(* Title: Pure/thm.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
The very core of Isabelle's Meta Logic: certified types and terms,
meta theorems, meta rules (including lifting and resolution).
*)
signature BASIC_THM =
sig
(*certified types*)
type ctyp
val rep_ctyp: ctyp ->
{thy: theory,
T: typ,
maxidx: int,
sorts: sort list}
val theory_of_ctyp: ctyp -> theory
val typ_of: ctyp -> typ
val ctyp_of: theory -> typ -> ctyp
(*certified terms*)
type cterm
exception CTERM of string * cterm list
val rep_cterm: cterm ->
{thy: theory,
t: term,
T: typ,
maxidx: int,
sorts: sort list}
val crep_cterm: cterm -> {thy: theory, t: term, T: ctyp, maxidx: int, sorts: sort list}
val theory_of_cterm: cterm -> theory
val term_of: cterm -> term
val cterm_of: theory -> term -> cterm
val ctyp_of_term: cterm -> ctyp
(*meta theorems*)
type thm
type conv = cterm -> thm
type attribute = Context.generic * thm -> Context.generic * thm
val rep_thm: thm ->
{thy: theory,
der: bool * Proofterm.proof,
tags: Markup.property list,
maxidx: int,
shyps: sort list,
hyps: term list,
tpairs: (term * term) list,
prop: term}
val crep_thm: thm ->
{thy: theory,
der: bool * Proofterm.proof,
tags: Markup.property list,
maxidx: int,
shyps: sort list,
hyps: cterm list,
tpairs: (cterm * cterm) list,
prop: cterm}
exception THM of string * int * thm list
val theory_of_thm: thm -> theory
val prop_of: thm -> term
val proof_of: thm -> Proofterm.proof
val tpairs_of: thm -> (term * term) list
val concl_of: thm -> term
val prems_of: thm -> term list
val nprems_of: thm -> int
val cprop_of: thm -> cterm
val cprem_of: thm -> int -> cterm
val transfer: theory -> thm -> thm
val weaken: cterm -> thm -> thm
val extra_shyps: thm -> sort list
val strip_shyps: thm -> thm
val get_axiom_i: theory -> string -> thm
val get_axiom: theory -> xstring -> thm
val def_name: string -> string
val def_name_optional: string -> string -> string
val get_def: theory -> xstring -> thm
val axioms_of: theory -> (string * thm) list
(*meta 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: thm -> thm Seq.seq
val generalize: string list * string list -> int -> thm -> thm
val instantiate: (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm
val instantiate_cterm: (ctyp * ctyp) list * (cterm * cterm) list -> cterm -> cterm
val trivial: cterm -> thm
val class_triv: theory -> class -> thm
val unconstrainT: ctyp -> 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: 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 rename_params_rule: string list * int -> thm -> thm
val compose_no_flatten: bool -> thm * int -> int -> thm -> thm Seq.seq
val bicompose: bool -> bool * thm * int -> int -> thm -> thm Seq.seq
val biresolution: bool -> (bool * thm) list -> int -> thm -> thm Seq.seq
val invoke_oracle: theory -> xstring -> theory * Object.T -> thm
val invoke_oracle_i: theory -> string -> theory * Object.T -> thm
end;
signature THM =
sig
include BASIC_THM
val dest_ctyp: ctyp -> ctyp list
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 adjust_maxidx_cterm: int -> cterm -> cterm
val capply: cterm -> cterm -> cterm
val cabs: cterm -> cterm -> cterm
val major_prem_of: thm -> term
val no_prems: thm -> bool
val terms_of_tpairs: (term * term) list -> term list
val maxidx_of: thm -> int
val maxidx_thm: thm -> int -> int
val hyps_of: thm -> term list
val full_prop_of: thm -> term
val get_name: thm -> string
val put_name: string -> thm -> thm
val get_tags: thm -> Markup.property list
val map_tags: (Markup.property list -> Markup.property list) -> thm -> thm
val compress: thm -> thm
val norm_proof: thm -> thm
val adjust_maxidx_thm: int -> thm -> thm
val rename_boundvars: term -> term -> thm -> thm
val match: cterm * cterm -> (ctyp * ctyp) list * (cterm * cterm) list
val first_order_match: cterm * cterm -> (ctyp * ctyp) list * (cterm * cterm) list
val incr_indexes_cterm: int -> cterm -> cterm
val varifyT: thm -> thm
val varifyT': (string * sort) list -> thm -> ((string * sort) * indexname) list * thm
val freezeT: thm -> thm
end;
structure Thm: THM =
struct
structure Pt = Proofterm;
(*** Certified terms and types ***)
(** collect occurrences of sorts -- unless all sorts non-empty **)
fun may_insert_typ_sorts thy T = if Sign.all_sorts_nonempty thy then I else Sorts.insert_typ T;
fun may_insert_term_sorts thy t = if Sign.all_sorts_nonempty thy then I else Sorts.insert_term t;
(*NB: type unification may invent new sorts*)
fun may_insert_env_sorts thy (env as Envir.Envir {iTs, ...}) =
if Sign.all_sorts_nonempty thy then I
else Vartab.fold (fn (_, (_, T)) => Sorts.insert_typ T) iTs;
(** certified types **)
abstype ctyp = Ctyp of
{thy_ref: theory_ref,
T: typ,
maxidx: int,
sorts: sort list}
with
fun rep_ctyp (Ctyp {thy_ref, T, maxidx, sorts}) =
let val thy = Theory.deref thy_ref
in {thy = thy, T = T, maxidx = maxidx, sorts = sorts} end;
fun theory_of_ctyp (Ctyp {thy_ref, ...}) = Theory.deref thy_ref;
fun typ_of (Ctyp {T, ...}) = T;
fun ctyp_of thy raw_T =
