rule a
(Cterm {t = x, T, sorts, ...})
(th as Thm (der, {cert, maxidx, hyps, shyps, tpairs, prop, ...})) =
let
val (t, u) = Logic.dest_equals prop
handle TERM _ => raise THM ("abstract_rule: premise not an equality", 0, [th]);
val result =
Thm (deriv_rule1 (Proofterm.abstract_rule x a) der,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = Sorts.union sorts shyps,
hyps = hyps,
tpairs = tpairs,
prop = Logic.mk_equals
(Abs (a, T, abstract_over (x, t)), Abs (a, T, abstract_over (x, u)))});
fun check_occs a x ts =
if exists (fn t => Logic.occs (x, t)) ts then
raise THM ("abstract_rule: variable " ^ quote a ^ " free in assumptions", 0, [th])
else ();
in
(case x of
Free (a, _) => (check_occs a x hyps; check_occs a x (terms_of_tpairs tpairs); result)
| Var ((a, _), _) => (check_occs a x (terms_of_tpairs tpairs); result)
| _ => raise THM ("abstract_rule: not a variable", 0, [th]))
end;
(*The combination rule
f == g t == u
--------------
f t == g u
*)
fun combination th1 th2 =
let
val Thm (der1, {maxidx = max1, shyps = shyps1, hyps = hyps1, tpairs = tpairs1,
prop = prop1, ...}) = th1
and Thm (der2, {maxidx = max2, shyps = shyps2, hyps = hyps2, tpairs = tpairs2,
prop = prop2, ...}) = th2;
fun chktypes fT tT =
(case fT of
Type ("fun", [T1, _]) =>
if T1 <> tT then
raise THM ("combination: types", 0, [th1, th2])
else ()
| _ => raise THM ("combination: not function type", 0, [th1, th2]));
in
(case (prop1, prop2) of
(Const ("Pure.eq", Type ("fun", [fT, _])) $ f $ g,
Const ("Pure.eq", Type ("fun", [tT, _])) $ t $ u) =>
(chktypes fT tT;
Thm (deriv_rule2 (Proofterm.combination f g t u fT) der1 der2,
{cert = join_certificate2 (th1, th2),
tags = [],
maxidx = Int.max (max1, max2),
shyps = Sorts.union shyps1 shyps2,
hyps = union_hyps hyps1 hyps2,
tpairs = union_tpairs tpairs1 tpairs2,
prop = Logic.mk_equals (f $ t, g $ u)}))
| _ => raise THM ("combination: premises", 0, [th1, th2]))
end;
(*Equality introduction
A ==> B B ==> A
----------------
A == B
*)
fun equal_intr th1 th2 =
let
val Thm (der1, {maxidx = max1, shyps = shyps1, hyps = hyps1, tpairs = tpairs1,
prop = prop1, ...}) = th1
and Thm (der2, {maxidx = max2, shyps = shyps2, hyps = hyps2, tpairs = tpairs2,
prop = prop2, ...}) = th2;
fun err msg = raise THM ("equal_intr: " ^ msg, 0, [th1, th2]);
in
(case (prop1, prop2) of
(Const("Pure.imp", _) $ A $ B, Const("Pure.imp", _) $ B' $ A') =>
if A aconv A' andalso B aconv B' then
Thm (deriv_rule2 (Proofterm.equal_intr A B) der1 der2,
{cert = join_certificate2 (th1, th2),
tags = [],
maxidx = Int.max (max1, max2),
shyps = Sorts.union shyps1 shyps2,
hyps = union_hyps hyps1 hyps2,
tpairs = union_tpairs tpairs1 tpairs2,
prop = Logic.mk_equals (A, B)})
else err "not equal"
| _ => err "premises")
end;
(*The equal propositions rule
A == B A
---------
B
*)
fun equal_elim th1 th2 =
let
val Thm (der1, {maxidx = max1, shyps = shyps1, hyps = hyps1,
tpairs = tpairs1, prop = prop1, ...}) = th1
and Thm (der2, {maxidx = max2, shyps = shyps2, hyps = hyps2,
tpairs = tpairs2, prop = prop2, ...}) = th2;
fun err msg = raise THM ("equal_elim: " ^ msg, 0, [th1, th2]);
in
(case prop1 of
Const ("Pure.eq", _) $ A $ B =>
if prop2 aconv A then
Thm (deriv_rule2 (Proofterm.equal_elim A B) der1 der2,
{cert = join_certificate2 (th1, th2),
tags = [],
maxidx = Int.max (max1, max2),
shyps = Sorts.union shyps1 shyps2,
hyps = union_hyps hyps1 hyps2,
tpairs = union_tpairs tpairs1 tpairs2,
prop = B})
else err "not equal"
| _ => err "major premise")
end;
(**** Derived rules ****)
(*Smash unifies the list of term pairs leaving no flex-flex pairs.
