(* Title: Pure/thm.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
The core of Isabelle's Meta Logic: certified types and terms, meta
theorems, meta rules (including resolution and simplification).
*)
signature THM =
sig
(*certified types*)
type ctyp
val rep_ctyp : ctyp -> {sign: Sign.sg, T: typ}
val typ_of : ctyp -> typ
val ctyp_of : Sign.sg -> typ -> ctyp
val read_ctyp : Sign.sg -> string -> ctyp
(*certified terms*)
type cterm
exception CTERM of string
val rep_cterm : cterm -> {sign: Sign.sg, t: term, T: typ,
maxidx: int}
val term_of : cterm -> term
val cterm_of : Sign.sg -> term -> cterm
val read_cterm : Sign.sg -> string * typ -> cterm
val read_cterms : Sign.sg -> string list * typ list -> cterm list
val cterm_fun : (term -> term) -> (cterm -> cterm)
val dest_comb : cterm -> cterm * cterm
val dest_abs : cterm -> cterm * cterm
val adjust_maxidx : cterm -> cterm
val capply : cterm -> cterm -> cterm
val cabs : cterm -> cterm -> cterm
val read_def_cterm :
Sign.sg * (indexname -> typ option) * (indexname -> sort option) ->
string list -> bool -> string * typ -> cterm * (indexname * typ) list
(*theories*)
(*proof terms [must duplicate declaration as a specification]*)
datatype deriv_kind = MinDeriv | ThmDeriv | FullDeriv;
val keep_derivs : deriv_kind ref
datatype rule =
MinProof
| Oracle of theory * Sign.sg * exn
| Axiom of theory * string
| Theorem of string
| Assume of cterm
| Implies_intr of cterm
| Implies_intr_shyps
| Implies_intr_hyps
| Implies_elim
| Forall_intr of cterm
| Forall_elim of cterm
| Reflexive of cterm
| Symmetric
| Transitive
| Beta_conversion of cterm
| Extensional
| Abstract_rule of string * cterm
| Combination
| Equal_intr
| Equal_elim
| Trivial of cterm
| Lift_rule of cterm * int
| Assumption of int * Envir.env option
| Instantiate of (indexname * ctyp) list * (cterm * cterm) list
| Bicompose of bool * bool * int * int * Envir.env
| Flexflex_rule of Envir.env
| Class_triv of theory * class
| VarifyT
| FreezeT
| RewriteC of cterm
| CongC of cterm
| Rewrite_cterm of cterm
| Rename_params_rule of string list * int;
type deriv (* = rule mtree *)
(*meta theorems*)
type thm
exception THM of string * int * thm list
val rep_thm : thm -> {sign: Sign.sg, der: deriv, maxidx: int,
shyps: sort list, hyps: term list,
prop: term}
val crep_thm : thm -> {sign: Sign.sg, der: deriv, maxidx: int,
shyps: sort list, hyps: cterm list,
prop: cterm}
val stamps_of_thm : thm -> string ref list
val tpairs_of : thm -> (term * term) list
val prems_of : thm -> term list
val nprems_of : thm -> int
val concl_of : thm -> term
val cprop_of : thm -> cterm
val extra_shyps : thm -> sort list
val force_strip_shyps : bool ref (* FIXME tmp *)
val strip_shyps : thm -> thm
val implies_intr_shyps: thm -> thm
val get_axiom : theory -> string -> thm
val name_thm : string * thm -> thm
val axioms_of : theory -> (string * thm) list
(*meta rules*)
val assume : cterm -> thm
val compress : thm -> 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 flexpair_def : thm
val reflexive : cterm -> thm
val symmetric : thm -> thm
val transitive : thm -> thm -> thm
val beta_conversion : cterm -> thm
val extensional : thm -> thm
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 implies_intr_hyps : thm -> thm
val flexflex_rule : thm -> thm Sequence.seq
val instantiate :
(indexname * ctyp) list * (cterm * cterm) list -> thm -> thm
val trivial : cterm -> thm
val class_triv : theory -> class -> thm
val varifyT : thm -> thm
val freezeT : thm -> thm
val dest_state : thm * int ->
(term * term) list * term list * term * term
val lift_rule : (thm * int) -> thm -> thm
val assumption : int -> thm -> thm Sequence.seq
val eq_assumption : int -> thm -> thm
val rename_params_rule: string list * int -> thm -> thm
val bicompose : bool -> bool * thm * int ->
int -> thm -> thm Sequence.seq
val biresolution : bool -> (bool * thm) list ->
int -> thm -> thm Sequence.seq
(*meta simplification*)
type meta_simpset
exception SIMPLIFIER of string * thm
val empty_mss : meta_simpset
val add_simps : meta_simpset * thm list -> meta_simpset
val del_simps : meta_simpset * thm list -> meta_simpset
val mss_of : thm list -> meta_simpset
val add_congs : meta_simpset * thm list -> meta_simpset
val add_prems : meta_simpset * thm list -> meta_simpset
val prems_of_mss : meta_simpset -> thm list
val set_mk_rews : meta_simpset * (thm -> thm list) -> meta_simpset
val mk_rews_of_mss : meta_simpset -> thm -> thm list
val trace_simp : bool ref
val rewrite_cterm : bool * bool -> meta_simpset ->
(meta_simpset -> thm -> thm option) -> cterm -> thm
val invoke_oracle : theory * Sign.sg * exn -> thm
end;
structure Thm : THM =
struct
(*** Certified terms and types ***)
(** certified types **)
(*certified typs under a signature*)
datatype ctyp = Ctyp of {sign: Sign.sg, T: typ};
fun rep_ctyp (Ctyp args) = args;
fun typ_of (Ctyp {T, ...}) = T;
fun ctyp_of sign T =
Ctyp {sign = sign, T = Sign.certify_typ sign T};
fun read_ctyp sign s =
Ctyp {sign = sign, T = Sign.read_typ (sign, K None) s};
(** certified terms **)
(*certified terms under a signature, with checked typ and maxidx of Vars*)
datatype cterm = Cterm of {sign: Sign.sg, t: term, T: typ, maxidx: int};
fun rep_cterm (Cterm args) = args;
fun term_of (Cterm {t, ...}) = t;
(*create a cterm by checking a "raw" term with respect to a signature*)
fun cterm_of sign tm =
let val (t, T, maxidx) = Sign.certify_term sign tm
in Cterm {sign = sign, t = t, T = T, maxidx = maxidx}
end;
fun cterm_fun f (Cterm {sign, t, ...}) = cterm_of sign (f t);
exception CTERM of string;
(*Destruct application in cterms*)
fun dest_comb (Cterm{sign, T, maxidx, t = A $ B}) =
let val typeA = fastype_of A;
val typeB =
case typeA of Type("fun",[S,T]) => S
| _ => error "Function type expected in dest_comb";
in
(Cterm {sign=sign, maxidx=maxidx, t=A, T=typeA},
Cterm {sign=sign, maxidx=maxidx, t=B, T=typeB})
end
| dest_comb _ = raise CTERM "dest_comb";
(*Destruct abstraction in cterms*)
fun dest_abs (Cterm {sign, T as Type("fun",[_,S]), maxidx, t=Abs(x,ty,M)}) =
let val (y,N) = variant_abs (x,ty,M)
in (Cterm {sign = sign, T = ty, maxidx = 0, t = Free(y,ty)},
Cterm {sign = sign, T = S, maxidx = maxidx, t = N})
end
| dest_abs _ = raise CTERM "dest_abs";
fun adjust_maxidx (Cterm {sign, T, t, ...}) =
Cterm {sign = sign, T = T, maxidx = maxidx_of_term t, t = t}
(*Form cterm out of a function and an argument*)
fun capply (Cterm {t=f, sign=sign1, T=Type("fun",[dty,rty]), maxidx=maxidx1})
(Cterm {t=x, sign=sign2, T, maxidx=maxidx2}) =
if T = dty then Cterm{t=f$x, sign=Sign.merge(sign1,sign2), T=rty,
maxidx=max[maxidx1, maxidx2]}
else raise CTERM "capply: types don't agree"
| capply _ _ = raise CTERM "capply: first arg is not a function"
fun cabs (Cterm {t=Free(a,ty), sign=sign1, T=T1, maxidx=maxidx1})
(Cterm {t=t2, sign=sign2, T=T2, maxidx=maxidx2}) =
Cterm {t=absfree(a,ty,t2), sign=Sign.merge(sign1,sign2),
T = ty --> T2, maxidx=max[maxidx1, maxidx2]}
| cabs _ _ = raise CTERM "cabs: first arg is not a free variable";
(** read cterms **) (*exception ERROR*)
(*read term, infer types, certify term*)
fun read_def_cterm (sign, types, sorts) used freeze (a, T) =
let
val T' = Sign.certify_typ sign T
handle TYPE (msg, _, _) => error msg;
val ts = Syntax.read (#syn (Sign.rep_sg sign)) T' a;
val (_, t', tye) =
Sign.infer_types sign types sorts used freeze (ts, T');
val ct = cterm_of sign t'
handle TYPE arg => error (Sign.exn_type_msg sign arg)
| TERM (msg, _) => error msg;
in (ct, tye) end;
fun read_cterm sign = #1 o read_def_cterm (sign, K None, K None) [] true;
(*read a list of terms, matching them against a list of expected types.
