signature BASIC_THM;
theorem / axiom deriv: added tags;
get/put_name_tags;
type attribute;
(* 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 BASIC_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 crep_cterm : cterm -> {sign: Sign.sg, t: term, T: ctyp, maxidx: int}
val term_of : cterm -> term
val cterm_of : Sign.sg -> term -> cterm
val ctyp_of_term : cterm -> ctyp
val read_cterm : Sign.sg -> string * typ -> cterm
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
val read_def_cterms :
Sign.sg * (indexname -> typ option) * (indexname -> sort option) ->
string list -> bool -> (string * typ)list
-> cterm list * (indexname * typ)list
(*proof terms [must DUPLICATE declaration as a specification]*)
datatype deriv_kind = MinDeriv | ThmDeriv | FullDeriv;
type tag (* = string * string list *)
val keep_derivs : deriv_kind ref
datatype rule =
MinProof
| Oracle of string * Sign.sg * Object.T
| Axiom of string * tag list
| Theorem of string * tag list
| 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
| Rotate_rule of int * int
| Instantiate of (indexname * ctyp) list * (cterm * cterm) list
| Bicompose of bool * bool * int * int * Envir.env
| Flexflex_rule of Envir.env
| Class_triv of 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
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}
exception THM of string * int * thm list
type 'a attribute (* = 'a * thm -> 'a * thm *)
val eq_thm : thm * thm -> bool
val sign_of_thm : thm -> Sign.sg
val transfer_sg : Sign.sg -> thm -> thm
val transfer : theory -> thm -> thm
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 (since 1995/08/01) *)
val strip_shyps : thm -> thm
val implies_intr_shyps: thm -> thm
val get_axiom : theory -> xstring -> thm
val get_def : theory -> xstring -> 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 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 Seq.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 Seq.seq
val eq_assumption : int -> thm -> thm
val rotate_rule : int -> int -> thm -> thm
val rename_params_rule: string list * int -> thm -> thm
val bicompose : bool -> bool * thm * int ->
int -> thm -> thm Seq.seq
val biresolution : bool -> (bool * thm) list ->
int -> thm -> thm Seq.seq
(*meta simplification*)
exception SIMPLIFIER of string * thm
type meta_simpset
val dest_mss : meta_simpset ->
{simps: thm list, congs: thm list, procs: (string * cterm list) list}
val empty_mss : meta_simpset
val merge_mss : meta_simpset * meta_simpset -> 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 del_congs : meta_simpset * thm list -> meta_simpset
val add_simprocs : meta_simpset *
(string * cterm list * (Sign.sg -> thm list -> term -> thm option) * stamp) list
-> meta_simpset
val del_simprocs : meta_simpset *
(string * cterm list * (Sign.sg -> thm list -> term -> thm option) * stamp) 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 set_mk_sym : meta_simpset * (thm -> thm option) -> meta_simpset
val set_mk_eq_True : meta_simpset * (thm -> thm option) -> meta_simpset
val set_termless : meta_simpset * (term * term -> bool) -> meta_simpset
val trace_simp : bool ref
val rewrite_cterm : bool * bool * bool -> meta_simpset ->
(meta_simpset -> thm -> thm option) -> cterm -> thm
val invoke_oracle : theory -> xstring -> Sign.sg * Object.T -> thm
end;
signature THM =
sig
include BASIC_THM
val no_attributes : 'a -> 'a * 'b attribute list
val apply_attributes : ('a * thm) * 'a attribute list -> ('a * thm)
val applys_attributes : ('a * thm list) * 'a attribute list -> ('a * thm list)
val get_name_tags : thm -> string * tag list
val put_name_tags : string * tag list -> thm -> thm
val name_of_thm : thm -> string
val tags_of_thm : thm -> tag list
val name_thm : string * thm -> thm
end;
structure Thm: THM =
struct
(*** Certified terms and types ***)
(** certified types **)
(*certified typs under a signature*)
datatype ctyp = Ctyp of {sign_ref: Sign.sg_ref, T: typ};
fun rep_ctyp (Ctyp {sign_ref, T}) = {sign = Sign.deref sign_ref, T = T};
fun typ_of (Ctyp {T, ...}) = T;
fun ctyp_of sign T =
Ctyp {sign_ref = Sign.self_ref sign, T = Sign.certify_typ sign T};
fun read_ctyp sign s =
Ctyp {sign_ref = Sign.self_ref 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_ref: Sign.sg_ref, t: term, T: typ, maxidx: int};
fun rep_cterm (Cterm {sign_ref, t, T, maxidx}) =
{sign = Sign.deref sign_ref, t = t, T = T, maxidx = maxidx};