Instantiates the theorem and deletes trivial tpairs. Resulting
sequence may contain multiple elements if the tpairs are not all
flex-flex.*)
fun flexflex_rule opt_ctxt (th as Thm (der, {cert, maxidx, shyps, hyps, tpairs, prop, ...})) =
Unify.smash_unifiers (make_context [th] opt_ctxt cert) tpairs (Envir.empty maxidx)
|> Seq.map (fn env =>
if Envir.is_empty env then th
else
let
val tpairs' = tpairs |> map (apply2 (Envir.norm_term env))
(*remove trivial tpairs, of the form t==t*)
|> filter_out (op aconv);
val der' = deriv_rule1 (Proofterm.norm_proof' env) der;
val prop' = Envir.norm_term env prop;
val maxidx = maxidx_tpairs tpairs' (maxidx_of_term prop');
val shyps = Envir.insert_sorts env shyps;
in
Thm (der', {cert = cert, tags = [], maxidx = maxidx,
shyps = shyps, hyps = hyps, tpairs = tpairs', prop = prop'})
end);
(*Generalization of fixed variables
A
--------------------
A[?'a/'a, ?x/x, ...]
*)
fun generalize ([], []) _ th = th
| generalize (tfrees, frees) idx th =
let
val Thm (der, {cert, maxidx, shyps, hyps, tpairs, prop, ...}) = th;
val _ = idx <= maxidx andalso raise THM ("generalize: bad index", idx, [th]);
val bad_type =
if null tfrees then K false
else Term.exists_subtype (fn TFree (a, _) => member (op =) tfrees a | _ => false);
fun bad_term (Free (x, T)) = bad_type T orelse member (op =) frees x
| bad_term (Var (_, T)) = bad_type T
| bad_term (Const (_, T)) = bad_type T
| bad_term (Abs (_, T, t)) = bad_type T orelse bad_term t
| bad_term (t $ u) = bad_term t orelse bad_term u
| bad_term (Bound _) = false;
val _ = exists bad_term hyps andalso
raise THM ("generalize: variable free in assumptions", 0, [th]);
val gen = Term_Subst.generalize (tfrees, frees) idx;
val prop' = gen prop;
val tpairs' = map (apply2 gen) tpairs;
val maxidx' = maxidx_tpairs tpairs' (maxidx_of_term prop');
in
Thm (deriv_rule1 (Proofterm.generalize (tfrees, frees) idx) der,
{cert = cert,
tags = [],
maxidx = maxidx',
shyps = shyps,
hyps = hyps,
tpairs = tpairs',
prop = prop'})
end;
(*Instantiation of schematic variables
A
--------------------
A[t1/v1, ..., tn/vn]
*)
local
fun pretty_typing thy t T = Pretty.block
[Syntax.pretty_term_global thy t, Pretty.str " ::", Pretty.brk 1, Syntax.pretty_typ_global thy T];
fun add_inst (v as (_, T), cu) (cert, sorts) =
let
val Cterm {t = u, T = U, cert = cert2, sorts = sorts_u, maxidx = maxidx_u, ...} = cu;
val cert' = Context.join_certificate (cert, cert2);
val sorts' = Sorts.union sorts_u sorts;
in
if T = U then ((v, (u, maxidx_u)), (cert', sorts'))
else
let
val msg =
(case cert' of
Context.Certificate thy' =>
Pretty.string_of (Pretty.block
[Pretty.str "instantiate: type conflict",
Pretty.fbrk, pretty_typing thy' (Var v) T,
Pretty.fbrk, pretty_typing thy' u U])
| Context.Certificate_Id _ => "instantiate: type conflict");
in raise TYPE (msg, [T, U], [Var v, u]) end
end;
fun add_instT (v as (_, S), cU) (cert, sorts) =
let
val Ctyp {T = U, cert = cert2, sorts = sorts_U, maxidx = maxidx_U, ...} = cU;
val cert' = Context.join_certificate (cert, cert2);
val thy' = Context.certificate_theory cert'
handle ERROR msg => raise CONTEXT (msg, [cU], [], [], NONE);
val sorts' = Sorts.union sorts_U sorts;
in
if Sign.of_sort thy' (U, S) then ((v, (U, maxidx_U)), (cert', sorts'))
else raise TYPE ("Type not of sort " ^ Syntax.string_of_sort_global thy' S, [U], [])
end;
in
(*Left-to-right replacements: ctpairs = [..., (vi, ti), ...].