NO disambiguation of alternative parses via type-checking -- it is just
not practical.*)
fun read_cterms sign (bs, Ts) =
let
val {tsig, syn, ...} = Sign.rep_sg sign
fun read (b,T) =
case Syntax.read syn T b of
[t] => t
| _ => error("Error or ambiguity in parsing of " ^ b)
val (us,_) = Type.infer_types(tsig, Sign.const_type sign,
K None, K None,
[], true,
map (Sign.certify_typ sign) Ts,
map read (bs~~Ts))
in map (cterm_of sign) us end
handle TYPE arg => error (Sign.exn_type_msg sign arg)
| TERM (msg, _) => error msg;
(*** Derivations ***)
(*Names of rules in derivations. Includes logically trivial rules, if
executed in ML.*)
datatype rule =
MinProof (*for building minimal proof terms*)
| Oracle of theory * Sign.sg * exn (*oracles*)
(*Axioms/theorems*)
| Axiom of theory * string
| Theorem of string
(*primitive inferences and compound versions of them*)
| Assume of cterm
| Implies_intr of cterm
| Implies_intr_shyps
| Implies_intr_hyps
| Implies_elim
| Forall_intr of cterm
| Forall_elim of cterm
| Reflexive of cterm
| Symmetric
| Transitive
| Beta_conversion of cterm
| Extensional
| Abstract_rule of string * cterm
| Combination
| Equal_intr
| Equal_elim
(*derived rules for tactical proof*)
| Trivial of cterm
(*For lift_rule, the proof state is not a premise.
Use cterm instead of thm to avoid mutual recursion.*)
| Lift_rule of cterm * int
| Assumption of int * Envir.env option (*includes eq_assumption*)
| Instantiate of (indexname * ctyp) list * (cterm * cterm) list
| Bicompose of bool * bool * int * int * Envir.env
| Flexflex_rule of Envir.env (*identifies unifier chosen*)
(*other derived rules*)
| Class_triv of theory * class (*derived rule????*)
| VarifyT
| FreezeT
(*for the simplifier*)
| RewriteC of cterm
| CongC of cterm
| Rewrite_cterm of cterm
(*Logical identities, recorded since they are part of the proof process*)
| Rename_params_rule of string list * int;
type deriv = rule mtree;
datatype deriv_kind = MinDeriv | ThmDeriv | FullDeriv;
val keep_derivs = ref MinDeriv;
(*Build a minimal derivation. Keep oracles; suppress atomic inferences;
retain Theorems or their underlying links; keep anything else*)
fun squash_derivs [] = []
| squash_derivs (der::ders) =
(case der of
Join (Oracle _, _) => der :: squash_derivs ders
| Join (Theorem _, [der']) => if !keep_derivs=ThmDeriv
then der :: squash_derivs ders
else squash_derivs (der'::ders)
| Join (Axiom _, _) => if !keep_derivs=ThmDeriv
then der :: squash_derivs ders
else squash_derivs ders
| Join (_, []) => squash_derivs ders
| _ => der :: squash_derivs ders);
(*Ensure sharing of the most likely derivation, the empty one!*)
val min_infer = Join (MinProof, []);
(*Make a minimal inference*)
fun make_min_infer [] = min_infer
| make_min_infer [der] = der
| make_min_infer ders = Join (MinProof, ders);
fun infer_derivs (rl, []) = Join (rl, [])
| infer_derivs (rl, ders) =
if !keep_derivs=FullDeriv then Join (rl, ders)
else make_min_infer (squash_derivs ders);
(*** Meta theorems ***)
datatype thm = Thm of
{sign: Sign.sg, (*signature for hyps and prop*)
der: deriv, (*derivation*)
maxidx: int, (*maximum index of any Var or TVar*)
shyps: sort list, (* FIXME comment *)
hyps: term list, (*hypotheses*)
prop: term}; (*conclusion*)
fun rep_thm (Thm args) = args;
(*Version of rep_thm returning cterms instead of terms*)
fun crep_thm (Thm {sign, der, maxidx, shyps, hyps, prop}) =
let fun ctermf max t = Cterm{sign=sign, t=t, T=propT, maxidx=max};
in {sign=sign, der=der, maxidx=maxidx, shyps=shyps,
hyps = map (ctermf ~1) hyps,
prop = ctermf maxidx prop}
end;
(*errors involving theorems*)
exception THM of string * int * thm list;
val stamps_of_thm = #stamps o Sign.rep_sg o #sign o rep_thm;
(*merge signatures of two theorems; raise exception if incompatible*)
fun merge_thm_sgs (th1, th2) =
Sign.merge (pairself (#sign o rep_thm) (th1, th2))
handle TERM (msg, _) => raise THM (msg, 0, [th1, th2]);
(*maps object-rule to tpairs*)
fun tpairs_of (Thm {prop, ...}) = #1 (Logic.strip_flexpairs prop);
(*maps object-rule to premises*)
fun prems_of (Thm {prop, ...}) =
Logic.strip_imp_prems (Logic.skip_flexpairs prop);
(*counts premises in a rule*)
fun nprems_of (Thm {prop, ...}) =
Logic.count_prems (Logic.skip_flexpairs prop, 0);
(*maps object-rule to conclusion*)
fun concl_of (Thm {prop, ...}) = Logic.strip_imp_concl prop;
(*the statement of any thm is a cterm*)
fun cprop_of (Thm {sign, maxidx, prop, ...}) =
Cterm {sign = sign, maxidx = maxidx, T = propT, t = prop};
(** sort contexts of theorems **)
(* basic utils *)
(*accumulate sorts suppressing duplicates; these are coded low level
to improve efficiency a bit*)
fun add_typ_sorts (Type (_, Ts), Ss) = add_typs_sorts (Ts, Ss)
| add_typ_sorts (TFree (_, S), Ss) = S ins Ss
| add_typ_sorts (TVar (_, S), Ss) = S ins Ss
and add_typs_sorts ([], Ss) = Ss
| add_typs_sorts (T :: Ts, Ss) = add_typs_sorts (Ts, add_typ_sorts (T, Ss));
fun add_term_sorts (Const (_, T), Ss) = add_typ_sorts (T, Ss)
| add_term_sorts (Free (_, T), Ss) = add_typ_sorts (T, Ss)
| add_term_sorts (Var (_, T), Ss) = add_typ_sorts (T, Ss)
| add_term_sorts (Bound _, Ss) = Ss
| add_term_sorts (Abs (_, T, t), Ss) = add_term_sorts (t, add_typ_sorts (T, Ss))
| add_term_sorts (t $ u, Ss) = add_term_sorts (t, add_term_sorts (u, Ss));
fun add_terms_sorts ([], Ss) = Ss
| add_terms_sorts (t :: ts, Ss) = add_terms_sorts (ts, add_term_sorts (t, Ss));
fun env_codT (Envir.Envir {iTs, ...}) = map snd iTs;
fun add_env_sorts (env, Ss) =
add_terms_sorts (map snd (Envir.alist_of env),