let
val T = Sign.certify_typ thy raw_T;
val maxidx = Term.maxidx_of_typ T;
val sorts = may_insert_typ_sorts thy T [];
in Ctyp {thy_ref = Theory.check_thy thy, T = T, maxidx = maxidx, sorts = sorts} end;
fun dest_ctyp (Ctyp {thy_ref, T = Type (s, Ts), maxidx, sorts}) =
map (fn T => Ctyp {thy_ref = thy_ref, 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
{thy_ref: theory_ref,
t: term,
T: typ,
maxidx: int,
sorts: sort list}
with
exception CTERM of string * cterm list;
fun rep_cterm (Cterm {thy_ref, t, T, maxidx, sorts}) =
let val thy = Theory.deref thy_ref
in {thy = thy, t = t, T = T, maxidx = maxidx, sorts = sorts} end;
fun crep_cterm (Cterm {thy_ref, t, T, maxidx, sorts}) =
let val thy = Theory.deref thy_ref in
{thy = thy, t = t,
T = Ctyp {thy_ref = thy_ref, T = T, maxidx = maxidx, sorts = sorts},
maxidx = maxidx, sorts = sorts}
end;
fun theory_of_cterm (Cterm {thy_ref, ...}) = Theory.deref thy_ref;
fun term_of (Cterm {t, ...}) = t;
fun ctyp_of_term (Cterm {thy_ref, T, maxidx, sorts, ...}) =
Ctyp {thy_ref = thy_ref, T = T, maxidx = maxidx, sorts = sorts};
fun cterm_of thy tm =
let
val (t, T, maxidx) = Sign.certify_term thy tm;
val sorts = may_insert_term_sorts thy t [];
in Cterm {thy_ref = Theory.check_thy thy, t = t, T = T, maxidx = maxidx, sorts = sorts} end;
fun merge_thys0 (Cterm {thy_ref = r1, t = t1, ...}) (Cterm {thy_ref = r2, t = t2, ...}) =
Theory.merge_refs (r1, r2);
(* destructors *)
fun dest_comb (ct as Cterm {t = c $ a, T, thy_ref, maxidx, sorts}) =
let val A = Term.argument_type_of c 0 in
(Cterm {t = c, T = A --> T, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts},
Cterm {t = a, T = A, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts})
end
| dest_comb ct = raise CTERM ("dest_comb", [ct]);
fun dest_fun (ct as Cterm {t = c $ _, T, thy_ref, maxidx, sorts}) =
let val A = Term.argument_type_of c 0
in Cterm {t = c, T = A --> T, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts} end
| dest_fun ct = raise CTERM ("dest_fun", [ct]);
fun dest_arg (ct as Cterm {t = c $ a, T = _, thy_ref, maxidx, sorts}) =
let val A = Term.argument_type_of c 0
in Cterm {t = a, T = A, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts} end
| dest_arg ct = raise CTERM ("dest_arg", [ct]);
fun dest_fun2 (Cterm {t = c $ a $ b, T, thy_ref, 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, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts} end
| dest_fun2 ct = raise CTERM ("dest_fun2", [ct]);
fun dest_arg1 (Cterm {t = c $ a $ _, T = _, thy_ref, maxidx, sorts}) =
let val A = Term.argument_type_of c 0
in Cterm {t = a, T = A, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts} end
| dest_arg1 ct = raise CTERM ("dest_arg1", [ct]);
fun dest_abs a (ct as
Cterm {t = Abs (x, T, t), T = Type ("fun", [_, U]), thy_ref, maxidx, sorts}) =
let val (y', t') = Term.dest_abs (the_default x a, T, t) in
(Cterm {t = Free (y', T), T = T, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts},
Cterm {t = t', T = U, thy_ref = thy_ref, maxidx = maxidx, sorts = sorts})
end
| dest_abs _ ct = raise CTERM ("dest_abs", [ct]);
(* constructors *)
fun capply
(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 {thy_ref = merge_thys0 cf cx,
t = f $ x,
T = rty,
maxidx = Int.max (maxidx1, maxidx2),
sorts = Sorts.union sorts1 sorts2}
else raise CTERM ("capply: types don't agree", [cf, cx])
| capply cf cx = raise CTERM ("capply: first arg is not a function", [cf, cx]);
fun cabs
(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 t1 t2 in
Cterm {thy_ref = merge_thys0 ct1 ct2,
t = t, T = T1 --> T2,
maxidx = Int.max (maxidx1, maxidx2),
sorts = Sorts.union sorts1 sorts2}
end;
(* indexes *)
fun adjust_maxidx_cterm i (ct as Cterm {thy_ref, t, T, maxidx, sorts}) =
if maxidx = i then ct
else if maxidx < i then
Cterm {maxidx = i, thy_ref = thy_ref, t = t, T = T, sorts = sorts}
else
Cterm {maxidx = Int.max (maxidx_of_term t, i), thy_ref = thy_ref, t = t, T = T, sorts = sorts};
fun incr_indexes_cterm i (ct as Cterm {thy_ref, t, T, maxidx, sorts}) =
if i < 0 then raise CTERM ("negative increment", [ct])
else if i = 0 then ct
else Cterm {thy_ref = thy_ref, t = Logic.incr_indexes ([], i) t,
T = Logic.incr_tvar i T, maxidx = maxidx + i, sorts = sorts};
(* matching *)
local
fun gen_match match
(ct1 as Cterm {t = t1, sorts = sorts1, ...},
ct2 as Cterm {t = t2, sorts = sorts2, maxidx = maxidx2, ...}) =
let
val thy = Theory.deref (merge_thys0 ct1 ct2);
val (Tinsts, tinsts) = match thy (t1, t2) (Vartab.empty, Vartab.empty);
val sorts = Sorts.union sorts1 sorts2;
fun mk_cTinst ((a, i), (S, T)) =
(Ctyp {T = TVar ((a, i), S), thy_ref = Theory.check_thy thy, maxidx = i, sorts = sorts},
Ctyp {T = T, thy_ref = Theory.check_thy thy, maxidx = maxidx2, sorts = sorts});
fun mk_ctinst ((x, i), (T, t)) =
let val T = Envir.typ_subst_TVars Tinsts T in
(Cterm {t = Var ((x, i), T), T = T, thy_ref = Theory.check_thy thy,
maxidx = i, sorts = sorts},