Instantiates distinct Vars by terms of same type.
Does NOT normalize the resulting theorem!*)
fun instantiate ([], []) th = th
| instantiate (instT, inst) th =
let
val Thm (der, {cert, hyps, shyps, tpairs, prop, ...}) = th;
val (inst', (instT', (cert', shyps'))) =
(cert, shyps) |> fold_map add_inst inst ||> fold_map add_instT instT
handle CONTEXT (msg, cTs, cts, ths, context) =>
raise CONTEXT (msg, cTs, cts, th :: ths, context);
val subst = Term_Subst.instantiate_maxidx (instT', inst');
val (prop', maxidx1) = subst prop ~1;
val (tpairs', maxidx') =
fold_map (fn (t, u) => fn i => subst t i ||>> subst u) tpairs maxidx1;
in
Thm (deriv_rule1
(fn d => Proofterm.instantiate (map (apsnd #1) instT', map (apsnd #1) inst') d) der,
{cert = cert',
tags = [],
maxidx = maxidx',
shyps = shyps',
hyps = hyps,
tpairs = tpairs',
prop = prop'})
end
handle TYPE (msg, _, _) => raise THM (msg, 0, [th]);
fun instantiate_cterm ([], []) ct = ct
| instantiate_cterm (instT, inst) ct =
let
val Cterm {cert, t, T, sorts, ...} = ct;
val (inst', (instT', (cert', sorts'))) =
(cert, sorts) |> fold_map add_inst inst ||> fold_map add_instT instT;
val subst = Term_Subst.instantiate_maxidx (instT', inst');
val substT = Term_Subst.instantiateT_maxidx instT';
val (t', maxidx1) = subst t ~1;
val (T', maxidx') = substT T maxidx1;
in Cterm {cert = cert', t = t', T = T', sorts = sorts', maxidx = maxidx'} end
handle TYPE (msg, _, _) => raise CTERM (msg, [ct]);
end;
(*The trivial implication A ==> A, justified by assume and forall rules.
A can contain Vars, not so for assume!*)
fun trivial (Cterm {cert, t = A, T, maxidx, sorts}) =
if T <> propT then
raise THM ("trivial: the term must have type prop", 0, [])
else
Thm (deriv_rule0 (Proofterm.AbsP ("H", NONE, Proofterm.PBound 0)),
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_implies (A, A)});
(*Axiom-scheme reflecting signature contents
T :: c
-------------------
OFCLASS(T, c_class)
*)
fun of_class (cT, raw_c) =
let
val Ctyp {cert, T, ...} = cT;
val thy = Context.certificate_theory cert
handle ERROR msg => raise CONTEXT (msg, [cT], [], [], NONE);
val c = Sign.certify_class thy raw_c;
val Cterm {t = prop, maxidx, sorts, ...} = global_cterm_of thy (Logic.mk_of_class (T, c));
in
if Sign.of_sort thy (T, [c]) then
Thm (deriv_rule0 (Proofterm.OfClass (T, c)),
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = prop})
else raise THM ("of_class: type not of class " ^ Syntax.string_of_sort_global thy [c], 0, [])
end;
(*Remove extra sorts that are witnessed by type signature information*)
fun strip_shyps (thm as Thm (_, {shyps = [], ...})) = thm
| strip_shyps (thm as Thm (der, {cert, tags, maxidx, shyps, hyps, tpairs, prop})) =
let
val thy = theory_of_thm thm;
val algebra = Sign.classes_of thy;
val present = (fold_terms o fold_types o fold_atyps_sorts) (insert (eq_fst op =)) thm [];
val extra = fold (Sorts.remove_sort o #2) present shyps;
val witnessed = Sign.witness_sorts thy present extra;
val extra' = fold (Sorts.remove_sort o #2) witnessed extra
|> Sorts.minimal_sorts algebra;
val shyps' = fold (Sorts.insert_sort o #2) present extra';
in
Thm (deriv_rule_unconditional
(Proofterm.strip_shyps_proof algebra present witnessed extra') der,
{cert = cert, tags = tags, maxidx = maxidx,
shyps = shyps', hyps = hyps, tpairs = tpairs, prop = prop})
end;
(*Internalize sort constraints of type variables*)
fun unconstrainT (thm as Thm (der, args)) =
let
val Deriv {promises, body} = der;
val {cert, shyps, hyps, tpairs, prop, ...} = args;
val thy = theory_of_thm thm;
fun err msg = raise THM ("unconstrainT: " ^ msg, 0, [thm]);