add_typs_sorts (env_codT env, Ss));
fun add_thm_sorts (Thm {hyps, prop, ...}, Ss) =
add_terms_sorts (hyps, add_term_sorts (prop, Ss));
fun add_thms_shyps ([], Ss) = Ss
| add_thms_shyps (Thm {shyps, ...} :: ths, Ss) =
add_thms_shyps (ths, shyps union Ss);
(*get 'dangling' sort constraints of a thm*)
fun extra_shyps (th as Thm {shyps, ...}) =
shyps \\ add_thm_sorts (th, []);
(* fix_shyps *)
(*preserve sort contexts of rule premises and substituted types*)
fun fix_shyps thms Ts thm =
let
val Thm {sign, der, maxidx, hyps, prop, ...} = thm;
val shyps =
add_thm_sorts (thm, add_typs_sorts (Ts, add_thms_shyps (thms, [])));
in
Thm {sign = sign,
der = der, (*No new derivation, as other rules call this*)
maxidx = maxidx,
shyps = shyps, hyps = hyps, prop = prop}
end;
(* strip_shyps *) (* FIXME improve? (e.g. only minimal extra sorts) *)
val force_strip_shyps = ref true; (* FIXME tmp *)
(*remove extra sorts that are known to be syntactically non-empty*)
fun strip_shyps thm =
let
val Thm {sign, der, maxidx, shyps, hyps, prop} = thm;
val sorts = add_thm_sorts (thm, []);
val maybe_empty = not o Sign.nonempty_sort sign sorts;
val shyps' = filter (fn S => S mem sorts orelse maybe_empty S) shyps;
in
Thm {sign = sign, der = der, maxidx = maxidx,
shyps =
(if eq_set (shyps',sorts) orelse not (!force_strip_shyps) then shyps'
else (* FIXME tmp *)
(writeln ("WARNING Removed sort hypotheses: " ^
commas (map Type.str_of_sort (shyps' \\ sorts)));
writeln "WARNING Let's hope these sorts are non-empty!";
sorts)),
hyps = hyps,
prop = prop}
end;
(* implies_intr_shyps *)
(*discharge all extra sort hypotheses*)
fun implies_intr_shyps thm =
(case extra_shyps thm of
[] => thm
| xshyps =>
let
val Thm {sign, der, maxidx, shyps, hyps, prop} = thm;
val shyps' = logicS ins (shyps \\ xshyps);
val used_names = foldr add_term_tfree_names (prop :: hyps, []);
val names =
tl (variantlist (replicate (length xshyps + 1) "'", used_names));
val tfrees = map (TFree o rpair logicS) names;
fun mk_insort (T, S) = map (Logic.mk_inclass o pair T) S;
val sort_hyps = flat (map2 mk_insort (tfrees, xshyps));
in
Thm {sign = sign,
der = infer_derivs (Implies_intr_shyps, [der]),
maxidx = maxidx,
shyps = shyps',
hyps = hyps,
prop = Logic.list_implies (sort_hyps, prop)}
end);
(** Axioms **)
(*look up the named axiom in the theory*)
fun get_axiom theory name =
let
fun get_ax [] = raise Match
| get_ax (thy :: thys) =
let val {sign, new_axioms, parents, ...} = rep_theory thy
in case Symtab.lookup (new_axioms, name) of
Some t => fix_shyps [] []
(Thm {sign = sign,
der = infer_derivs (Axiom(theory,name), []),
maxidx = maxidx_of_term t,
shyps = [],
hyps = [],
prop = t})
| None => get_ax parents handle Match => get_ax thys
end;
in
get_ax [theory] handle Match
=> raise THEORY ("get_axiom: no axiom " ^ quote name, [theory])
end;
(*return additional axioms of this theory node*)
fun axioms_of thy =
map (fn (s, _) => (s, get_axiom thy s))
(Symtab.dest (#new_axioms (rep_theory thy)));
(*Attach a label to a theorem to make proof objects more readable*)
fun name_thm (name, th as Thm {sign, der, maxidx, shyps, hyps, prop}) =
Thm {sign = sign,
der = Join (Theorem name, [der]),
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
prop = prop};
(*Compression of theorems -- a separate rule, not integrated with the others,
as it could be slow.*)
fun compress (Thm {sign, der, maxidx, shyps, hyps, prop}) =
Thm {sign = sign,
der = der, (*No derivation recorded!*)
maxidx = maxidx,
shyps = shyps,
hyps = map Term.compress_term hyps,
prop = Term.compress_term prop};
(*** Meta rules ***)
(* check that term does not contain same var with different typing/sorting *)
fun nodup_Vars(thm as Thm{prop,...}) s =
Sign.nodup_Vars prop handle TYPE(msg,_,_) => raise THM(s^": "^msg,0,[thm]);
(** 'primitive' rules **)
(*discharge all assumptions t from ts*)
val disch = gen_rem (op aconv);
(*The assumption rule A|-A in a theory*)
fun assume ct : thm =
let val {sign, t=prop, T, maxidx} = rep_cterm ct
in if T<>propT then
raise THM("assume: assumptions must have type prop", 0, [])
else if maxidx <> ~1 then
raise THM("assume: assumptions may not contain scheme variables",
maxidx, [])
else Thm{sign = sign,
der = infer_derivs (Assume ct, []),
maxidx = ~1,
shyps = add_term_sorts(prop,[]),
hyps = [prop],
prop = prop}
end;
(*Implication introduction
A |- B
-------
A ==> B
*)
fun implies_intr cA (thB as Thm{sign,der,maxidx,hyps,prop,...}) : thm =
let val {sign=signA, t=A, T, maxidx=maxidxA} = rep_cterm cA
in if T<>propT then
raise THM("implies_intr: assumptions must have type prop", 0, [thB])
else fix_shyps [thB] []
(Thm{sign = Sign.merge (sign,signA),
der = infer_derivs (Implies_intr cA, [der]),
maxidx = max[maxidxA, maxidx],
shyps = [],
hyps = disch(hyps,A),
prop = implies$A$prop})
handle TERM _ =>
raise THM("implies_intr: incompatible signatures", 0, [thB])
end;
(*Implication elimination
A ==> B A
------------
B
*)
fun implies_elim thAB thA : thm =
let val Thm{maxidx=maxA, der=derA, hyps=hypsA, prop=propA,...} = thA
and Thm{sign, der, maxidx, hyps, prop,...} = thAB;
fun err(a) = raise THM("implies_elim: "^a, 0, [thAB,thA])
in case prop of
imp$A$B =>
if imp=implies andalso A aconv propA
then fix_shyps [thAB, thA] []
(Thm{sign= merge_thm_sgs(thAB,thA),
der = infer_derivs (Implies_elim, [der,derA]),
maxidx = max[maxA,maxidx],
shyps = [],
hyps = hypsA union hyps, (*dups suppressed*)
prop = B})
else err("major premise")
| _ => err("major premise")
end;
(*Forall introduction. The Free or Var x must not be free in the hypotheses.