fun crep_cterm (Cterm {sign_ref, t, T, maxidx}) =
{sign = Sign.deref sign_ref, t = t, T = Ctyp {sign_ref = sign_ref, T = T},
maxidx = maxidx};
fun term_of (Cterm {t, ...}) = t;
fun ctyp_of_term (Cterm {sign_ref, T, ...}) = Ctyp {sign_ref = sign_ref, 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_ref = Sign.self_ref sign, t = t, T = T, maxidx = maxidx}
end;
fun cterm_fun f (Cterm {sign_ref, t, ...}) = cterm_of (Sign.deref sign_ref) (f t);
exception CTERM of string;
(*Destruct application in cterms*)
fun dest_comb (Cterm {sign_ref, 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_ref=sign_ref, maxidx=maxidx, t=A, T=typeA},
Cterm {sign_ref=sign_ref, maxidx=maxidx, t=B, T=typeB})
end
| dest_comb _ = raise CTERM "dest_comb";
(*Destruct abstraction in cterms*)
fun dest_abs (Cterm {sign_ref, T as Type("fun",[_,S]), maxidx, t=Abs(x,ty,M)}) =
let val (y,N) = variant_abs (x,ty,M)
in (Cterm {sign_ref = sign_ref, T = ty, maxidx = 0, t = Free(y,ty)},
Cterm {sign_ref = sign_ref, T = S, maxidx = maxidx, t = N})
end
| dest_abs _ = raise CTERM "dest_abs";
(*Makes maxidx precise: it is often too big*)
fun adjust_maxidx (ct as Cterm {sign_ref, T, t, maxidx, ...}) =
if maxidx = ~1 then ct
else Cterm {sign_ref = sign_ref, T = T, maxidx = maxidx_of_term t, t = t};
(*Form cterm out of a function and an argument*)
fun capply (Cterm {t=f, sign_ref=sign_ref1, T=Type("fun",[dty,rty]), maxidx=maxidx1})
(Cterm {t=x, sign_ref=sign_ref2, T, maxidx=maxidx2}) =
if T = dty then Cterm{t=f$x, sign_ref=Sign.merge_refs(sign_ref1,sign_ref2), T=rty,
maxidx=Int.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_ref=sign_ref1, T=T1, maxidx=maxidx1})
(Cterm {t=t2, sign_ref=sign_ref2, T=T2, maxidx=maxidx2}) =
Cterm {t=absfree(a,ty,t2), sign_ref=Sign.merge_refs(sign_ref1,sign_ref2),
T = ty --> T2, maxidx=Int.max(maxidx1, maxidx2)}
| cabs _ _ = raise CTERM "cabs: first arg is not a free variable";
(** read cterms **) (*exception ERROR*)
(*read terms, infer types, certify terms*)
fun read_def_cterms (sign, types, sorts) used freeze sTs =
let
val syn = #syn (Sign.rep_sg sign)
fun read(s,T) =
let val T' = Sign.certify_typ sign T
handle TYPE (msg, _, _) => error msg
in (Syntax.read syn T' s, T') end
val tsTs = map read sTs
val (ts',tye) = Sign.infer_types_simult sign types sorts used freeze tsTs;
val cts = map (cterm_of sign) ts'
handle TYPE (msg, _, _) => error msg
| TERM (msg, _) => error msg;
in (cts, tye) end;
(*read term, infer types, certify term*)
fun read_def_cterm args used freeze aT =
let val ([ct],tye) = read_def_cterms args used freeze [aT]
in (ct,tye) end;
fun read_cterm sign = #1 o read_def_cterm (sign, K None, K None) [] true;
(*** Derivations ***)
(*tags provide additional comment, apart from the axiom/theorem name*)
type tag = string * string list;
(*Names of rules in derivations. Includes logically trivial rules, if
executed in ML.*)
datatype rule =
MinProof (*for building minimal proof terms*)
| Oracle of string * Sign.sg * Object.T (*oracles*)
(*Axioms/theorems*)
| Axiom of string * tag list
| Theorem of string * tag list
(*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*)
| Rotate_rule of int * int
| 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 class
| 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_ref: Sign.sg_ref, (*mutable reference to signature*)
der: deriv, (*derivation*)
maxidx: int, (*maximum index of any Var or TVar*)
shyps: sort list, (*sort hypotheses*)
hyps: term list, (*hypotheses*)
prop: term}; (*conclusion*)
fun rep_thm (Thm {sign_ref, der, maxidx, shyps, hyps, prop}) =
{sign = Sign.deref sign_ref, der = der, maxidx = maxidx,
shyps = shyps, hyps = hyps, prop = prop};
(*Version of rep_thm returning cterms instead of terms*)
fun crep_thm (Thm {sign_ref, der, maxidx, shyps, hyps, prop}) =
let fun ctermf max t = Cterm{sign_ref=sign_ref, t=t, T=propT, maxidx=max};
in {sign = Sign.deref sign_ref, 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;
(*attributes subsume any kind of rules or addXXXs modifiers*)
type 'a attribute = 'a * thm -> 'a * thm;
fun no_attributes x = (x, []);
fun apply_attributes (x_th, atts) = Library.apply atts x_th;
fun applys_attributes (x_ths, atts) = foldl_map (Library.apply atts) x_ths;
(*equality of theorems uses equality of signatures and the
a-convertible test for terms*)
fun eq_thm (th1, th2) =
let
val {sign = sg1, shyps = shyps1, hyps = hyps1, prop = prop1, ...} = rep_thm th1;