val _ = null hyps orelse err "illegal hyps";
val _ = null tpairs orelse err "unsolved flex-flex constraints";
val tfrees = rev (Term.add_tfree_names prop []);
val _ = null tfrees orelse err ("illegal free type variables " ^ commas_quote tfrees);
val ps = map (apsnd (Future.map fulfill_body)) promises;
val (pthm as (_, (_, prop', _)), proof) =
Proofterm.unconstrain_thm_proof thy shyps prop ps body;
val der' = make_deriv [] [] [pthm] proof;
in
Thm (der',
{cert = cert,
tags = [],
maxidx = maxidx_of_term prop',
shyps = [[]], (*potentially redundant*)
hyps = [],
tpairs = [],
prop = prop'})
end;
(* Replace all TFrees not fixed or in the hyps by new TVars *)
fun varifyT_global' fixed (Thm (der, {cert, maxidx, shyps, hyps, tpairs, prop, ...})) =
let
val tfrees = fold Term.add_tfrees hyps fixed;
val prop1 = attach_tpairs tpairs prop;
val (al, prop2) = Type.varify_global tfrees prop1;
val (ts, prop3) = Logic.strip_prems (length tpairs, [], prop2);
in
(al, Thm (deriv_rule1 (Proofterm.varify_proof prop tfrees) der,
{cert = cert,
tags = [],
maxidx = Int.max (0, maxidx),
shyps = shyps,
hyps = hyps,
tpairs = rev (map Logic.dest_equals ts),
prop = prop3}))
end;
val varifyT_global = #2 o varifyT_global' [];
(* Replace all TVars by TFrees that are often new *)
fun legacy_freezeT (Thm (der, {cert, shyps, hyps, tpairs, prop, ...})) =
let
val prop1 = attach_tpairs tpairs prop;
val prop2 = Type.legacy_freeze prop1;
val (ts, prop3) = Logic.strip_prems (length tpairs, [], prop2);
in
Thm (deriv_rule1 (Proofterm.legacy_freezeT prop1) der,
{cert = cert,
tags = [],
maxidx = maxidx_of_term prop2,
shyps = shyps,
hyps = hyps,
tpairs = rev (map Logic.dest_equals ts),
prop = prop3})
end;
(*** Inference rules for tactics ***)
(*Destruct proof state into constraints, other goals, goal(i), rest *)
fun dest_state (state as Thm (_, {prop,tpairs,...}), i) =
(case Logic.strip_prems(i, [], prop) of
(B::rBs, C) => (tpairs, rev rBs, B, C)
| _ => raise THM("dest_state", i, [state]))
handle TERM _ => raise THM("dest_state", i, [state]);
(*Prepare orule for resolution by lifting it over the parameters and
assumptions of goal.*)
fun lift_rule goal orule =
let
val Cterm {t = gprop, T, maxidx = gmax, sorts, ...} = goal;
val inc = gmax + 1;
val lift_abs = Logic.lift_abs inc gprop;
val lift_all = Logic.lift_all inc gprop;
val Thm (der, {maxidx, shyps, hyps, tpairs, prop, ...}) = orule;
val (As, B) = Logic.strip_horn prop;
in
if T <> propT then raise THM ("lift_rule: the term must have type prop", 0, [])
else
Thm (deriv_rule1 (Proofterm.lift_proof gprop inc prop) der,
{cert = join_certificate1 (goal, orule),
tags = [],
maxidx = maxidx + inc,
shyps = Sorts.union shyps sorts, (*sic!*)
hyps = hyps,
tpairs = map (apply2 lift_abs) tpairs,
prop = Logic.list_implies (map lift_all As, lift_all B)})
end;
fun incr_indexes i (thm as Thm (der, {cert, maxidx, shyps, hyps, tpairs, prop, ...})) =
if i < 0 then raise THM ("negative increment", 0, [thm])
else if i = 0 then thm
else
Thm (deriv_rule1 (Proofterm.incr_indexes i) der,
{cert = cert,
tags = [],
maxidx = maxidx + i,
shyps = shyps,
hyps = hyps,
tpairs = map (apply2 (Logic.incr_indexes ([], [], i))) tpairs,
prop = Logic.incr_indexes ([], [], i) prop});
(*Solve subgoal Bi of proof state B1...Bn/C by assumption. *)
fun assumption opt_ctxt i state =
let
val Thm (der, {cert, maxidx, shyps, hyps, ...}) = state;
val context = make_context [state] opt_ctxt cert;
val (tpairs, Bs, Bi, C) = dest_state (state, i);
fun newth n (env, tpairs) =
Thm (deriv_rule1
((if Envir.is_empty env then I else (Proofterm.norm_proof' env)) o