A
-----
!!x.A
*)
fun forall_intr cx (th as Thm{sign,der,maxidx,hyps,prop,...}) =
let val x = term_of cx;
fun result(a,T) = fix_shyps [th] []
(Thm{sign = sign,
der = infer_derivs (Forall_intr cx, [der]),
maxidx = maxidx,
shyps = [],
hyps = hyps,
prop = all(T) $ Abs(a, T, abstract_over (x,prop))})
in case x of
Free(a,T) =>
if exists (apl(x, Logic.occs)) hyps
then raise THM("forall_intr: variable free in assumptions", 0, [th])
else result(a,T)
| Var((a,_),T) => result(a,T)
| _ => raise THM("forall_intr: not a variable", 0, [th])
end;
(*Forall elimination
!!x.A
------
A[t/x]
*)
fun forall_elim ct (th as Thm{sign,der,maxidx,hyps,prop,...}) : thm =
let val {sign=signt, t, T, maxidx=maxt} = rep_cterm ct
in case prop of
Const("all",Type("fun",[Type("fun",[qary,_]),_])) $ A =>
if T<>qary then
raise THM("forall_elim: type mismatch", 0, [th])
else let val thm = fix_shyps [th] []
(Thm{sign= Sign.merge(sign,signt),
der = infer_derivs (Forall_elim ct, [der]),
maxidx = max[maxidx, maxt],
shyps = [],
hyps = hyps,
prop = betapply(A,t)})
in nodup_Vars thm "forall_elim"; thm end
| _ => raise THM("forall_elim: not quantified", 0, [th])
end
handle TERM _ =>
raise THM("forall_elim: incompatible signatures", 0, [th]);
(* Equality *)
(* Definition of the relation =?= *)
val flexpair_def = fix_shyps [] []
(Thm{sign= Sign.proto_pure,
der = Join(Axiom(pure_thy, "flexpair_def"), []),
shyps = [],
hyps = [],
maxidx = 0,
prop = term_of (read_cterm Sign.proto_pure
("(?t =?= ?u) == (?t == ?u::?'a::{})", propT))});
(*The reflexivity rule: maps t to the theorem t==t *)
fun reflexive ct =
let val {sign, t, T, maxidx} = rep_cterm ct
in fix_shyps [] []
(Thm{sign= sign,
der = infer_derivs (Reflexive ct, []),
shyps = [],
hyps = [],
maxidx = maxidx,
prop = Logic.mk_equals(t,t)})
end;
(*The symmetry rule
t==u
----
u==t
*)
fun symmetric (th as Thm{sign,der,maxidx,shyps,hyps,prop}) =
case prop of
(eq as Const("==",_)) $ t $ u =>
(*no fix_shyps*)
Thm{sign = sign,
der = infer_derivs (Symmetric, [der]),
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
prop = eq$u$t}
| _ => raise THM("symmetric", 0, [th]);
(*The transitive rule
t1==u u==t2
--------------
t1==t2
*)
fun transitive th1 th2 =
let val Thm{der=der1, maxidx=max1, hyps=hyps1, prop=prop1,...} = th1
and Thm{der=der2, maxidx=max2, hyps=hyps2, prop=prop2,...} = th2;
fun err(msg) = raise THM("transitive: "^msg, 0, [th1,th2])
in case (prop1,prop2) of
((eq as Const("==",_)) $ t1 $ u, Const("==",_) $ u' $ t2) =>
if not (u aconv u') then err"middle term"
else let val thm =
fix_shyps [th1, th2] []
(Thm{sign= merge_thm_sgs(th1,th2),
der = infer_derivs (Transitive, [der1, der2]),
maxidx = max[max1,max2],
shyps = [],
hyps = hyps1 union hyps2,
prop = eq$t1$t2})
in nodup_Vars thm "transitive"; thm end
| _ => err"premises"
end;
(*Beta-conversion: maps (%x.t)(u) to the theorem (%x.t)(u) == t[u/x] *)
fun beta_conversion ct =
let val {sign, t, T, maxidx} = rep_cterm ct
in case t of
Abs(_,_,bodt) $ u => fix_shyps [] []
(Thm{sign = sign,
der = infer_derivs (Beta_conversion ct, []),
maxidx = maxidx_of_term t,
shyps = [],
hyps = [],
prop = Logic.mk_equals(t, subst_bounds([u],bodt))})
| _ => raise THM("beta_conversion: not a redex", 0, [])
end;
(*The extensionality rule (proviso: x not free in f, g, or hypotheses)
f(x) == g(x)
------------
f == g
*)
fun extensional (th as Thm{sign, der, maxidx,shyps,hyps,prop}) =
case prop of
(Const("==",_)) $ (f$x) $ (g$y) =>
let fun err(msg) = raise THM("extensional: "^msg, 0, [th])
in (if x<>y then err"different variables" else
case y of
Free _ =>
if exists (apl(y, Logic.occs)) (f::g::hyps)
then err"variable free in hyps or functions" else ()
| Var _ =>
if Logic.occs(y,f) orelse Logic.occs(y,g)
then err"variable free in functions" else ()
| _ => err"not a variable");
(*no fix_shyps*)
Thm{sign = sign,
der = infer_derivs (Extensional, [der]),
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
prop = Logic.mk_equals(f,g)}
end
| _ => raise THM("extensional: premise", 0, [th]);
(*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 cx (th as Thm{sign,der,maxidx,hyps,prop,...}) =
let val x = term_of cx;
val (t,u) = Logic.dest_equals prop
handle TERM _ =>
raise THM("abstract_rule: premise not an equality", 0, [th])
fun result T = fix_shyps [th] []
(Thm{sign = sign,
der = infer_derivs (Abstract_rule (a,cx), [der]),
maxidx = maxidx,
shyps = [],
hyps = hyps,
prop = Logic.mk_equals(Abs(a, T, abstract_over (x,t)),
Abs(a, T, abstract_over (x,u)))})
in case x of
Free(_,T) =>
if exists (apl(x, Logic.occs)) hyps
then raise THM("abstract_rule: variable free in assumptions", 0, [th])
else result T
| Var(_,T) => result T
| _ => 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,
prop=prop1,...} = th1
and Thm{der=der2, maxidx=max2, shyps=shyps2, hyps=hyps2,
prop=prop2,...} = th2
fun chktypes (f,t) =
(case fastype_of f of
Type("fun",[T1,T2]) =>
if T1 <> fastype_of t then
raise THM("combination: types", 0, [th1,th2])
else ()
| _ => raise THM("combination: not function type", 0,
[th1,th2]))
in case (prop1,prop2) of
(Const("==",_) $ f $ g, Const("==",_) $ t $ u) =>
let val _ = chktypes (f,t)
val thm = (*no fix_shyps*)
Thm{sign = merge_thm_sgs(th1,th2),
der = infer_derivs (Combination, [der1, der2]),
maxidx = max[max1,max2],
shyps = shyps1 union shyps2,
hyps = hyps1 union hyps2,
prop = Logic.mk_equals(f$t, g$u)}
in nodup_Vars thm "combination"; thm end
| _ => 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,
prop=prop1,...} = th1
and Thm{der=der2, maxidx=max2, shyps=shyps2, hyps=hyps2,
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
(*no fix_shyps*)
Thm{sign = merge_thm_sgs(th1,th2),
der = infer_derivs (Equal_intr, [der1, der2]),
maxidx = max[max1,max2],
shyps = shyps1 union shyps2,
hyps = hyps1 union hyps2,
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, hyps=hyps1, prop=prop1,...} = th1
and Thm{der=der2, maxidx=max2, hyps=hyps2, prop=prop2,...} = th2;
fun err(msg) = raise THM("equal_elim: "^msg, 0, [th1,th2])
in case prop1 of
Const("==",_) $ A $ B =>
if not (prop2 aconv A) then err"not equal" else
fix_shyps [th1, th2] []
(Thm{sign= merge_thm_sgs(th1,th2),
der = infer_derivs (Equal_elim, [der1, der2]),
maxidx = max[max1,max2],
shyps = [],
hyps = hyps1 union hyps2,
prop = B})
| _ => err"major premise"
end;
(**** Derived rules ****)
(*Discharge all hypotheses. Need not verify cterms or call fix_shyps.