val {sign = sg2, shyps = shyps2, hyps = hyps2, prop = prop2, ...} = rep_thm th2;
in
Sign.eq_sg (sg1, sg2) andalso
eq_set_sort (shyps1, shyps2) andalso
aconvs (hyps1, hyps2) andalso
prop1 aconv prop2
end;
fun sign_of_thm (Thm {sign_ref, ...}) = Sign.deref sign_ref;
(*merge signatures of two theorems; raise exception if incompatible*)
fun merge_thm_sgs
(th1 as Thm {sign_ref = sgr1, ...}, th2 as Thm {sign_ref = sgr2, ...}) =
Sign.merge_refs (sgr1, sgr2) handle TERM (msg, _) => raise THM (msg, 0, [th1, th2]);
(*transfer thm to super theory (non-destructive)*)
fun transfer_sg sign' thm =
let
val Thm {sign_ref, der, maxidx, shyps, hyps, prop} = thm;
val sign = Sign.deref sign_ref;
in
if Sign.eq_sg (sign, sign') then thm
else if Sign.subsig (sign, sign') then
Thm {sign_ref = Sign.self_ref sign', der = der, maxidx = maxidx,
shyps = shyps, hyps = hyps, prop = prop}
else raise THM ("transfer: not a super theory", 0, [thm])
end;
val transfer = transfer_sg o sign_of;
(*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_ref, maxidx, prop, ...}) =
Cterm {sign_ref = sign_ref, maxidx = maxidx, T = propT, t = prop};
(** sort contexts of theorems **)
(* basic utils *)
(*accumulate sorts suppressing duplicates; these are coded low levelly
to improve efficiency a bit*)
fun add_typ_sorts (Type (_, Ts), Ss) = add_typs_sorts (Ts, Ss)
| add_typ_sorts (TFree (_, S), Ss) = ins_sort(S,Ss)
| add_typ_sorts (TVar (_, S), Ss) = ins_sort(S,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, union_sort(shyps,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_ref, der, maxidx, hyps, prop, ...} = thm;
val shyps =
add_thm_sorts (thm, add_typs_sorts (Ts, add_thms_shyps (thms, [])));
in
Thm {sign_ref = sign_ref,
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 (since 1995/08/01) *)
(*remove extra sorts that are known to be syntactically non-empty*)
fun strip_shyps thm =
let
val Thm {sign_ref, der, maxidx, shyps, hyps, prop} = thm;
val sorts = add_thm_sorts (thm, []);
val maybe_empty = not o Sign.nonempty_sort (Sign.deref sign_ref) sorts;
val shyps' = filter (fn S => mem_sort(S,sorts) orelse maybe_empty S) shyps;
in
Thm {sign_ref = sign_ref, der = der, maxidx = maxidx,
shyps =
(if eq_set_sort (shyps',sorts) orelse
not (!force_strip_shyps) then shyps'
else (* FIXME tmp (since 1995/08/01) *)
(warning ("Removed sort hypotheses: " ^
commas (map Sorts.str_of_sort (shyps' \\ sorts)));
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_ref, der, maxidx, shyps, hyps, prop} = thm;
val shyps' = ins_sort (logicS, 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 = List.concat (map2 mk_insort (tfrees, xshyps));
in
Thm {sign_ref = sign_ref,
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 raw_name =
let
val name = Sign.intern (Theory.sign_of theory) Theory.axiomK raw_name;
fun get_ax [] = None
| get_ax (thy :: thys) =
let val {sign, axioms, ...} = Theory.rep_theory thy in
(case Symtab.lookup (axioms, name) of
Some t =>
Some (fix_shyps [] []
(Thm {sign_ref = Sign.self_ref sign,
der = infer_derivs (Axiom (name, []), []),
maxidx = maxidx_of_term t,
shyps = [],
hyps = [],
prop = t}))
| None => get_ax thys)
end;
in
(case get_ax (theory :: Theory.ancestors_of theory) of
Some thm => thm
| None => raise THEORY ("No axiom " ^ quote name, [theory]))
end;
fun get_def thy name = get_axiom thy (name ^ "_def");
(*return additional axioms of this theory node*)
fun axioms_of thy =
map (fn (s, _) => (s, get_axiom thy s))
(Symtab.dest (#axioms (rep_theory thy)));
(* name and tags -- make proof objects more readable *)
fun get_name_tags (Thm {der, ...}) =
(case der of
Join (Theorem x, _) => x
| Join (Axiom x, _) => x
| _ => ("", []));
fun put_name_tags x (Thm {sign_ref, der, maxidx, shyps, hyps, prop}) =
let
val der' =
(case der of
Join (Theorem _, ds) => Join (Theorem x, ds)
| Join (Axiom _, ds) => Join (Axiom x, ds)
| _ => Join (Theorem x, [der]));
in
Thm {sign_ref = sign_ref, der = der', maxidx = maxidx,
shyps = shyps, hyps = hyps, prop = prop}
end;
val name_of_thm = #1 o get_name_tags;
val tags_of_thm = #2 o get_name_tags;
fun name_thm (name, thm) = put_name_tags (name, tags_of_thm thm) thm;
(*Compression of theorems -- a separate rule, not integrated with the others,
as it could be slow.*)
fun compress (Thm {sign_ref, der, maxidx, shyps, hyps, prop}) =
Thm {sign_ref = sign_ref,
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.