Cterm {t = t, T = T, thy_ref = Theory.check_thy thy, 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;
(*** Meta theorems ***)
abstype thm = Thm of
{thy_ref: theory_ref, (*dynamic reference to theory*)
der: bool * Pt.proof, (*derivation*)
tags: Markup.property list, (*additional annotations/comments*)
maxidx: int, (*maximum index of any Var or TVar*)
shyps: sort list, (*sort hypotheses as ordered list*)
hyps: term list, (*hypotheses as ordered list*)
tpairs: (term * term) list, (*flex-flex pairs*)
prop: term} (*conclusion*)
with
type conv = cterm -> thm;
(*attributes subsume any kind of rules or context modifiers*)
type attribute = Context.generic * thm -> Context.generic * thm;
(*errors involving theorems*)
exception THM of string * int * thm list;
fun rep_thm (Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop}) =
let val thy = Theory.deref thy_ref in
{thy = thy, der = der, tags = tags, maxidx = maxidx,
shyps = shyps, hyps = hyps, tpairs = tpairs, prop = prop}
end;
(*version of rep_thm returning cterms instead of terms*)
fun crep_thm (Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop}) =
let
val thy = Theory.deref thy_ref;
fun cterm max t = Cterm {thy_ref = thy_ref, t = t, T = propT, maxidx = max, sorts = shyps};
in
{thy = thy, der = der, tags = tags, maxidx = maxidx, shyps = shyps,
hyps = map (cterm ~1) hyps,
tpairs = map (pairself (cterm maxidx)) tpairs,
prop = cterm maxidx prop}
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 = OrdList.union Term.fast_term_ord;
(* merge theories of cterms/thms -- trivial absorption only *)
fun merge_thys1 (Cterm {thy_ref = r1, ...}) (th as Thm {thy_ref = r2, ...}) =
Theory.merge_refs (r1, r2);
fun merge_thys2 (th1 as Thm {thy_ref = r1, ...}) (th2 as Thm {thy_ref = r2, ...}) =
Theory.merge_refs (r1, r2);
(* basic components *)
fun theory_of_thm (Thm {thy_ref, ...}) = Theory.deref thy_ref;
fun maxidx_of (Thm {maxidx, ...}) = maxidx;
fun maxidx_thm th i = Int.max (maxidx_of th, i);
fun hyps_of (Thm {hyps, ...}) = hyps;
fun prop_of (Thm {prop, ...}) = prop;
fun proof_of (Thm {der = (_, proof), ...}) = proof;
fun tpairs_of (Thm {tpairs, ...}) = tpairs;
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]));
(*the statement of any thm is a cterm*)
fun cprop_of (Thm {thy_ref, maxidx, shyps, prop, ...}) =
Cterm {thy_ref = thy_ref, maxidx = maxidx, T = propT, t = prop, sorts = shyps};
fun cprem_of (th as Thm {thy_ref, maxidx, shyps, prop, ...}) i =
Cterm {thy_ref = thy_ref, maxidx = maxidx, T = propT, sorts = shyps,
t = Logic.nth_prem (i, prop) handle TERM _ => raise THM ("cprem_of", i, [th])};
(*explicit transfer to a super theory*)
fun transfer thy' thm =
let
val Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop} = thm;
val thy = Theory.deref thy_ref;
val _ = subthy (thy, thy') orelse raise THM ("transfer: not a super theory", 0, [thm]);
val is_eq = eq_thy (thy, thy');
val _ = Theory.check_thy thy;
in
if is_eq then thm
else
Thm {thy_ref = Theory.check_thy thy',
der = der,
tags = tags,
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
tpairs = tpairs,
prop = prop}
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 {thy_ref = merge_thys1 ct th,
der = der,
tags = tags,
maxidx = maxidx,
shyps = Sorts.union sorts shyps,
hyps = OrdList.insert Term.fast_term_ord A hyps,
tpairs = tpairs,
prop = prop}
end;
(** sort contexts of theorems **)
fun present_sorts (Thm {hyps, tpairs, prop, ...}) =
fold (fn (t, u) => Sorts.insert_term t o Sorts.insert_term u) tpairs
(Sorts.insert_terms hyps (Sorts.insert_term prop []));
(*remove extra sorts that are non-empty by virtue of type signature information*)
fun strip_shyps (thm as Thm {shyps = [], ...}) = thm
| strip_shyps (thm as Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop}) =
let
val thy = Theory.deref thy_ref;
val shyps' =
if Sign.all_sorts_nonempty thy then []
else
let
val present = present_sorts thm;
val extra = Sorts.subtract present shyps;
val witnessed = map #2 (Sign.witness_sorts thy present extra);
in Sorts.subtract witnessed shyps end;
in
Thm {thy_ref = Theory.check_thy thy, der = der, tags = tags, maxidx = maxidx,
shyps = shyps', hyps = hyps, tpairs = tpairs, prop = prop}
end;
(*dangling sort constraints of a thm*)
fun extra_shyps (th as Thm {shyps, ...}) = Sorts.subtract (present_sorts th) shyps;
(** Axioms **)
(*look up the named axiom in the theory or its ancestors*)
fun get_axiom_i theory name =
let
fun get_ax thy =
Symtab.lookup (Theory.axiom_table thy) name
|> Option.map (fn prop =>
let
val der = Pt.infer_derivs' I (false, Pt.axm_proof name prop);
val maxidx = maxidx_of_term prop;
val shyps = may_insert_term_sorts thy prop [];
in
Thm {thy_ref = Theory.check_thy thy, der = der, tags = [],
maxidx = maxidx, shyps = shyps, hyps = [], tpairs = [], prop = prop}
end);
in
(case get_first get_ax (theory :: Theory.ancestors_of theory) of
SOME thm => thm
| NONE => raise THEORY ("No axiom " ^ quote name, [theory]))
end;
fun get_axiom thy =
get_axiom_i thy o NameSpace.intern (Theory.axiom_space thy);