Proofterm.assumption_proof Bs Bi n) der,
{tags = [],
maxidx = Envir.maxidx_of env,
shyps = Envir.insert_sorts env shyps,
hyps = hyps,
tpairs =
if Envir.is_empty env then tpairs
else map (apply2 (Envir.norm_term env)) tpairs,
prop =
if Envir.is_empty env then (*avoid wasted normalizations*)
Logic.list_implies (Bs, C)
else (*normalize the new rule fully*)
Envir.norm_term env (Logic.list_implies (Bs, C)),
cert = cert});
val (close, asms, concl) = Logic.assum_problems (~1, Bi);
val concl' = close concl;
fun addprfs [] _ = Seq.empty
| addprfs (asm :: rest) n = Seq.make (fn () => Seq.pull
(Seq.mapp (newth n)
(if Term.could_unify (asm, concl) then
(Unify.unifiers (context, Envir.empty maxidx, (close asm, concl') :: tpairs))
else Seq.empty)
(addprfs rest (n + 1))))
in addprfs asms 1 end;
(*Solve subgoal Bi of proof state B1...Bn/C by assumption.
Checks if Bi's conclusion is alpha/eta-convertible to one of its assumptions*)
fun eq_assumption i state =
let
val Thm (der, {cert, maxidx, shyps, hyps, ...}) = state;
val (tpairs, Bs, Bi, C) = dest_state (state, i);
val (_, asms, concl) = Logic.assum_problems (~1, Bi);
in
(case find_index (fn asm => Envir.aeconv (asm, concl)) asms of
~1 => raise THM ("eq_assumption", 0, [state])
| n =>
Thm (deriv_rule1 (Proofterm.assumption_proof Bs Bi (n + 1)) der,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
tpairs = tpairs,
prop = Logic.list_implies (Bs, C)}))
end;
(*For rotate_tac: fast rotation of assumptions of subgoal i*)
fun rotate_rule k i state =
let
val Thm (der, {cert, maxidx, shyps, hyps, ...}) = state;
val (tpairs, Bs, Bi, C) = dest_state (state, i);
val params = Term.strip_all_vars Bi;
val rest = Term.strip_all_body Bi;
val asms = Logic.strip_imp_prems rest
val concl = Logic.strip_imp_concl rest;
val n = length asms;
val m = if k < 0 then n + k else k;
val Bi' =
if 0 = m orelse m = n then Bi
else if 0 < m andalso m < n then
let val (ps, qs) = chop m asms
in Logic.list_all (params, Logic.list_implies (qs @ ps, concl)) end
else raise THM ("rotate_rule", k, [state]);
in
Thm (deriv_rule1 (Proofterm.rotate_proof Bs Bi m) der,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
tpairs = tpairs,
prop = Logic.list_implies (Bs @ [Bi'], C)})
end;
(*Rotates a rule's premises to the left by k, leaving the first j premises
unchanged. Does nothing if k=0 or if k equals n-j, where n is the
number of premises. Useful with eresolve_tac and underlies defer_tac*)
fun permute_prems j k rl =
let
val Thm (der, {cert, maxidx, shyps, hyps, tpairs, prop, ...}) = rl;
val prems = Logic.strip_imp_prems prop
and concl = Logic.strip_imp_concl prop;
val moved_prems = List.drop (prems, j)
and fixed_prems = List.take (prems, j)
handle General.Subscript => raise THM ("permute_prems: j", j, [rl]);
val n_j = length moved_prems;
val m = if k < 0 then n_j + k else k;
val prop' =
if 0 = m orelse m = n_j then prop
else if 0 < m andalso m < n_j then
let val (ps, qs) = chop m moved_prems
in Logic.list_implies (fixed_prems @ qs @ ps, concl) end
else raise THM ("permute_prems: k", k, [rl]);
in
Thm (deriv_rule1 (Proofterm.permute_prems_proof prems j m) der,
{cert = cert,
tags = [],
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
tpairs = tpairs,
prop = prop'})
end;
(* strip_apply f B A strips off all assumptions/parameters from A
introduced by lifting over B, and applies f to remaining part of A*)
fun strip_apply f =
let fun strip (Const ("Pure.imp", _) $ _ $ B1)
(Const ("Pure.imp", _) $ A2 $ B2) = Logic.mk_implies (A2, strip B1 B2)
| strip ((c as Const ("Pure.all", _)) $ Abs (_, _, t1))