If this check must be made, recalculate maxidx in hope of preventing its
recurrence.*)
fun nodup_Vars (thm as Thm{sign_ref, der, maxidx, shyps, hyps, prop}) s =
(Sign.nodup_Vars prop;
Thm {sign_ref = sign_ref,
der = der,
maxidx = maxidx_of_term prop,
shyps = shyps,
hyps = hyps,
prop = prop})
handle TYPE(msg,Ts,ts) => raise TYPE(s^": "^msg,Ts,ts);
(** 'primitive' rules **)
(*discharge all assumptions t from ts*)
val disch = gen_rem (op aconv);
(*The assumption rule A|-A in a theory*)
fun assume raw_ct : thm =
let val ct as Cterm {sign_ref, t=prop, T, maxidx} = adjust_maxidx raw_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_ref = sign_ref,
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_ref,der,maxidx,hyps,prop,...}) : thm =
let val Cterm {sign_ref=sign_refA, t=A, T, maxidx=maxidxA} = cA
in if T<>propT then
raise THM("implies_intr: assumptions must have type prop", 0, [thB])
else fix_shyps [thB] []
(Thm{sign_ref = Sign.merge_refs (sign_ref,sign_refA),
der = infer_derivs (Implies_intr cA, [der]),
maxidx = Int.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_ref, 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_ref= merge_thm_sgs(thAB,thA),
der = infer_derivs (Implies_elim, [der,derA]),
maxidx = Int.max(maxA,maxidx),
shyps = [],
hyps = union_term(hypsA,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_ref,der,maxidx,hyps,prop,...}) =
let val x = term_of cx;
fun result(a,T) = fix_shyps [th] []
(Thm{sign_ref = sign_ref,
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_ref,der,maxidx,hyps,prop,...}) : thm =
let val Cterm {sign_ref=sign_reft, t, T, maxidx=maxt} = 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_ref= Sign.merge_refs(sign_ref,sign_reft),
der = infer_derivs (Forall_elim ct, [der]),
maxidx = Int.max(maxidx, maxt),
shyps = [],
hyps = hyps,
prop = betapply(A,t)})
in if maxt >= 0 andalso maxidx >= 0
then nodup_Vars thm "forall_elim"
else thm (*no new Vars: no expensive check!*)
end
| _ => raise THM("forall_elim: not quantified", 0, [th])
end
handle TERM _ =>
raise THM("forall_elim: incompatible signatures", 0, [th]);
(* Equality *)
(*The reflexivity rule: maps t to the theorem t==t *)
fun reflexive ct =
let val Cterm {sign_ref, t, T, maxidx} = ct
in fix_shyps [] []
(Thm{sign_ref= sign_ref,
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_ref,der,maxidx,shyps,hyps,prop}) =
case prop of
(eq as Const("==",_)) $ t $ u =>
(*no fix_shyps*)
Thm{sign_ref = sign_ref,
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_ref= merge_thm_sgs(th1,th2),
der = infer_derivs (Transitive, [der1, der2]),
maxidx = Int.max(max1,max2),
shyps = [],
hyps = union_term(hyps1,hyps2),
prop = eq$t1$t2})
in if max1 >= 0 andalso max2 >= 0
then nodup_Vars thm "transitive"
else thm (*no new Vars: no expensive check!*)
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 Cterm {sign_ref, t, T, maxidx} = ct
in case t of
Abs(_,_,bodt) $ u => fix_shyps [] []
(Thm{sign_ref = sign_ref,
der = infer_derivs (Beta_conversion ct, []),
maxidx = maxidx,
shyps = [],
hyps = [],
prop = Logic.mk_equals(t, subst_bound (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_ref, 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_ref = sign_ref,
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_ref,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_ref = sign_ref,
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_ref = merge_thm_sgs(th1,th2),
der = infer_derivs (Combination, [der1, der2]),
maxidx = Int.max(max1,max2),
shyps = union_sort(shyps1,shyps2),
hyps = union_term(hyps1,hyps2),
prop = Logic.mk_equals(f$t, g$u)}
in if max1 >= 0 andalso max2 >= 0
then nodup_Vars thm "combination"
else thm (*no new Vars: no expensive check!*)
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_ref = merge_thm_sgs(th1,th2),
der = infer_derivs (Equal_intr, [der1, der2]),
maxidx = Int.max(max1,max2),
shyps = union_sort(shyps1,shyps2),
hyps = union_term(hyps1,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_ref= merge_thm_sgs(th1,th2),
der = infer_derivs (Equal_elim, [der1, der2]),
maxidx = Int.max(max1,max2),
shyps = [],
hyps = union_term(hyps1,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_ref, der, maxidx, shyps, hyps=A::As, prop}) =
implies_intr_hyps (*no fix_shyps*)
(Thm{sign_ref = sign_ref,
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_ref, 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_ref = sign_ref,
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 Seq.map newthm
(Unify.smash_unifiers(Sign.deref sign_ref, 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_ref,tpairs)) =
let val Cterm {sign_ref=sign_reft, t=t, T= T, ...} = ct
and Cterm {sign_ref=sign_refu, t=u, T= U, ...} = cu
in
if T=U then
(Sign.merge_refs (sign_ref, Sign.merge_refs (sign_reft, sign_refu)), (t,u)::tpairs)
else raise TYPE("add_ctpair", [T,U], [t,u])
end;
fun add_ctyp ((v,ctyp), (sign_ref',vTs)) =
let val Ctyp {T,sign_ref} = ctyp
in (Sign.merge_refs(sign_ref,sign_ref'), (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_ref,der,maxidx,hyps,prop,...}) =
let val (newsign_ref,tpairs) = foldr add_ctpair (ctpairs, (sign_ref,[]));
val (newsign_ref,vTs) = foldr add_ctyp (vcTs, (newsign_ref,[]));
val newprop =
Envir.norm_term (Envir.empty 0)
(subst_atomic tpairs
(Type.inst_term_tvars(Sign.tsig_of (Sign.deref newsign_ref),vTs) prop))
val newth =
fix_shyps [th] (map snd vTs)
(Thm{sign_ref = newsign_ref,
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"
end
handle TERM _ =>
raise THM("instantiate: incompatible signatures",0,[th])
| TYPE (msg,_,_) => raise THM("instantiate: type conflict: " ^ msg,
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 Cterm {sign_ref, t=A, T, maxidx} = ct
in if T<>propT then
raise THM("trivial: the term must have type prop", 0, [])
else fix_shyps [] []
(Thm{sign_ref = sign_ref,