fun def_name c = c ^ "_def";
fun def_name_optional c "" = def_name c
| def_name_optional _ name = name;
fun get_def thy = get_axiom thy o def_name;
(*return additional axioms of this theory node*)
fun axioms_of thy =
map (fn s => (s, get_axiom_i thy s)) (Symtab.keys (Theory.axiom_table thy));
(* official name and additional tags *)
fun get_name (Thm {hyps, prop, der = (_, prf), ...}) =
Pt.get_name hyps prop prf;
fun put_name name (Thm {thy_ref, der = (ora, prf), tags, maxidx, shyps, hyps, tpairs = [], prop}) =
let
val thy = Theory.deref thy_ref;
val der = (ora, Pt.thm_proof thy name hyps prop prf);
in
Thm {thy_ref = Theory.check_thy thy, der = der, tags = tags, maxidx = maxidx,
shyps = shyps, hyps = hyps, tpairs = [], prop = prop}
end
| put_name _ thm = raise THM ("name_thm: unsolved flex-flex constraints", 0, [thm]);
val get_tags = #tags o rep_thm;
fun map_tags f (Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop}) =
Thm {thy_ref = thy_ref, der = der, tags = f tags, maxidx = maxidx,
shyps = shyps, hyps = hyps, tpairs = tpairs, prop = prop};
(*Compression of theorems -- a separate rule, not integrated with the others,
as it could be slow.*)
fun compress (Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop}) =
let
val thy = Theory.deref thy_ref;
val hyps' = map (Compress.term thy) hyps;
val tpairs' = map (pairself (Compress.term thy)) tpairs;
val prop' = Compress.term thy prop;
in
Thm {thy_ref = Theory.check_thy thy,
der = der,
tags = tags,
maxidx = maxidx,
shyps = shyps,
hyps = hyps',
tpairs = tpairs',
prop = prop'}
end;
fun norm_proof (Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop}) =
let
val thy = Theory.deref thy_ref;
val der = Pt.infer_derivs' (Pt.rew_proof thy) der;
in
Thm {thy_ref = Theory.check_thy thy, der = der, tags = tags, maxidx = maxidx,
shyps = shyps, hyps = hyps, tpairs = tpairs, prop = prop}
end;
fun adjust_maxidx_thm i (th as Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop}) =
if maxidx = i then th
else if maxidx < i then
Thm {maxidx = i, thy_ref = thy_ref, der = der, tags = tags, shyps = shyps,
hyps = hyps, tpairs = tpairs, prop = prop}
else
Thm {maxidx = Int.max (maxidx_tpairs tpairs (maxidx_of_term prop), i), thy_ref = thy_ref,
der = der, 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 {thy_ref, 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 {thy_ref = thy_ref,
der = Pt.infer_derivs' I (false, Pt.Hyp prop),
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 {thy_ref = merge_thys1 ct th,
der = Pt.infer_derivs' (Pt.implies_intr_proof A) der,
tags = [],
maxidx = Int.max (maxidxA, maxidx),
shyps = Sorts.union sorts shyps,
hyps = OrdList.remove Term.fast_term_ord A hyps,
tpairs = tpairs,
prop = implies $ A $ prop};
(*Implication elimination
A ==> B A
------------
B
*)
fun implies_elim thAB thA =
let
val Thm {maxidx = maxA, der = derA, 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 ("==>", _) $ A $ B =>
if A aconv propA then
Thm {thy_ref = merge_thys2 thAB thA,
der = Pt.infer_derivs (curry Pt.%%) der derA,
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 {thy_ref = merge_thys1 ct th,
der = Pt.infer_derivs' (Pt.forall_intr_proof x a) der,
tags = [],
maxidx = maxidx,
shyps = Sorts.union sorts shyps,
hyps = hyps,
tpairs = tpairs,
prop = all 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 ("all", Type ("fun", [Type ("fun", [qary, _]), _])) $ A =>
if T <> qary then
raise THM ("forall_elim: type mismatch", 0, [th])
else
Thm {thy_ref = merge_thys1 ct th,
der = Pt.infer_derivs' (Pt.% o rpair (SOME t)) der,
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 (ct as Cterm {thy_ref, t, T, maxidx, sorts}) =
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' I (false, Pt.reflexive),
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_equals (t, t)};
(*Symmetry
t == u
------
u == t
*)
fun symmetric (th as Thm {thy_ref, der, maxidx, shyps, hyps, tpairs, prop, ...}) =
(case prop of
(eq as Const ("==", Type (_, [T, _]))) $ t $ u =>
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' Pt.symmetric der,
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 {der = der1, maxidx = max1, hyps = hyps1, shyps = shyps1, tpairs = tpairs1,
prop = prop1, ...} = th1
and Thm {der = 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 ("==", Type (_, [T, _]))) $ t1 $ u, Const ("==", _) $ u' $ t2) =>
if not (u aconv u') then err "middle term"
else
Thm {thy_ref = merge_thys2 th1 th2,
der = Pt.infer_derivs (Pt.transitive u T) der1 der2,
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 {thy_ref, 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 {thy_ref = thy_ref,
der = Pt.infer_derivs' I (false, Pt.reflexive),
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_equals (t, t')}
end;
fun eta_conversion (Cterm {thy_ref, t, T, maxidx, sorts}) =
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' I (false, Pt.reflexive),
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_equals (t, Envir.eta_contract t)};