( Const ("Pure.all", _) $ Abs (a, T, t2)) = c $ Abs (a, T, strip t1 t2)
| strip _ A = f A
in strip end;
fun strip_lifted (Const ("Pure.imp", _) $ _ $ B1)
(Const ("Pure.imp", _) $ _ $ B2) = strip_lifted B1 B2
| strip_lifted (Const ("Pure.all", _) $ Abs (_, _, t1))
(Const ("Pure.all", _) $ Abs (_, _, t2)) = strip_lifted t1 t2
| strip_lifted _ A = A;
(*Use the alist to rename all bound variables and some unknowns in a term
dpairs = current disagreement pairs; tpairs = permanent ones (flexflex);
Preserves unknowns in tpairs and on lhs of dpairs. *)
fun rename_bvs [] _ _ _ _ = K I
| rename_bvs al dpairs tpairs B As =
let
val add_var = fold_aterms (fn Var ((x, _), _) => insert (op =) x | _ => I);
val vids = []
|> fold (add_var o fst) dpairs
|> fold (add_var o fst) tpairs
|> fold (add_var o snd) tpairs;
val vids' = fold (add_var o strip_lifted B) As [];
(*unknowns appearing elsewhere be preserved!*)
val al' = distinct ((op =) o apply2 fst)
(filter_out (fn (x, y) =>
not (member (op =) vids' x) orelse
member (op =) vids x orelse member (op =) vids y) al);
val unchanged = filter_out (AList.defined (op =) al') vids';
fun del_clashing clash xs _ [] qs =
if clash then del_clashing false xs xs qs [] else qs
| del_clashing clash xs ys ((p as (x, y)) :: ps) qs =
if member (op =) ys y
then del_clashing true (x :: xs) (x :: ys) ps qs
else del_clashing clash xs (y :: ys) ps (p :: qs);
val al'' = del_clashing false unchanged unchanged al' [];
fun rename (t as Var ((x, i), T)) =
(case AList.lookup (op =) al'' x of
SOME y => Var ((y, i), T)
| NONE => t)
| rename (Abs (x, T, t)) =
Abs (the_default x (AList.lookup (op =) al x), T, rename t)
| rename (f $ t) = rename f $ rename t
| rename t = t;
fun strip_ren f Ai = f rename B Ai
in strip_ren end;
(*Function to rename bounds/unknowns in the argument, lifted over B*)
fun rename_bvars dpairs =
rename_bvs (fold_rev Term.match_bvars dpairs []) dpairs;
(*** RESOLUTION ***)
(** Lifting optimizations **)
(*strip off pairs of assumptions/parameters in parallel -- they are
identical because of lifting*)
fun strip_assums2 (Const("Pure.imp", _) $ _ $ B1,
Const("Pure.imp", _) $ _ $ B2) = strip_assums2 (B1,B2)
| strip_assums2 (Const("Pure.all",_)$Abs(a,T,t1),
Const("Pure.all",_)$Abs(_,_,t2)) =
let val (B1,B2) = strip_assums2 (t1,t2)
in (Abs(a,T,B1), Abs(a,T,B2)) end
| strip_assums2 BB = BB;
(*Faster normalization: skip assumptions that were lifted over*)
fun norm_term_skip env 0 t = Envir.norm_term env t
| norm_term_skip env n (Const ("Pure.all", _) $ Abs (a, T, t)) =
let
val T' = Envir.norm_type (Envir.type_env env) T
(*Must instantiate types of parameters because they are flattened;
this could be a NEW parameter*)
in Logic.all_const T' $ Abs (a, T', norm_term_skip env n t) end
| norm_term_skip env n (Const ("Pure.imp", _) $ A $ B) =
Logic.mk_implies (A, norm_term_skip env (n - 1) B)
| norm_term_skip _ _ _ = error "norm_term_skip: too few assumptions??";
(*unify types of schematic variables (non-lifted case)*)
fun unify_var_types context (th1, th2) env =
let
fun unify_vars (T :: Us) = fold (fn U => Pattern.unify_types context (T, U)) Us
| unify_vars _ = I;
val add_vars =
full_prop_of #>
fold_aterms (fn Var v => Vartab.insert_list (op =) v | _ => I);
val vars = Vartab.empty |> add_vars th1 |> add_vars th2;
in SOME (Vartab.fold (unify_vars o #2) vars env) end
handle Pattern.Unif => NONE;
(*Composition of object rule r=(A1...Am/B) with proof state s=(B1...Bn/C)
Unifies B with Bi, replacing subgoal i (1 <= i <= n)
If match then forbid instantiations in proof state
If lifted then shorten the dpair using strip_assums2.