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 {sign_ref, 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_ref = sign_ref,
der = infer_derivs (Class_triv c, []),
maxidx = maxidx,
shyps = [],
hyps = [],
prop = t})
end;
(* Replace all TFrees not in the hyps by new TVars *)
fun varifyT(Thm{sign_ref,der,maxidx,shyps,hyps,prop}) =
let val tfrees = foldr add_term_tfree_names (hyps,[])
in let val thm = (*no fix_shyps*)
Thm{sign_ref = sign_ref,
der = infer_derivs (VarifyT, [der]),
maxidx = Int.max(0,maxidx),
shyps = shyps,
hyps = hyps,
prop = Type.varify(prop,tfrees)}
in nodup_Vars thm "varifyT" 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_ref,der,maxidx,shyps,hyps,prop}) =
let val (prop',_) = Type.freeze_thaw prop
in (*no fix_shyps*)
Thm{sign_ref = sign_ref,
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_ref=ssign_ref,...} = 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_ref=ssign_ref, maxidx=smax, T=propT, t=Bi}
val (lift_abs,lift_all) = Logic.lift_fns(Bi,smax+1)
val (Thm{sign_ref, der, maxidx,shyps,hyps,prop}) = orule
val (tpairs,As,B) = Logic.strip_horn prop
in (*no fix_shyps*)
Thm{sign_ref = merge_thm_sgs(state,orule),
der = infer_derivs (Lift_rule(ct_Bi, i), [der]),
maxidx = maxidx+smax+1,
shyps=union_sort(sshyps,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_ref,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_ref = sign_ref,
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 [] = Seq.empty
| addprfs ((t,u)::apairs) = Seq.make (fn()=> Seq.pull
(Seq.mapp newth
(Unify.unifiers(Sign.deref sign_ref,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_ref,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_ref = sign_ref,
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;
(*For rotate_tac: fast rotation of assumptions of subgoal i*)
fun rotate_rule k i state =
let val Thm{sign_ref,der,maxidx,hyps,prop,shyps} = state;
val (tpairs, Bs, Bi, C) = dest_state(state,i)
val params = Logic.strip_params Bi
and asms = Logic.strip_assums_hyp Bi
and concl = Logic.strip_assums_concl Bi
val n = length asms
fun rot m = if 0=m orelse m=n then Bi
else if 0<m andalso m<n
then list_all
(params,
Logic.list_implies(List.drop(asms, m) @
List.take(asms, m),
concl))
else raise THM("rotate_rule", m, [state])
in Thm{sign_ref = sign_ref,
der = infer_derivs (Rotate_rule (k,i), [der]),
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
prop = Logic.rule_of(tpairs, Bs@[rot (if k<0 then n+k else k)], C)}
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_ref,der,maxidx,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 variantlist(take (short,iparams), cs) @ cs
val freenames = map (#1 o dest_Free) (term_frees Bi)
val newBi = Logic.list_rename_params (newnames, Bi)
in
case findrep cs of
c::_ => (warning ("Can't rename. Bound variables not distinct: " ^ c);
state)
| [] => (case cs inter_string freenames of
a::_ => (warning ("Can't rename. Bound/Free variable clash: " ^ a);
state)
| [] => fix_shyps [state] []
(Thm{sign_ref = sign_ref,
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 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_ref = merge_thm_sgs(state,orule);
val sign = Sign.deref sign_ref;
(** 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 Envir.above (smax, env) 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_ref = sign_ref,
der = infer_derivs (Bicompose(match, eres_flg,
1 + length Bs, nsubgoal, env),
[rder,sder]),
maxidx = maxidx,
shyps = add_env_sorts (env, union_sort(rshyps,sshyps)),
hyps = union_term(rhyps,shyps),
prop = Logic.rule_of normp}
in Seq.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(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 (As, n, (t,u)::apairs) =
(case Seq.pull(Unify.unifiers(sign, env, (t,u)::dpairs)) of
None => tryasms (As, n+1, apairs)
| cell as Some((_,tpairs),_) =>
Seq.it_right (addth (newAs(As, n, [BBi,(u,t)], tpairs)))
(Seq.make (fn()=> cell),
Seq.make (fn()=> Seq.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) = Seq.empty
| res(cell as Some((_,tpairs),_)) =
Seq.it_right (addth(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(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 [] = Seq.empty
| res ((eres_flg, rule)::brules) =
if could_bires (Hs, B, eres_flg, rule)
then Seq.make (*delay processing remainder till needed*)
(fn()=> Some(comp (eres_flg, lift rule, nprems_of rule),
res brules))
else res brules
in Seq.flat (res brules) end;
(*** Meta Simplification ***)
(** diagnostics **)
exception SIMPLIFIER of string * thm;
fun prnt warn a = if warn then warning a else writeln a;
fun prtm warn a sign t =
(prnt warn a; prnt warn (Sign.string_of_term sign t));
fun prthm warn a (thm as Thm{sign_ref, prop, ...}) =
(prtm warn a (Sign.deref sign_ref) prop);
val trace_simp = ref false;
fun trace warn a = if !trace_simp then prnt warn a else ();
fun trace_term warn a sign t =
if !trace_simp then prtm warn a sign t else ();
fun trace_thm warn a (thm as Thm{sign_ref, prop, ...}) =
(trace_term warn a (Sign.deref sign_ref) prop);
(** meta simp sets **)
(* basic components *)
type rrule = {thm: thm, lhs: term, elhs: term, fo: bool, perm: bool};
type cong = {thm: thm, lhs: term};
type simproc =
{name: string, proc: Sign.sg -> thm list -> term -> thm option, lhs: cterm, id: stamp};
fun eq_rrule ({thm = Thm {prop = p1, ...}, ...}: rrule,
{thm = Thm {prop = p2, ...}, ...}: rrule) = p1 aconv p2;
fun eq_cong ({thm = Thm {prop = p1, ...}, ...}: cong,
{thm = Thm {prop = p2, ...}, ...}: cong) = p1 aconv p2;
fun eq_prem (Thm {prop = p1, ...}, Thm {prop = p2, ...}) = p1 aconv p2;
fun eq_simproc ({id = s1, ...}:simproc, {id = s2, ...}:simproc) = (s1 = s2);
fun mk_simproc (name, proc, lhs, id) =
{name = name, proc = proc, lhs = lhs, id = id};
(* datatype mss *)
(*
A "mss" contains data needed during conversion:
rules: discrimination net of rewrite rules;
congs: association list of congruence rules and
a list of `weak' congruence constants.