Repeated hypotheses are discharged only once; fold cannot do this*)
fun implies_intr_hyps (Thm{sign, der, maxidx, shyps, hyps=A::As, prop}) =
implies_intr_hyps (*no fix_shyps*)
(Thm{sign = sign,
der = infer_derivs (Implies_intr_hyps, [der]),
maxidx = maxidx,
shyps = shyps,
hyps = disch(As,A),
prop = implies$A$prop})
| implies_intr_hyps th = th;
(*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{sign, der, maxidx, hyps, prop,...}) =
let fun newthm env =
if Envir.is_empty env then th
else
let val (tpairs,horn) =
Logic.strip_flexpairs (Envir.norm_term env prop)
(*Remove trivial tpairs, of the form t=t*)
val distpairs = filter (not o op aconv) tpairs
val newprop = Logic.list_flexpairs(distpairs, horn)
in fix_shyps [th] (env_codT env)
(Thm{sign = sign,
der = infer_derivs (Flexflex_rule env, [der]),
maxidx = maxidx_of_term newprop,
shyps = [],
hyps = hyps,
prop = newprop})
end;
val (tpairs,_) = Logic.strip_flexpairs prop
in Sequence.maps newthm
(Unify.smash_unifiers(sign, Envir.empty maxidx, tpairs))
end;
(*Instantiation of Vars
A
-------------------
A[t1/v1,....,tn/vn]
*)
(*Check that all the terms are Vars and are distinct*)
fun instl_ok ts = forall is_Var ts andalso null(findrep ts);
(*For instantiate: process pair of cterms, merge theories*)
fun add_ctpair ((ct,cu), (sign,tpairs)) =
let val {sign=signt, t=t, T= T, ...} = rep_cterm ct
and {sign=signu, t=u, T= U, ...} = rep_cterm cu
in if T=U then (Sign.merge(sign, Sign.merge(signt, signu)), (t,u)::tpairs)
else raise TYPE("add_ctpair", [T,U], [t,u])
end;
fun add_ctyp ((v,ctyp), (sign',vTs)) =
let val {T,sign} = rep_ctyp ctyp
in (Sign.merge(sign,sign'), (v,T)::vTs) end;
(*Left-to-right replacements: ctpairs = [...,(vi,ti),...].
Instantiates distinct Vars by terms of same type.
Normalizes the new theorem! *)
fun instantiate ([], []) th = th
| instantiate (vcTs,ctpairs) (th as Thm{sign,der,maxidx,hyps,prop,...}) =
let val (newsign,tpairs) = foldr add_ctpair (ctpairs, (sign,[]));
val (newsign,vTs) = foldr add_ctyp (vcTs, (newsign,[]));
val newprop =
Envir.norm_term (Envir.empty 0)
(subst_atomic tpairs
(Type.inst_term_tvars(#tsig(Sign.rep_sg newsign),vTs) prop))
val newth =
fix_shyps [th] (map snd vTs)
(Thm{sign = newsign,
der = infer_derivs (Instantiate(vcTs,ctpairs), [der]),
maxidx = maxidx_of_term newprop,
shyps = [],
hyps = hyps,
prop = newprop})
in if not(instl_ok(map #1 tpairs))
then raise THM("instantiate: variables not distinct", 0, [th])
else if not(null(findrep(map #1 vTs)))
then raise THM("instantiate: type variables not distinct", 0, [th])
else nodup_Vars newth "instantiate";
newth
end
handle TERM _ =>
raise THM("instantiate: incompatible signatures",0,[th])
| TYPE _ => raise THM("instantiate: type conflict", 0, [th]);
(*The trivial implication A==>A, justified by assume and forall rules.
A can contain Vars, not so for assume! *)
fun trivial ct : thm =
let val {sign, t=A, T, maxidx} = rep_cterm ct
in if T<>propT then
raise THM("trivial: the term must have type prop", 0, [])
else fix_shyps [] []
(Thm{sign = sign,
der = infer_derivs (Trivial ct, []),
maxidx = maxidx,
shyps = [],
hyps = [],
prop = implies$A$A})
end;
(*Axiom-scheme reflecting signature contents: "OFCLASS(?'a::c, c_class)" *)
fun class_triv thy c =
let val sign = sign_of thy;
val Cterm {t, maxidx, ...} =
cterm_of sign (Logic.mk_inclass (TVar (("'a", 0), [c]), c))
handle TERM (msg, _) => raise THM ("class_triv: " ^ msg, 0, []);
in
fix_shyps [] []
(Thm {sign = sign,
der = infer_derivs (Class_triv(thy,c), []),
maxidx = maxidx,
shyps = [],
hyps = [],
prop = t})
end;
(* Replace all TFrees not in the hyps by new TVars *)
fun varifyT(Thm{sign,der,maxidx,shyps,hyps,prop}) =
let val tfrees = foldr add_term_tfree_names (hyps,[])
in let val thm = (*no fix_shyps*)
Thm{sign = sign,
der = infer_derivs (VarifyT, [der]),
maxidx = max[0,maxidx],
shyps = shyps,
hyps = hyps,
prop = Type.varify(prop,tfrees)}
in nodup_Vars thm "varifyT"; thm end
(* this nodup_Vars check can be removed if thms are guaranteed not to contain
duplicate TVars with differnt sorts *)
end;
(* Replace all TVars by new TFrees *)
fun freezeT(Thm{sign,der,maxidx,shyps,hyps,prop}) =
let val prop' = Type.freeze prop
in (*no fix_shyps*)
Thm{sign = sign,
der = infer_derivs (FreezeT, [der]),
maxidx = maxidx_of_term prop',
shyps = shyps,
hyps = hyps,
prop = prop'}
end;
(*** Inference rules for tactics ***)
(*Destruct proof state into constraints, other goals, goal(i), rest *)
fun dest_state (state as Thm{prop,...}, i) =
let val (tpairs,horn) = Logic.strip_flexpairs prop
in case Logic.strip_prems(i, [], horn) of
(B::rBs, C) => (tpairs, rev rBs, B, C)
| _ => raise THM("dest_state", i, [state])
end
handle TERM _ => raise THM("dest_state", i, [state]);
(*Increment variables and parameters of orule as required for
resolution with goal i of state. *)
fun lift_rule (state, i) orule =
let val Thm{shyps=sshyps, prop=sprop, maxidx=smax, sign=ssign,...} = state
val (Bi::_, _) = Logic.strip_prems(i, [], Logic.skip_flexpairs sprop)
handle TERM _ => raise THM("lift_rule", i, [orule,state])
val ct_Bi = Cterm {sign=ssign, maxidx=smax, T=propT, t=Bi}
val (lift_abs,lift_all) = Logic.lift_fns(Bi,smax+1)
val (Thm{sign, der, maxidx,shyps,hyps,prop}) = orule
val (tpairs,As,B) = Logic.strip_horn prop
in (*no fix_shyps*)
Thm{sign = merge_thm_sgs(state,orule),
der = infer_derivs (Lift_rule(ct_Bi, i), [der]),
maxidx = maxidx+smax+1,
shyps=sshyps union shyps,
hyps=hyps,
prop = Logic.rule_of (map (pairself lift_abs) tpairs,
map lift_all As,
lift_all B)}
end;
(*Solve subgoal Bi of proof state B1...Bn/C by assumption. *)
fun assumption i state =
let val Thm{sign,der,maxidx,hyps,prop,...} = state;
val (tpairs, Bs, Bi, C) = dest_state(state,i)
fun newth (env as Envir.Envir{maxidx, ...}, tpairs) =
fix_shyps [state] (env_codT env)
(Thm{sign = sign,
der = infer_derivs (Assumption (i, Some env), [der]),
maxidx = maxidx,
shyps = [],
hyps = hyps,
prop =
if Envir.is_empty env then (*avoid wasted normalizations*)
Logic.rule_of (tpairs, Bs, C)
else (*normalize the new rule fully*)
Envir.norm_term env (Logic.rule_of (tpairs, Bs, C))});
fun addprfs [] = Sequence.null
| addprfs ((t,u)::apairs) = Sequence.seqof (fn()=> Sequence.pull
(Sequence.mapp newth
(Unify.unifiers(sign,Envir.empty maxidx, (t,u)::tpairs))
(addprfs apairs)))