fun eta_long_conversion (Cterm {thy_ref, t, T, maxidx, sorts}) =
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' I (false, Pt.reflexive),
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = Logic.mk_equals (t, Pattern.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 {thy_ref, der, 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 {thy_ref = thy_ref,
der = Pt.infer_derivs' (Pt.abstract_rule x a) der,
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 {der = der1, maxidx = max1, shyps = shyps1, hyps = hyps1, tpairs = tpairs1,
prop = prop1, ...} = th1
and Thm {der = der2, maxidx = max2, shyps = shyps2, hyps = hyps2, tpairs = tpairs2,
prop = prop2, ...} = th2;
fun chktypes fT tT =
(case fT of
Type ("fun", [T1, T2]) =>
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 ("==", Type ("fun", [fT, _])) $ f $ g,
Const ("==", Type ("fun", [tT, _])) $ t $ u) =>
(chktypes fT tT;
Thm {thy_ref = merge_thys2 th1 th2,
der = Pt.infer_derivs (Pt.combination f g t u fT) der1 der2,
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 {der = der1, maxidx = max1, shyps = shyps1, hyps = hyps1, tpairs = tpairs1,
prop = prop1, ...} = th1
and Thm {der = 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("==>", _) $ A $ B, Const("==>", _) $ B' $ A') =>
if A aconv A' andalso B aconv B' then
Thm {thy_ref = merge_thys2 th1 th2,
der = Pt.infer_derivs (Pt.equal_intr A B) der1 der2,
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 {der = der1, maxidx = max1, shyps = shyps1, hyps = hyps1,
tpairs = tpairs1, prop = prop1, ...} = th1
and Thm {der = 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 ("==", _) $ A $ B =>
if prop2 aconv A then
Thm {thy_ref = merge_thys2 th1 th2,
der = Pt.infer_derivs (Pt.equal_elim A B) der1 der2,
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 (th as Thm {thy_ref, der, maxidx, shyps, hyps, tpairs, prop, ...}) =
let val thy = Theory.deref thy_ref in
Unify.smash_unifiers thy tpairs (Envir.empty maxidx)
|> Seq.map (fn env =>
if Envir.is_empty env then th
else
let
val tpairs' = tpairs |> map (pairself (Envir.norm_term env))
(*remove trivial tpairs, of the form t==t*)
|> filter_out (op aconv);
val der = Pt.infer_derivs' (Pt.norm_proof' env) der;
val prop' = Envir.norm_term env prop;
val maxidx = maxidx_tpairs tpairs' (maxidx_of_term prop');
val shyps = may_insert_env_sorts thy env shyps;
in
Thm {thy_ref = Theory.check_thy thy, der = der, tags = [], maxidx = maxidx,
shyps = shyps, hyps = hyps, tpairs = tpairs', prop = prop'}
end)
end;
(*Generalization of fixed variables
A
--------------------
A[?'a/'a, ?x/x, ...]
*)
fun generalize ([], []) _ th = th
| generalize (tfrees, frees) idx th =
let
val Thm {thy_ref, der, 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 = TermSubst.generalize (tfrees, frees) idx;
val prop' = gen prop;
val tpairs' = map (pairself gen) tpairs;
val maxidx' = maxidx_tpairs tpairs' (maxidx_of_term prop');
in
Thm {
thy_ref = thy_ref,
der = Pt.infer_derivs' (Pt.generalize (tfrees, frees) idx) der,
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 [Sign.pretty_term thy t, Pretty.str " ::", Pretty.brk 1, Sign.pretty_typ thy T];
fun add_inst (ct, cu) (thy_ref, sorts) =
let
val Cterm {t = t, T = T, ...} = ct
and Cterm {t = u, T = U, sorts = sorts_u, maxidx = maxidx_u, ...} = cu;
val thy_ref' = Theory.merge_refs (thy_ref, merge_thys0 ct cu);
val sorts' = Sorts.union sorts_u sorts;
in
(case t of Var v =>
if T = U then ((v, (u, maxidx_u)), (thy_ref', sorts'))
else raise TYPE (Pretty.string_of (Pretty.block
[Pretty.str "instantiate: type conflict",
Pretty.fbrk, pretty_typing (Theory.deref thy_ref') t T,
Pretty.fbrk, pretty_typing (Theory.deref thy_ref') u U]), [T, U], [t, u])
| _ => raise TYPE (Pretty.string_of (Pretty.block
[Pretty.str "instantiate: not a variable",
Pretty.fbrk, Sign.pretty_term (Theory.deref thy_ref') t]), [], [t]))
end;
fun add_instT (cT, cU) (thy_ref, sorts) =
let
val Ctyp {T, thy_ref = thy_ref1, ...} = cT
and Ctyp {T = U, thy_ref = thy_ref2, sorts = sorts_U, maxidx = maxidx_U, ...} = cU;
val thy' = Theory.deref (Theory.merge_refs (thy_ref, Theory.merge_refs (thy_ref1, thy_ref2)));
val sorts' = Sorts.union sorts_U sorts;
in
(case T of TVar (v as (_, S)) =>
if Sign.of_sort thy' (U, S) then ((v, (U, maxidx_U)), (Theory.check_thy thy', sorts'))
else raise TYPE ("Type not of sort " ^ Sign.string_of_sort thy' S, [U], [])
| _ => raise TYPE (Pretty.string_of (Pretty.block
[Pretty.str "instantiate: not a type variable",
Pretty.fbrk, Sign.pretty_typ thy' T]), [T], []))
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 {thy_ref, der, hyps, shyps, tpairs, prop, ...} = th;
val (inst', (instT', (thy_ref', shyps'))) =
(thy_ref, shyps) |> fold_map add_inst inst ||> fold_map add_instT instT;
val subst = TermSubst.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 {thy_ref = thy_ref',
der = Pt.infer_derivs' (fn d =>
Pt.instantiate (map (apsnd #1) instT', map (apsnd #1) inst') d) der,