A congruence is `weak' if it avoids normalization of some argument.
procs: discrimination net of simplification procedures
(functions that prove rewrite rules on the fly);
bounds: names of bound variables already used
(for generating new names when rewriting under lambda abstractions);
prems: current premises;
mk_rews: mk: turns simplification thms into rewrite rules;
mk_sym: turns == around; (needs Drule!)
mk_eq_True: turns P into P == True - logic specific;
termless: relation for ordered rewriting;
*)
datatype meta_simpset =
Mss of {
rules: rrule Net.net,
congs: (string * cong) list * string list,
procs: simproc Net.net,
bounds: string list,
prems: thm list,
mk_rews: {mk: thm -> thm list,
mk_sym: thm -> thm option,
mk_eq_True: thm -> thm option},
termless: term * term -> bool};
fun mk_mss (rules, congs, procs, bounds, prems, mk_rews, termless) =
Mss {rules = rules, congs = congs, procs = procs, bounds = bounds,
prems=prems, mk_rews=mk_rews, termless=termless};
fun upd_rules(Mss{rules,congs,procs,bounds,prems,mk_rews,termless}, rules') =
mk_mss(rules',congs,procs,bounds,prems,mk_rews,termless);
val empty_mss =
let val mk_rews = {mk = K [], mk_sym = K None, mk_eq_True = K None}
in mk_mss (Net.empty, ([],[]), Net.empty, [], [], mk_rews, Term.termless)
end;
(** simpset operations **)
(* dest_mss *)
fun dest_mss (Mss {rules, congs, procs, ...}) =
{simps = map (fn (_, {thm, ...}) => thm) (Net.dest rules),
congs = map (fn (_, {thm, ...}) => thm) (fst congs),
procs =
map (fn (_, {name, lhs, id, ...}) => ((name, lhs), id)) (Net.dest procs)
|> partition_eq eq_snd
|> map (fn ps => (#1 (#1 (hd ps)), map (#2 o #1) ps))};
(* merge_mss *) (*NOTE: ignores mk_rews and termless of 2nd mss*)
fun merge_mss
(Mss {rules = rules1, congs = (congs1,weak1), procs = procs1,
bounds = bounds1, prems = prems1, mk_rews, termless},
Mss {rules = rules2, congs = (congs2,weak2), procs = procs2,
bounds = bounds2, prems = prems2, ...}) =
mk_mss
(Net.merge (rules1, rules2, eq_rrule),
(generic_merge (eq_cong o pairself snd) I I congs1 congs2,
merge_lists weak1 weak2),
Net.merge (procs1, procs2, eq_simproc),
merge_lists bounds1 bounds2,
generic_merge eq_prem I I prems1 prems2,
mk_rews, termless);
(* add_simps *)
fun mk_rrule2{thm,lhs,perm} =
let val elhs = Pattern.eta_contract lhs
val fo = Pattern.first_order elhs orelse not(Pattern.pattern elhs)
in {thm=thm,lhs=lhs,elhs=elhs,fo=fo,perm=perm} end
fun insert_rrule(mss as Mss {rules,...},
rrule as {thm,lhs,perm}) =
(trace_thm false "Adding rewrite rule:" thm;
let val rrule2 as {elhs,...} = mk_rrule2 rrule
val rules' = Net.insert_term ((elhs, rrule2), rules, eq_rrule)
in upd_rules(mss,rules') end
handle Net.INSERT =>
(prthm true "Ignoring duplicate rewrite rule:" thm; mss));
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_term (term_vars t, term_vars u);
(* FIXME: it seems that the conditions on extra variables are too liberal if
prems are nonempty: does solving the prems really guarantee instantiation of
all its Vars? Better: a dynamic check each time a rule is applied.