in addprfs (Logic.assum_pairs Bi) 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{sign,der,maxidx,hyps,prop,...} = state;
val (tpairs, Bs, Bi, C) = dest_state(state,i)
in if exists (op aconv) (Logic.assum_pairs Bi)
then fix_shyps [state] []
(Thm{sign = sign,
der = infer_derivs (Assumption (i,None), [der]),
maxidx = maxidx,
shyps = [],
hyps = hyps,
prop = Logic.rule_of(tpairs, Bs, C)})
else raise THM("eq_assumption", 0, [state])
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{sign,der,maxidx,hyps,prop,...} = 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 variantlist(take (short,iparams), cs) @ cs
val freenames = map (#1 o dest_Free) (term_frees prop)
val newBi = Logic.list_rename_params (newnames, Bi)
in
case findrep cs of
c::_ => error ("Bound variables not distinct: " ^ c)
| [] => (case cs inter_string freenames of
a::_ => error ("Bound/Free variable clash: " ^ a)
| [] => fix_shyps [state] []
(Thm{sign = sign,
der = infer_derivs (Rename_params_rule(cs,i), [der]),
maxidx = maxidx,
shyps = [],
hyps = hyps,
prop = Logic.rule_of(tpairs, Bs@[newBi], C)}))
end;
(*** Preservation of bound variable names ***)
(*Scan a pair of terms; while they are similar,
accumulate corresponding bound vars in "al"*)
fun match_bvs(Abs(x,_,s),Abs(y,_,t), al) =
match_bvs(s, t, if x="" orelse y="" then al
else (x,y)::al)
| match_bvs(f$s, g$t, al) = match_bvs(f,g,match_bvs(s,t,al))
| match_bvs(_,_,al) = al;
(* strip abstractions created by parameters *)
fun match_bvars((s,t),al) = match_bvs(strip_abs_body s, strip_abs_body t, al);
(* 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$Abs(a,T,strip(t1,t2))
| strip(A,_) = f A
in strip end;
(*Use the alist to rename all bound variables and some unknowns in a term
dpairs = current disagreement pairs; tpairs = permanent ones (flexflex);
Preserves unknowns in tpairs and on lhs of dpairs. *)
fun rename_bvs([],_,_,_) = I
| rename_bvs(al,dpairs,tpairs,B) =
let val vars = foldr add_term_vars
(map fst dpairs @ map fst tpairs @ map snd tpairs, [])
(*unknowns appearing elsewhere be preserved!*)
val vids = map (#1 o #1 o dest_Var) vars;
fun rename(t as Var((x,i),T)) =
(case assoc(al,x) of
Some(y) => if x mem_string vids orelse y mem_string vids then t
else Var((y,i),T)
| None=> t)
| rename(Abs(x,T,t)) =
Abs(case assoc_string(al,x) of Some(y) => y | None => x,
T, rename t)
| rename(f$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(foldr 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' = 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 open Sequence; exception COMPOSE
in
fun bicompose_aux 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,
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 sign = merge_thm_sgs(state,orule);
(** Add new theorem with prop = '[| Bs; As |] ==> C' to thq **)
fun addth As ((env as Envir.Envir {maxidx, ...}, tpairs), thq) =
let val normt = Envir.norm_term env;
(*perform minimal copying here by examining env*)
val normp =
if Envir.is_empty env then (tpairs, Bs @ As, C)
else
let val ntps = map (pairself normt) tpairs
in if the (Envir.minidx 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 = (*tuned fix_shyps*)
Thm{sign = sign,
der = infer_derivs (Bicompose(match, eres_flg,
1 + length Bs, nsubgoal, env),
[rder,sder]),
maxidx = maxidx,
shyps = add_env_sorts (env, rshyps union sshyps),
hyps = rhyps union shyps,
prop = Logic.rule_of normp}
in cons(th, thq) end handle COMPOSE => thq
val (rtpairs,rhorn) = Logic.strip_flexpairs(rprop);
val (rAs,B) = Logic.strip_prems(nsubgoal, [], rhorn)
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 = if !Logic.auto_rename orelse not lifted then As0
else map (rename_bvars(dpairs,tpairs,B)) As0
in (map (Logic.flatten_params n) As1)
handle TERM _ =>
raise THM("bicompose: 1st premise", 0, [orule])
end;
val env = Envir.empty(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 (_, _, []) = null
| tryasms (As, n, (t,u)::apairs) =
(case pull(Unify.unifiers(sign, env, (t,u)::dpairs)) of
None => tryasms (As, n+1, apairs)
| cell as Some((_,tpairs),_) =>
its_right (addth (newAs(As, n, [BBi,(u,t)], tpairs)))
(seqof (fn()=> cell),
seqof (fn()=> pull (tryasms (As, n+1, apairs)))));
fun eres [] = raise THM("bicompose: no premises", 0, [orule,state])
| eres (A1::As) = tryasms (As, 1, Logic.assum_pairs A1);
(*ordinary resolution*)
fun res(None) = null
| res(cell as Some((_,tpairs),_)) =
its_right (addth(newAs(rev rAs, 0, [BBi], tpairs)))
(seqof (fn()=> cell), null)
in if eres_flg then eres(rev rAs)
else res(pull(Unify.unifiers(sign, env, dpairs)))
end;
end; (*open Sequence*)
fun bicompose match arg i state =
bicompose_aux 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 (apl(A1,could_unify)) 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 lift = lift_rule(state, i);
val (stpairs, Bs, Bi, C) = dest_state(state,i)
val B = Logic.strip_assums_concl Bi;
val Hs = Logic.strip_assums_hyp Bi;
val comp = bicompose_aux match (state, (stpairs, Bs, Bi, C), true);
fun res [] = Sequence.null
| res ((eres_flg, rule)::brules) =
if could_bires (Hs, B, eres_flg, rule)
then Sequence.seqof (*delay processing remainder till needed*)
(fn()=> Some(comp (eres_flg, lift rule, nprems_of rule),
res brules))
else res brules
in Sequence.flats (res brules) end;
(*** Meta simp sets ***)
type rrule = {thm:thm, lhs:term, perm:bool};
type cong = {thm:thm, lhs:term};
datatype meta_simpset =
Mss of {net:rrule Net.net, congs:(string * cong)list, bounds:string list,
prems: thm list, mk_rews: thm -> thm list};
(*A "mss" contains data needed during conversion:
net: discrimination net of rewrite rules
congs: association list of congruence rules
bounds: names of bound variables already used;
for generating new names when rewriting under lambda abstractions
mk_rews: used when local assumptions are added
*)
val empty_mss = Mss{net = Net.empty, congs = [], bounds=[], prems = [],
mk_rews = K[]};
exception SIMPLIFIER of string * thm;
fun prtm a sign t = (writeln a; writeln(Sign.string_of_term sign t));
fun prtm_warning a sign t = warning (a ^ "\n" ^ (Sign.string_of_term sign t));
val trace_simp = ref false;
fun trace_term a sign t = if !trace_simp then prtm a sign t else ();
fun trace_thm a (Thm{sign,prop,...}) = trace_term a sign prop;
fun trace_term_warning a sign t = if !trace_simp then prtm_warning a sign t else ();
fun trace_thm_warning a (Thm{sign,prop,...}) = trace_term_warning a sign prop;