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 {thy_ref, t, T, sorts, ...} = ct;
val (inst', (instT', (thy_ref', sorts'))) =
(thy_ref, sorts) |> fold_map add_inst inst ||> fold_map add_instT instT;
val subst = TermSubst.instantiate_maxidx (instT', inst');
val substT = TermSubst.instantiateT_maxidx instT';
val (t', maxidx1) = subst t ~1;
val (T', maxidx') = substT T maxidx1;
in Cterm {thy_ref = thy_ref', 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 {thy_ref, t =A, T, maxidx, sorts}) =
if T <> propT then
raise THM ("trivial: the term must have type prop", 0, [])
else
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' I (false, Pt.AbsP ("H", NONE, Pt.PBound 0)),
tags = [],
maxidx = maxidx,
shyps = sorts,
hyps = [],
tpairs = [],
prop = implies $ A $ A};
(*Axiom-scheme reflecting signature contents: "OFCLASS(?'a::c, c_class)" *)
fun class_triv thy c =
let
val Cterm {t, maxidx, sorts, ...} =
cterm_of thy (Logic.mk_inclass (TVar ((Name.aT, 0), [c]), Sign.certify_class thy c))
handle TERM (msg, _) => raise THM ("class_triv: " ^ msg, 0, []);
val der = Pt.infer_derivs' I (false, Pt.PAxm ("Pure.class_triv:" ^ c, t, SOME []));
in
Thm {thy_ref = Theory.check_thy thy, der = der, tags = [], maxidx = maxidx,
shyps = sorts, hyps = [], tpairs = [], prop = t}
end;
(*Internalize sort constraints of type variable*)
fun unconstrainT
(Ctyp {thy_ref = thy_ref1, T, ...})
(th as Thm {thy_ref = thy_ref2, der, maxidx, shyps, hyps, tpairs, prop, ...}) =
let
val ((x, i), S) = Term.dest_TVar T handle TYPE _ =>
raise THM ("unconstrainT: not a type variable", 0, [th]);
val T' = TVar ((x, i), []);
val unconstrain = Term.map_types (Term.map_atyps (fn U => if U = T then T' else U));
val constraints = map (curry Logic.mk_inclass T') S;
in
Thm {thy_ref = Theory.merge_refs (thy_ref1, thy_ref2),
der = Pt.infer_derivs' I (false, Pt.PAxm ("Pure.unconstrainT", prop, SOME [])),
tags = [],
maxidx = Int.max (maxidx, i),
shyps = Sorts.remove_sort S shyps,
hyps = hyps,
tpairs = map (pairself unconstrain) tpairs,
prop = Logic.list_implies (constraints, unconstrain prop)}
end;
(* Replace all TFrees not fixed or in the hyps by new TVars *)
fun varifyT' fixed (Thm {thy_ref, der, maxidx, shyps, hyps, tpairs, prop, ...}) =
let
val tfrees = List.foldr add_term_tfrees fixed hyps;
val prop1 = attach_tpairs tpairs prop;
val (al, prop2) = Type.varify tfrees prop1;
val (ts, prop3) = Logic.strip_prems (length tpairs, [], prop2);
in
(al, Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' (Pt.varify_proof prop tfrees) der,
tags = [],
maxidx = Int.max (0, maxidx),
shyps = shyps,
hyps = hyps,
tpairs = rev (map Logic.dest_equals ts),
prop = prop3})
end;
val varifyT = #2 o varifyT' [];
(* Replace all TVars by new TFrees *)
fun freezeT (Thm {thy_ref, der, maxidx, shyps, hyps, tpairs, prop, ...}) =
let
val prop1 = attach_tpairs tpairs prop;
val prop2 = Type.freeze prop1;
val (ts, prop3) = Logic.strip_prems (length tpairs, [], prop2);
in
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' (Pt.freezeT prop1) der,
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]);
(*Increment variables and parameters of orule as required for
resolution with a 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 {thy_ref = merge_thys1 goal orule,
der = Pt.infer_derivs' (Pt.lift_proof gprop inc prop) der,
tags = [],
maxidx = maxidx + inc,
shyps = Sorts.union shyps sorts, (*sic!*)
hyps = hyps,
tpairs = map (pairself lift_abs) tpairs,
prop = Logic.list_implies (map lift_all As, lift_all B)}
end;
fun incr_indexes i (thm as Thm {thy_ref, der, maxidx, shyps, hyps, tpairs, prop, ...}) =
if i < 0 then raise THM ("negative increment", 0, [thm])
else if i = 0 then thm
else
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs'
(Pt.map_proof_terms (Logic.incr_indexes ([], i)) (Logic.incr_tvar i)) der,
tags = [],
maxidx = maxidx + i,
shyps = shyps,
hyps = hyps,
tpairs = map (pairself (Logic.incr_indexes ([], i))) tpairs,
prop = Logic.incr_indexes ([], i) prop};
(*Solve subgoal Bi of proof state B1...Bn/C by assumption. *)
fun assumption i state =
let
val Thm {thy_ref, der, maxidx, shyps, hyps, prop, ...} = state;
val thy = Theory.deref thy_ref;
val (tpairs, Bs, Bi, C) = dest_state (state, i);
fun newth n (env as Envir.Envir {maxidx, ...}, tpairs) =
Thm {
der = Pt.infer_derivs'
((if Envir.is_empty env then I else (Pt.norm_proof' env)) o
Pt.assumption_proof Bs Bi n) der,
tags = [],
maxidx = maxidx,
shyps = may_insert_env_sorts thy env shyps,
hyps = hyps,
tpairs =
if Envir.is_empty env then tpairs
else map (pairself (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)),
thy_ref = Theory.check_thy thy};
fun addprfs [] _ = Seq.empty
| addprfs ((t, u) :: apairs) n = Seq.make (fn () => Seq.pull
(Seq.mapp (newth n)
(Unify.unifiers (thy, Envir.empty maxidx, (t, u) :: tpairs))
(addprfs apairs (n + 1))))
in addprfs (Logic.assum_pairs (~1, Bi)) 1 end;
(*Solve subgoal Bi of proof state B1...Bn/C by assumption.