*)
fun rewrite_rule_extra_vars prems elhs erhs =
not ((term_vars erhs) subset
(union_term (term_vars elhs, List.concat(map term_vars prems))))
orelse
not ((term_tvars erhs) subset
(term_tvars elhs union List.concat(map term_tvars prems)));
(*Simple test for looping rewrite rules and stupid orientations*)
fun reorient sign prems lhs rhs =
rewrite_rule_extra_vars prems lhs rhs
orelse
is_Var (head_of 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" 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)*)
orelse
(is_Const lhs andalso not(is_Const rhs))
fun decomp_simp(thm as Thm {sign_ref, prop, ...}) =
let val sign = Sign.deref sign_ref;
val prems = Logic.strip_imp_prems prop;
val concl = Logic.strip_imp_concl prop;
val (lhs, rhs) = Logic.dest_equals concl handle TERM _ =>
raise SIMPLIFIER ("Rewrite rule not a meta-equality", thm)
val elhs = Pattern.eta_contract lhs;
val erhs = Pattern.eta_contract rhs;
val perm = var_perm (elhs, erhs) andalso not (elhs aconv erhs)
andalso not (is_Var elhs)
in (sign,prems,lhs,rhs,perm) end;
fun mk_eq_True (Mss{mk_rews={mk_eq_True,...},...}) thm =
case mk_eq_True thm of
None => []
| Some eq_True => let val (_,_,lhs,_,_) = decomp_simp eq_True
in [{thm=eq_True, lhs=lhs, perm=false}] end;
(* create the rewrite rule and possibly also the ==True variant,
in case there are extra vars on the rhs *)
fun rrule_eq_True(thm,lhs,rhs,mss,thm2) =
let val rrule = {thm=thm, lhs=lhs, perm=false}
in if (term_vars rhs) subset (term_vars lhs) andalso
(term_tvars rhs) subset (term_tvars lhs)
then [rrule]
else mk_eq_True mss thm2 @ [rrule]
end;
fun mk_rrule mss thm =
let val (_,prems,lhs,rhs,perm) = decomp_simp thm
in if perm then [{thm=thm, lhs=lhs, perm=true}] else
(* weak test for loops: *)
if rewrite_rule_extra_vars prems lhs rhs orelse
is_Var (head_of lhs) (* mk_cases may do this! *)
then mk_eq_True mss thm
else rrule_eq_True(thm,lhs,rhs,mss,thm)
end;
fun orient_rrule mss thm =
let val (sign,prems,lhs,rhs,perm) = decomp_simp thm
in if perm then [{thm=thm,lhs=lhs,perm=true}]
else if reorient sign prems lhs rhs
then if reorient sign prems rhs lhs
then mk_eq_True mss thm
else let val Mss{mk_rews={mk_sym,...},...} = mss
in case mk_sym thm of
None => []
| Some thm' =>
let val (_,_,lhs',rhs',_) = decomp_simp thm'
in rrule_eq_True(thm',lhs',rhs',mss,thm) end
end
else rrule_eq_True(thm,lhs,rhs,mss,thm)
end;
fun extract_rews(Mss{mk_rews = {mk,...},...},thms) = flat(map mk thms);
fun orient_comb_simps comb mk_rrule (mss,thms) =
let val rews = extract_rews(mss,thms)
val rrules = flat (map mk_rrule rews)
in foldl comb (mss,rrules) end
(* Add rewrite rules explicitly; do not reorient! *)
fun add_simps(mss,thms) =
orient_comb_simps insert_rrule (mk_rrule mss) (mss,thms);
fun mss_of thms =
foldl insert_rrule (empty_mss, flat(map (mk_rrule empty_mss) thms));
fun extract_safe_rrules(mss,thm) =
flat (map (orient_rrule mss) (extract_rews(mss,[thm])));
fun add_safe_simp(mss,thm) =
foldl insert_rrule (mss, extract_safe_rrules(mss,thm))
(* del_simps *)
fun del_rrule(mss as Mss {rules,...},
rrule as {thm, elhs, ...}) =
(upd_rules(mss, Net.delete_term ((elhs, rrule), rules, eq_rrule))
handle Net.DELETE =>
(prthm true "Rewrite rule not in simpset:" thm; mss));
fun del_simps(mss,thms) =
orient_comb_simps del_rrule (map mk_rrule2 o mk_rrule mss) (mss,thms);
(* add_congs *)
(*FIXME -> term.ML *)
fun is_Bound (Bound _) = true
fun is_Bound _ = false;
fun is_full_cong_prems [] varpairs = null varpairs
| is_full_cong_prems (p::prems) varpairs =
(case Logic.strip_assums_concl p of
Const("==",_) $ lhs $ rhs =>
let val (x,xs) = strip_comb lhs and (y,ys) = strip_comb rhs
in is_Var x andalso forall is_Bound xs andalso
null(findrep(xs)) andalso xs=ys andalso
(x,y) mem varpairs andalso
is_full_cong_prems (p::prems) (varpairs\(x,y))
end
| _ => false);
fun is_full_cong (Thm{prop,...}) =
let val prems = Logic.strip_imp_prems prop
and concl = Logic.strip_imp_concl prop
val (lhs,rhs) = Logic.dest_equals concl
val (f,xs) = strip_comb lhs
and (g,ys) = strip_comb rhs
in
f=g andalso null(findrep(xs@ys)) andalso length xs = length ys andalso
is_full_cong_prems prems (xs ~~ ys)
end
fun add_cong (Mss {rules,congs,procs,bounds,prems,mk_rews,termless}, 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);
val (alist,weak) = congs
val weak2 = if is_full_cong thm then weak else a::weak
in
mk_mss (rules, ((a, {lhs = lhs, thm = thm}) :: alist, weak2),
procs, bounds, prems, mk_rews, termless)
end;
val (op add_congs) = foldl add_cong;
(* del_congs *)
fun del_cong (Mss {rules,congs,procs,bounds,prems,mk_rews,termless}, 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);
val (alist,_) = congs
val alist2 = filter (fn (x,_)=> x<>a) alist