If eres_flg then simultaneously proves A1 by assumption.
nsubgoal is the number of new subgoals (written m above).
Curried so that resolution calls dest_state only once.
*)
local exception COMPOSE
in
fun bicompose_aux opt_ctxt {flatten, match, incremented} (state, (stpairs, Bs, Bi, C), lifted)
(eres_flg, orule, nsubgoal) =
let val Thm (sder, {maxidx=smax, shyps=sshyps, hyps=shyps, ...}) = state
and Thm (rder, {maxidx=rmax, shyps=rshyps, hyps=rhyps,
tpairs=rtpairs, prop=rprop,...}) = orule
(*How many hyps to skip over during normalization*)
and nlift = Logic.count_prems (strip_all_body Bi) + (if eres_flg then ~1 else 0)
val cert = join_certificate2 (state, orule);
val context = make_context [state, orule] opt_ctxt cert;
(** Add new theorem with prop = '[| Bs; As |] ==> C' to thq **)
fun addth A (As, oldAs, rder', n) ((env, tpairs), thq) =
let val normt = Envir.norm_term env;
(*perform minimal copying here by examining env*)
val (ntpairs, normp) =
if Envir.is_empty env then (tpairs, (Bs @ As, C))
else
let val ntps = map (apply2 normt) tpairs
in if Envir.above env smax then
(*no assignments in state; normalize the rule only*)
if lifted
then (ntps, (Bs @ map (norm_term_skip env nlift) As, C))
else (ntps, (Bs @ map normt As, C))
else if match then raise COMPOSE
else (*normalize the new rule fully*)
(ntps, (map normt (Bs @ As), normt C))
end
val th =
Thm (deriv_rule2
((if Envir.is_empty env then I
else if Envir.above env smax then
(fn f => fn der => f (Proofterm.norm_proof' env der))
else
curry op oo (Proofterm.norm_proof' env))
(Proofterm.bicompose_proof flatten Bs oldAs As A n (nlift+1))) rder' sder,
{tags = [],
maxidx = Envir.maxidx_of env,
shyps = Envir.insert_sorts env (Sorts.union rshyps sshyps),
hyps = union_hyps rhyps shyps,
tpairs = ntpairs,
prop = Logic.list_implies normp,
cert = cert})
in Seq.cons th thq end handle COMPOSE => thq;
val (rAs,B) = Logic.strip_prems(nsubgoal, [], rprop)
handle TERM _ => raise THM("bicompose: rule", 0, [orule,state]);
(*Modify assumptions, deleting n-th if n>0 for e-resolution*)
fun newAs(As0, n, dpairs, tpairs) =
let val (As1, rder') =
if not lifted then (As0, rder)
else
let val rename = rename_bvars dpairs tpairs B As0
in (map (rename strip_apply) As0,
deriv_rule1 (Proofterm.map_proof_terms (rename K) I) rder)
end;
in (map (if flatten then (Logic.flatten_params n) else I) As1, As1, rder', n)
handle TERM _ =>
raise THM("bicompose: 1st premise", 0, [orule])
end;
val BBi = if lifted then strip_assums2(B,Bi) else (B,Bi);
val dpairs = BBi :: (rtpairs@stpairs);
(*elim-resolution: try each assumption in turn*)
fun eres _ [] = raise THM ("bicompose: no premises", 0, [orule, state])
| eres env (A1 :: As) =
let
val A = SOME A1;
val (close, asms, concl) = Logic.assum_problems (nlift + 1, A1);
val concl' = close concl;
fun tryasms [] _ = Seq.empty
| tryasms (asm :: rest) n =
if Term.could_unify (asm, concl) then
let val asm' = close asm in
(case Seq.pull (Unify.unifiers (context, env, (asm', concl') :: dpairs)) of
NONE => tryasms rest (n + 1)
| cell as SOME ((_, tpairs), _) =>
Seq.it_right (addth A (newAs (As, n, [BBi, (concl', asm')], tpairs)))