Checks if Bi's conclusion is alpha-convertible to one of its assumptions*)
fun eq_assumption i state =
let
val Thm {thy_ref, der, maxidx, shyps, hyps, prop, ...} = state;
val (tpairs, Bs, Bi, C) = dest_state (state, i);
in
(case find_index (op aconv) (Logic.assum_pairs (~1, Bi)) of
~1 => raise THM ("eq_assumption", 0, [state])
| n =>
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' (Pt.assumption_proof Bs Bi (n + 1)) der,
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 {thy_ref, der, maxidx, shyps, hyps, prop, ...} = state;
val (tpairs, Bs, Bi, C) = dest_state (state, i);
val params = Term.strip_all_vars Bi
and rest = Term.strip_all_body Bi;
val asms = Logic.strip_imp_prems rest
and 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 list_all (params, Logic.list_implies (qs @ ps, concl)) end
else raise THM ("rotate_rule", k, [state]);
in
Thm {thy_ref = thy_ref,
der = Pt.infer_derivs' (Pt.rotate_proof Bs Bi m) der,
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 etac and underlies defer_tac*)
fun permute_prems j k rl =
let
val Thm {thy_ref, der, 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 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 {thy_ref = thy_ref,
der = Pt.infer_derivs' (Pt.permute_prems_prf prems j m) der,
tags = [],
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
tpairs = tpairs,
prop = prop'}
end;
(** User renaming of parameters in a subgoal **)
(*Calls error rather than raising an exception because it is intended
for top-level use -- exception handling would not make sense here.
The names in cs, if distinct, are used for the innermost parameters;
preceding parameters may be renamed to make all params distinct.*)
fun rename_params_rule (cs, i) state =
let
val Thm {thy_ref, der, tags, maxidx, shyps, hyps, ...} = state;
val (tpairs, Bs, Bi, C) = dest_state (state, i);
val iparams = map #1 (Logic.strip_params Bi);
val short = length iparams - length cs;
val newnames =
if short < 0 then error "More names than abstractions!"
else Name.variant_list cs (Library.take (short, iparams)) @ cs;
val freenames = Term.fold_aterms (fn Free (x, _) => insert (op =) x | _ => I) Bi [];
val newBi = Logic.list_rename_params (newnames, Bi);
in
(case duplicates (op =) cs of
a :: _ => (warning ("Can't rename. Bound variables not distinct: " ^ a); state)
| [] =>
(case cs inter_string freenames of
a :: _ => (warning ("Can't rename. Bound/Free variable clash: " ^ a); state)
| [] =>
Thm {thy_ref = thy_ref,
der = der,
tags = tags,
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
tpairs = tpairs,
prop = Logic.list_implies (Bs @ [newBi], C)}))
end;
(*** Preservation of bound variable names ***)
fun rename_boundvars pat obj (thm as Thm {thy_ref, der, tags, maxidx, shyps, hyps, tpairs, prop}) =
(case Term.rename_abs pat obj prop of
NONE => thm
| SOME prop' => Thm
{thy_ref = thy_ref,
der = der,
tags = tags,
maxidx = maxidx,
hyps = hyps,
shyps = shyps,
tpairs = tpairs,
prop = prop'});
(* strip_apply f (A, B) 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("==>",_)$ A1 $ B1,
Const("==>",_)$ _ $ B2) = implies $ A1 $ strip(B1,B2)
| strip((c as Const("all",_)) $ Abs(a,T,t1),
Const("all",_) $ Abs(_,_,t2)) = c$Abs(a,T,strip(t1,t2))
| strip(A,_) = f A
in strip end;
(*Use the alist to rename all bound variables and some unknowns in a term
dpairs = current disagreement pairs; tpairs = permanent ones (flexflex);
Preserves unknowns in tpairs and on lhs of dpairs. *)
fun rename_bvs([],_,_,_) = I
| rename_bvs(al,dpairs,tpairs,B) =
let
val add_var = fold_aterms (fn Var ((x, _), _) => insert (op =) x | _ => I);
val vids = []
|> fold (add_var o fst) dpairs
|> fold (add_var o fst) tpairs
|> fold (add_var o snd) tpairs;
(*unknowns appearing elsewhere be preserved!*)
fun rename(t as Var((x,i),T)) =
(case AList.lookup (op =) al x of
SOME y =>
if member (op =) vids x orelse member (op =) vids y then t
else Var((y,i),T)
| NONE=> t)
| rename(Abs(x,T,t)) =
Abs (the_default x (AList.lookup (op =) al x), T, rename t)
| rename(f$t) = rename f $ rename t
| rename(t) = t;
fun strip_ren Ai = strip_apply rename (Ai,B)
in strip_ren end;
(*Function to rename bounds/unknowns in the argument, lifted over B*)
fun rename_bvars(dpairs, tpairs, B) =
rename_bvs(List.foldr Term.match_bvars [] dpairs, dpairs, tpairs, B);
(*** RESOLUTION ***)
(** Lifting optimizations **)
(*strip off pairs of assumptions/parameters in parallel -- they are