fun vperm(Var _, Var _) = true
| vperm(Abs(_,_,s), Abs(_,_,t)) = vperm(s,t)
| vperm(t1$t2, u1$u2) = vperm(t1,u1) andalso vperm(t2,u2)
| vperm(t,u) = (t=u);
fun var_perm(t,u) = vperm(t,u) andalso
eq_set(add_term_vars(t,[]), add_term_vars(u,[]))
(*simple test for looping rewrite*)
fun loops sign prems (lhs,rhs) =
is_Var(lhs)
orelse
(exists (apl(lhs, Logic.occs)) (rhs::prems))
orelse
(null(prems) andalso
Pattern.matches (#tsig(Sign.rep_sg sign)) (lhs,rhs));
(* the condition "null(prems)" in the last case is necessary because
conditional rewrites with extra variables in the conditions may terminate
although the rhs is an instance of the lhs. Example:
?m < ?n ==> f(?n) == f(?m)
*)
fun mk_rrule raw_thm =
let
val thm = strip_shyps raw_thm;
val Thm{sign,prop,maxidx,...} = thm;
val prems = Logic.strip_imp_prems prop
val concl = Logic.strip_imp_concl prop
val (lhs,_) = Logic.dest_equals concl handle TERM _ =>
raise SIMPLIFIER("Rewrite rule not a meta-equality",thm)
val econcl = Pattern.eta_contract concl
val (elhs,erhs) = Logic.dest_equals econcl
val perm = var_perm(elhs,erhs) andalso not(elhs aconv erhs)
andalso not(is_Var(elhs))
in
if not perm andalso loops sign prems (elhs,erhs) then
(prtm_warning "ignoring looping rewrite rule" sign prop; None)
else Some{thm=thm,lhs=lhs,perm=perm}
end;
local
fun eq({thm=Thm{prop=p1,...},...}:rrule,
{thm=Thm{prop=p2,...},...}:rrule) = p1 aconv p2
in
fun add_simp(mss as Mss{net,congs,bounds,prems,mk_rews},
thm as Thm{sign,prop,...}) =
case mk_rrule thm of
None => mss
| Some(rrule as {lhs,...}) =>
(trace_thm "Adding rewrite rule:" thm;
Mss{net = (Net.insert_term((lhs,rrule),net,eq)
handle Net.INSERT =>
(prtm_warning "ignoring duplicate rewrite rule" sign prop;
net)),
congs=congs, bounds=bounds, prems=prems,mk_rews=mk_rews});
fun del_simp(mss as Mss{net,congs,bounds,prems,mk_rews},
thm as Thm{sign,prop,...}) =
case mk_rrule thm of
None => mss
| Some(rrule as {lhs,...}) =>
Mss{net = (Net.delete_term((lhs,rrule),net,eq)
handle Net.DELETE =>
(prtm_warning "rewrite rule not in simpset" sign prop;
net)),
congs=congs, bounds=bounds, prems=prems,mk_rews=mk_rews}
end;
val add_simps = foldl add_simp;
val del_simps = foldl del_simp;
fun mss_of thms = add_simps(empty_mss,thms);
fun add_cong(Mss{net,congs,bounds,prems,mk_rews},thm) =
let val (lhs,_) = Logic.dest_equals(concl_of thm) handle TERM _ =>
raise SIMPLIFIER("Congruence not a meta-equality",thm)
(* val lhs = Pattern.eta_contract lhs*)
val (a,_) = dest_Const (head_of lhs) handle TERM _ =>
raise SIMPLIFIER("Congruence must start with a constant",thm)
in Mss{net=net, congs=(a,{lhs=lhs,thm=thm})::congs, bounds=bounds,
prems=prems, mk_rews=mk_rews}
end;
val (op add_congs) = foldl add_cong;
fun add_prems(Mss{net,congs,bounds,prems,mk_rews},thms) =
Mss{net=net, congs=congs, bounds=bounds, prems=thms@prems, mk_rews=mk_rews};
fun prems_of_mss(Mss{prems,...}) = prems;
fun set_mk_rews(Mss{net,congs,bounds,prems,...},mk_rews) =
Mss{net=net, congs=congs, bounds=bounds, prems=prems, mk_rews=mk_rews};
fun mk_rews_of_mss(Mss{mk_rews,...}) = mk_rews;
(*** Meta-level rewriting
uses conversions, omitting proofs for efficiency. See
L C Paulson, A higher-order implementation of rewriting,
Science of Computer Programming 3 (1983), pages 119-149. ***)
type prover = meta_simpset -> thm -> thm option;
type termrec = (Sign.sg * term list) * term;
type conv = meta_simpset -> termrec -> termrec;
datatype order = LESS | EQUAL | GREATER;
fun stringord(a,b:string) = if a<b then LESS else
if a=b then EQUAL else GREATER;
fun intord(i,j:int) = if i<j then LESS else
if i=j then EQUAL else GREATER;
(* NB: non-linearity of the ordering is not a soundness problem *)
(* FIXME: "***ABSTRACTION***" is a hack and makes the ordering non-linear *)
fun string_of_hd(Const(a,_)) = a
| string_of_hd(Free(a,_)) = a
| string_of_hd(Var(v,_)) = Syntax.string_of_vname v
| string_of_hd(Bound i) = string_of_int i
| string_of_hd(Abs _) = "***ABSTRACTION***";
(* a strict (not reflexive) linear well-founded AC-compatible ordering
* for terms:
* s < t <=> 1. size(s) < size(t) or
2. size(s) = size(t) and s=f(...) and t = g(...) and f<g or
3. size(s) = size(t) and s=f(s1..sn) and t=f(t1..tn) and
(s1..sn) < (t1..tn) (lexicographically)
*)
(* FIXME: should really take types into account as well.
* Otherwise non-linear *)
fun termord(Abs(_,_,t),Abs(_,_,u)) = termord(t,u)
| termord(t,u) =
(case intord(size_of_term t,size_of_term u) of
EQUAL => let val (f,ts) = strip_comb t and (g,us) = strip_comb u
in case stringord(string_of_hd f, string_of_hd g) of
EQUAL => lextermord(ts,us)
| ord => ord
end
| ord => ord)
and lextermord(t::ts,u::us) =
(case termord(t,u) of
EQUAL => lextermord(ts,us)
| ord => ord)
| lextermord([],[]) = EQUAL
| lextermord _ = error("lextermord");
fun termless tu = (termord tu = LESS);
fun check_conv (thm as Thm{shyps,hyps,prop,sign,der,maxidx,...}, prop0, ders) =
let fun err() = (trace_thm "Proved wrong thm (Check subgoaler?)" thm;
trace_term "Should have proved" sign prop0;
None)
val (lhs0,_) = Logic.dest_equals(Logic.strip_imp_concl prop0)
in case prop of
Const("==",_) $ lhs $ rhs =>
if (lhs = lhs0) orelse
(lhs aconv Envir.norm_term (Envir.empty 0) lhs0)
then (trace_thm "SUCCEEDED" thm;
Some(shyps, hyps, maxidx, rhs, der::ders))
else err()
| _ => err()
end;
fun ren_inst(insts,prop,pat,obj) =
let val ren = match_bvs(pat,obj,[])
fun renAbs(Abs(x,T,b)) =
Abs(case assoc_string(ren,x) of None => x | Some(y) => y, T, renAbs(b))
| renAbs(f$t) = renAbs(f) $ renAbs(t)
| renAbs(t) = t
in subst_vars insts (if null(ren) then prop else renAbs(prop)) end;
fun add_insts_sorts ((iTs, is), Ss) =
add_typs_sorts (map snd iTs, add_terms_sorts (map snd is, Ss));
(*Conversion to apply the meta simpset to a term*)
fun rewritec (prover,signt) (mss as Mss{net,...})
(shypst,hypst,maxidxt,t,ders) =
let val etat = Pattern.eta_contract t;
fun rew {thm as Thm{sign,der,maxidx,shyps,hyps,prop,...}, lhs, perm} =
let val unit = if Sign.subsig(sign,signt) then ()
else (trace_thm_warning "rewrite rule from different theory"
thm;
raise Pattern.MATCH)
val rprop = if maxidxt = ~1 then prop
else Logic.incr_indexes([],maxidxt+1) prop;
val rlhs = if maxidxt = ~1 then lhs
else fst(Logic.dest_equals(Logic.strip_imp_concl rprop))
val insts = Pattern.match (#tsig(Sign.rep_sg signt)) (rlhs,etat)