(Seq.make (fn () => cell),
Seq.make (fn () => Seq.pull (tryasms rest (n + 1)))))
end
else tryasms rest (n + 1);
in tryasms asms 1 end;
(*ordinary resolution*)
fun res env =
(case Seq.pull (Unify.unifiers (context, env, dpairs)) of
NONE => Seq.empty
| cell as SOME ((_, tpairs), _) =>
Seq.it_right (addth NONE (newAs (rev rAs, 0, [BBi], tpairs)))
(Seq.make (fn () => cell), Seq.empty));
val env0 = Envir.empty (Int.max (rmax, smax));
in
(case if incremented then SOME env0 else unify_var_types context (state, orule) env0 of
NONE => Seq.empty
| SOME env => if eres_flg then eres env (rev rAs) else res env)
end;
end;
fun bicompose opt_ctxt flags arg i state =
bicompose_aux opt_ctxt flags (state, dest_state (state,i), false) arg;
(*Quick test whether rule is resolvable with the subgoal with hyps Hs
and conclusion B. If eres_flg then checks 1st premise of rule also*)
fun could_bires (Hs, B, eres_flg, rule) =
let fun could_reshyp (A1::_) = exists (fn H => Term.could_unify (A1, H)) Hs
| could_reshyp [] = false; (*no premise -- illegal*)
in Term.could_unify(concl_of rule, B) andalso
(not eres_flg orelse could_reshyp (prems_of rule))
end;
(*Bi-resolution of a state with a list of (flag,rule) pairs.
Puts the rule above: rule/state. Renames vars in the rules. *)
fun biresolution opt_ctxt match brules i state =
let val (stpairs, Bs, Bi, C) = dest_state(state,i);
val lift = lift_rule (cprem_of state i);
val B = Logic.strip_assums_concl Bi;
val Hs = Logic.strip_assums_hyp Bi;
val compose =
bicompose_aux opt_ctxt {flatten = true, match = match, incremented = true}
(state, (stpairs, Bs, Bi, C), true);
fun res [] = Seq.empty
| res ((eres_flg, rule)::brules) =
if Config.get_generic (make_context [state] opt_ctxt (cert_of state))
Pattern.unify_trace_failure orelse could_bires (Hs, B, eres_flg, rule)
then Seq.make (*delay processing remainder till needed*)
(fn()=> SOME(compose (eres_flg, lift rule, nprems_of rule),
res brules))
else res brules
in Seq.flat (res brules) end;
(*** Oracles ***)
(* oracle rule *)
fun invoke_oracle thy1 name oracle arg =
let val Cterm {cert = cert2, t = prop, T, maxidx, sorts} = oracle arg in
if T <> propT then
raise THM ("Oracle's result must have type prop: " ^ name, 0, [])
else
let val (ora, prf) = Proofterm.oracle_proof name prop in
Thm (make_deriv [] [ora] [] prf,
{cert = Context.join_certificate (Context.Certificate thy1, cert2),
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = prop})
end
end;
end;
end;
end;
(* authentic derivation names *)
structure Oracles = Theory_Data
(
type T = unit Name_Space.table;
val empty : T = Name_Space.empty_table "oracle";
val extend = I;
fun merge data : T = Name_Space.merge_tables data;
);
fun extern_oracles verbose ctxt =
map #1 (Name_Space.markup_table verbose ctxt (Oracles.get (Proof_Context.theory_of ctxt)));
fun add_oracle (b, oracle) thy =
let
val (name, tab') = Name_Space.define (Context.Theory thy) true (b, ()) (Oracles.get thy);
val thy' = Oracles.put tab' thy;
in ((name, invoke_oracle thy' name oracle), thy') end;
end;
structure Basic_Thm: BASIC_THM = Thm;
open Basic_Thm;