identical because of lifting*)
fun strip_assums2 (Const("==>", _) $ _ $ B1,
Const("==>", _) $ _ $ B2) = strip_assums2 (B1,B2)
| strip_assums2 (Const("all",_)$Abs(a,T,t1),
Const("all",_)$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("all",_)$Abs(a,T,t)) =
let val Envir.Envir{iTs, ...} = env
val T' = Envir.typ_subst_TVars iTs T
(*Must instantiate types of parameters because they are flattened;
this could be a NEW parameter*)
in all T' $ Abs(a, T', norm_term_skip env n t) end
| norm_term_skip env n (Const("==>", _) $ A $ B) =
implies $ A $ norm_term_skip env (n-1) B
| norm_term_skip env n t = error"norm_term_skip: too few assumptions??";
(*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 flatten match (state, (stpairs, Bs, Bi, C), lifted)
(eres_flg, orule, nsubgoal) =
let val Thm{der=sder, maxidx=smax, shyps=sshyps, hyps=shyps, ...} = state
and Thm{der=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 thy = Theory.deref (merge_thys2 state orule);
(** Add new theorem with prop = '[| Bs; As |] ==> C' to thq **)
fun addth A (As, oldAs, rder', n) ((env as Envir.Envir {maxidx, ...}, 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 (pairself 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{der = Pt.infer_derivs
((if Envir.is_empty env then I
else if Envir.above env smax then
(fn f => fn der => f (Pt.norm_proof' env der))
else
curry op oo (Pt.norm_proof' env))
(Pt.bicompose_proof flatten Bs oldAs As A n (nlift+1))) rder' sder,
tags = [],
maxidx = maxidx,
shyps = may_insert_env_sorts thy env (Sorts.union rshyps sshyps),
hyps = union_hyps rhyps shyps,
tpairs = ntpairs,
prop = Logic.list_implies normp,
thy_ref = Theory.check_thy thy}
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 (map (rename_bvars(dpairs,tpairs,B)) As0,
Pt.infer_derivs' (Pt.map_proof_terms
(rename_bvars (dpairs, tpairs, Bound 0)) I) rder);
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 env = Envir.empty(Int.max(rmax,smax));
val BBi = if lifted then strip_assums2(B,Bi) else (B,Bi);
val dpairs = BBi :: (rtpairs@stpairs);
(*elim-resolution: try each assumption in turn. Initially n=1*)
fun tryasms (_, _, _, []) = Seq.empty
| tryasms (A, As, n, (t,u)::apairs) =
(case Seq.pull(Unify.unifiers(thy, env, (t,u)::dpairs)) of
NONE => tryasms (A, As, n+1, apairs)
| cell as SOME((_,tpairs),_) =>
Seq.it_right (addth A (newAs(As, n, [BBi,(u,t)], tpairs)))
(Seq.make(fn()=> cell),
Seq.make(fn()=> Seq.pull (tryasms(A, As, n+1, apairs)))))
fun eres [] = raise THM("bicompose: no premises", 0, [orule,state])
| eres (A1::As) = tryasms(SOME A1, As, 1, Logic.assum_pairs(nlift+1,A1))
(*ordinary resolution*)
fun res(NONE) = Seq.empty
| res(cell as SOME((_,tpairs),_)) =
Seq.it_right (addth NONE (newAs(rev rAs, 0, [BBi], tpairs)))
(Seq.make (fn()=> cell), Seq.empty)
in if eres_flg then eres(rev rAs)
else res(Seq.pull(Unify.unifiers(thy, env, dpairs)))
end;
end;
fun compose_no_flatten match (orule, nsubgoal) i state =
bicompose_aux false match (state, dest_state (state, i), false) (false, orule, nsubgoal);
fun bicompose match arg i state =
bicompose_aux true match (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 => could_unify (A1, H)) Hs
| could_reshyp [] = false; (*no premise -- illegal*)
in 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 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 true match (state, (stpairs, Bs, Bi, C), true);
fun res [] = Seq.empty
| res ((eres_flg, rule)::brules) =
if !Pattern.trace_unify_fail 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 ***)
fun invoke_oracle_i thy1 name =
let
val oracle =
(case Symtab.lookup (Theory.oracle_table thy1) name of
NONE => raise THM ("Unknown oracle: " ^ name, 0, [])
| SOME (f, _) => f);
val thy_ref1 = Theory.check_thy thy1;
in
fn (thy2, data) =>
let
val thy' = Theory.merge (Theory.deref thy_ref1, thy2);
val (prop, T, maxidx) = Sign.certify_term thy' (oracle (thy', data));
val shyps' = may_insert_term_sorts thy' prop [];
val der = (true, Pt.oracle_proof name prop);
in
if T <> propT then
raise THM ("Oracle's result must have type prop: " ^ name, 0, [])
else
Thm {thy_ref = Theory.check_thy thy', der = der, tags = [],
maxidx = maxidx, shyps = shyps', hyps = [], tpairs = [], prop = prop}
end
end;
fun invoke_oracle thy =
invoke_oracle_i thy o NameSpace.intern (Theory.oracle_space thy);
end;
end;
end;
end;
structure BasicThm: BASIC_THM = Thm;
open BasicThm;