val prop' = ren_inst(insts,rprop,rlhs,t);
val hyps' = hyps union hypst;
val shyps' = add_insts_sorts (insts, shyps union shypst);
val maxidx' = maxidx_of_term prop'
val ct' = Cterm{sign = signt, (*used for deriv only*)
t = prop',
T = propT,
maxidx = maxidx'}
val der' = infer_derivs (RewriteC ct', [der])
val thm' = Thm{sign = signt,
der = der',
shyps = shyps',
hyps = hyps',
prop = prop',
maxidx = maxidx'}
val (lhs',rhs') = Logic.dest_equals(Logic.strip_imp_concl prop')
in if perm andalso not(termless(rhs',lhs')) then None else
if Logic.count_prems(prop',0) = 0
then (trace_thm "Rewriting:" thm';
Some(shyps', hyps', maxidx', rhs', der'::ders))
else (trace_thm "Trying to rewrite:" thm';
case prover mss thm' of
None => (trace_thm "FAILED" thm'; None)
| Some(thm2) => check_conv(thm2,prop',ders))
end
fun rews [] = None
| rews (rrule::rrules) =
let val opt = rew rrule handle Pattern.MATCH => None
in case opt of None => rews rrules | some => some end;
fun sort_rrules rrs = let
fun is_simple {thm as Thm{prop,...}, lhs, perm} = case prop of
Const("==",_) $ _ $ _ => true
| _ => false
fun sort [] (re1,re2) = re1 @ re2
| sort (rr::rrs) (re1,re2) = if is_simple rr
then sort rrs (rr::re1,re2)
else sort rrs (re1,rr::re2)
in sort rrs ([],[])
end
in case etat of
Abs(_,_,body) $ u => Some(shypst, hypst, maxidxt,
subst_bounds([u], body),
ders)
| _ => rews (sort_rrules (Net.match_term net etat))
end;
(*Conversion to apply a congruence rule to a term*)
fun congc (prover,signt) {thm=cong,lhs=lhs} (shypst,hypst,maxidxt,t,ders) =
let val Thm{sign,der,shyps,hyps,maxidx,prop,...} = cong
val unit = if Sign.subsig(sign,signt) then ()
else error("Congruence rule from different theory")
val tsig = #tsig(Sign.rep_sg signt)
val rprop = if maxidxt = ~1 then prop
else Logic.incr_indexes([],maxidxt+1) prop;
val rlhs = if maxidxt = ~1 then lhs
else fst(Logic.dest_equals(Logic.strip_imp_concl rprop))
val insts = Pattern.match tsig (rlhs,t)
(* Pattern.match can raise Pattern.MATCH;
is handled when congc is called *)
val prop' = ren_inst(insts,rprop,rlhs,t);
val shyps' = add_insts_sorts (insts, shyps union shypst)
val maxidx' = maxidx_of_term prop'
val ct' = Cterm{sign = signt, (*used for deriv only*)
t = prop',
T = propT,
maxidx = maxidx'}
val thm' = Thm{sign = signt,
der = infer_derivs (CongC ct', [der]),
shyps = shyps',
hyps = hyps union hypst,
prop = prop',
maxidx = maxidx'};
val unit = trace_thm "Applying congruence rule" thm';
fun err() = error("Failed congruence proof!")
in case prover thm' of
None => err()
| Some(thm2) => (case check_conv(thm2,prop',ders) of
None => err() | some => some)
end;
fun bottomc ((simprem,useprem),prover,sign) =
let fun botc fail mss trec =
(case subc mss trec of
some as Some(trec1) =>
(case rewritec (prover,sign) mss trec1 of
Some(trec2) => botc false mss trec2
| None => some)
| None =>
(case rewritec (prover,sign) mss trec of
Some(trec2) => botc false mss trec2
| None => if fail then None else Some(trec)))
and try_botc mss trec = (case botc true mss trec of
Some(trec1) => trec1
| None => trec)
and subc (mss as Mss{net,congs,bounds,prems,mk_rews})
(trec as (shyps,hyps,maxidx,t0,ders)) =
(case t0 of
Abs(a,T,t) =>
let val b = variant bounds a
val v = Free("." ^ b,T)
val mss' = Mss{net=net, congs=congs, bounds=b::bounds,
prems=prems,mk_rews=mk_rews}
in case botc true mss'
(shyps,hyps,maxidx,subst_bounds([v],t),ders) of
Some(shyps',hyps',maxidx',t',ders') =>
Some(shyps', hyps', maxidx',
Abs(a, T, abstract_over(v,t')),
ders')
| None => None
end
| t$u => (case t of
Const("==>",_)$s => Some(impc(shyps,hyps,maxidx,s,u,mss,ders))
| Abs(_,_,body) =>
let val trec = (shyps,hyps,maxidx,subst_bounds([u],body),ders)
in case subc mss trec of
None => Some(trec)
| trec => trec
end
| _ =>
let fun appc() =
(case botc true mss (shyps,hyps,maxidx,t,ders) of
Some(shyps1,hyps1,maxidx1,t1,ders1) =>
(case botc true mss (shyps1,hyps1,maxidx,u,ders1) of
Some(shyps2,hyps2,maxidx2,u1,ders2) =>
Some(shyps2, hyps2, max[maxidx1,maxidx2],
t1$u1, ders2)
| None =>
Some(shyps1, hyps1, max[maxidx1,maxidx], t1$u,
ders1))
| None =>
(case botc true mss (shyps,hyps,maxidx,u,ders) of
Some(shyps1,hyps1,maxidx1,u1,ders1) =>
Some(shyps1, hyps1, max[maxidx,maxidx1],
t$u1, ders1)
| None => None))
val (h,ts) = strip_comb t
in case h of
Const(a,_) =>
(case assoc_string(congs,a) of
None => appc()
| Some(cong) => (congc (prover mss,sign) cong trec
handle Pattern.MATCH => appc() ) )
| _ => appc()
end)
| _ => None)
and impc(shyps, hyps, maxidx, s, u, mss as Mss{mk_rews,...}, ders) =
let val (shyps1,hyps1,_,s1,ders1) =
if simprem then try_botc mss (shyps,hyps,maxidx,s,ders)
else (shyps,hyps,0,s,ders);
val maxidx1 = maxidx_of_term s1
val mss1 =
if not useprem orelse maxidx1 <> ~1 then mss
else let val thm = assume (Cterm{sign=sign, t=s1,
T=propT, maxidx=maxidx1})
in add_simps(add_prems(mss,[thm]), mk_rews thm) end
val (shyps2,hyps2,maxidx2,u1,ders2) =
try_botc mss1 (shyps1,hyps1,maxidx,u,ders1)
val hyps3 = if s1 mem hyps1 then hyps2 else hyps2\s1
in (shyps2, hyps3, max[maxidx1,maxidx2],
Logic.mk_implies(s1,u1), ders2)
end
in try_botc end;
(*** Meta-rewriting: rewrites t to u and returns the theorem t==u ***)
(* Parameters:
mode = (simplify A, use A in simplifying B) when simplifying A ==> B
mss: contains equality theorems of the form [|p1,...|] ==> t==u
prover: how to solve premises in conditional rewrites and congruences
*)
(*** FIXME: check that #bounds(mss) does not "occur" in ct alread ***)
fun rewrite_cterm mode mss prover ct =
let val {sign, t, T, maxidx} = rep_cterm ct;
val (shyps,hyps,maxidxu,u,ders) =
bottomc (mode,prover,sign) mss
(add_term_sorts(t,[]), [], maxidx, t, []);
val prop = Logic.mk_equals(t,u)
in
Thm{sign = sign,
der = infer_derivs (Rewrite_cterm ct, ders),
maxidx = max[maxidx,maxidxu],
shyps = shyps,
hyps = hyps,
prop = prop}
end
fun invoke_oracle (thy, sign, exn) =
case #oraopt(rep_theory thy) of
None => raise THM ("No oracle in supplied theory", 0, [])
| Some oracle =>
let val sign' = Sign.merge(sign_of thy, sign)
val (prop, T, maxidx) =
Sign.certify_term sign' (oracle (sign', exn))
in if T<>propT then
raise THM("Oracle's result must have type prop", 0, [])
else fix_shyps [] []
(Thm {sign = sign',
der = Join (Oracle(thy,sign,exn), []),
maxidx = maxidx,
shyps = [],
hyps = [],
prop = prop})
end;
end;
open Thm;