val weak2 = mapfilter (fn(a,{thm,...}) => if is_full_cong thm then None
else Some a)
alist2
in
mk_mss (rules, (alist2,weak2), procs, bounds, prems, mk_rews, termless)
end;
val (op del_congs) = foldl del_cong;
(* add_simprocs *)
fun add_proc (mss as Mss {rules,congs,procs,bounds,prems,mk_rews,termless},
(name, lhs as Cterm {sign_ref, t, ...}, proc, id)) =
(trace_term false ("Adding simplification procedure " ^ quote name ^ " for")
(Sign.deref sign_ref) t;
mk_mss (rules, congs,
Net.insert_term ((t, mk_simproc (name, proc, lhs, id)), procs, eq_simproc)
handle Net.INSERT =>
(warning ("Ignoring duplicate simplification procedure \""
^ name ^ "\"");
procs),
bounds, prems, mk_rews, termless));
fun add_simproc (mss, (name, lhss, proc, id)) =
foldl add_proc (mss, map (fn lhs => (name, lhs, proc, id)) lhss);
val add_simprocs = foldl add_simproc;
(* del_simprocs *)
fun del_proc (mss as Mss {rules,congs,procs,bounds,prems,mk_rews,termless},
(name, lhs as Cterm {t, ...}, proc, id)) =
mk_mss (rules, congs,
Net.delete_term ((t, mk_simproc (name, proc, lhs, id)), procs, eq_simproc)
handle Net.DELETE =>
(warning ("Simplification procedure \"" ^ name ^
"\" not in simpset"); procs),
bounds, prems, mk_rews, termless);
fun del_simproc (mss, (name, lhss, proc, id)) =
foldl del_proc (mss, map (fn lhs => (name, lhs, proc, id)) lhss);
val del_simprocs = foldl del_simproc;
(* prems *)
fun add_prems (Mss {rules,congs,procs,bounds,prems,mk_rews,termless}, thms) =
mk_mss (rules, congs, procs, bounds, thms @ prems, mk_rews, termless);
fun prems_of_mss (Mss {prems, ...}) = prems;
(* mk_rews *)
fun set_mk_rews
(Mss {rules, congs, procs, bounds, prems, mk_rews, termless}, mk) =
mk_mss (rules, congs, procs, bounds, prems,
{mk=mk, mk_sym= #mk_sym mk_rews, mk_eq_True= #mk_eq_True mk_rews},
termless);
fun set_mk_sym
(Mss {rules, congs, procs, bounds, prems, mk_rews, termless}, mk_sym) =
mk_mss (rules, congs, procs, bounds, prems,
{mk= #mk mk_rews, mk_sym= mk_sym, mk_eq_True= #mk_eq_True mk_rews},
termless);
fun set_mk_eq_True
(Mss {rules, congs, procs, bounds, prems, mk_rews, termless}, mk_eq_True) =
mk_mss (rules, congs, procs, bounds, prems,
{mk= #mk mk_rews, mk_sym= #mk_sym mk_rews, mk_eq_True= mk_eq_True},
termless);
(* termless *)
fun set_termless
(Mss {rules, congs, procs, bounds, prems, mk_rews, termless = _}, termless) =
mk_mss (rules, congs, procs, bounds, prems, mk_rews, termless);
(** 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_ref * term list) * term;
type conv = meta_simpset -> termrec -> termrec;
fun check_conv
(thm as Thm{shyps,hyps,prop,sign_ref,der,...}, prop0, ders) =
let fun err() = (trace_thm false "Proved wrong thm (Check subgoaler?)" thm;
trace_term false "Should have proved:" (Sign.deref sign_ref) 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 false "SUCCEEDED" thm;
Some(rhs, (shyps, hyps, 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 incr_insts i (in1:(indexname*typ)list,in2:(indexname*term)list) =
let fun incr ((a,n),x) = ((a,n+i),x)
in (map incr in1, map incr in2) end;
fun add_insts_sorts ((iTs, is), Ss) =
add_typs_sorts (map snd iTs, add_terms_sorts (map snd is, Ss));
(* mk_procrule *)
fun mk_procrule thm =
let val (_,prems,lhs,rhs,_) = decomp_simp thm
in if rewrite_rule_extra_vars prems lhs rhs
then (prthm true "Extra vars on rhs:" thm; [])
else [mk_rrule2{thm = thm, lhs = lhs, perm = false}]
end;
(* conversion to apply the meta simpset to a term *)
(* Since the rewriting strategy is bottom-up, we avoid re-normalizing already
normalized terms by carrying around the rhs of the rewrite rule just
applied. This is called the `skeleton'. It is decomposed in parallel
with the term. Once a Var is encountered, the corresponding term is
already in normal form.
skel0 is a dummy skeleton that is to enforce complete normalization.
*)
val skel0 = Bound 0;
(* Use rhs as skeleton only if the lhs does not contain unnormalized bits.
The latter may happen iff there are weak congruence rules for constants
in the lhs.
*)
fun uncond_skel((_,weak),(lhs,rhs)) =
if null weak then rhs (* optimization *)
else if exists_Const (fn (c,_) => c mem weak) lhs then skel0
else rhs;
(* Behaves like unconditional rule if rhs does not contain vars not in the lhs.
Otherwise those vars may become instantiated with unnormalized terms
while the premises are solved.
*)
fun cond_skel(args as (congs,(lhs,rhs))) =
if term_vars rhs subset term_vars lhs then uncond_skel(args)
else skel0;
(*
we try in order:
(1) beta reduction
(2) unconditional rewrite rules
(3) conditional rewrite rules
(4) simplification procedures
IMPORTANT: rewrite rules must not introduce new Vars or TVars!
*)
fun rewritec (prover,sign_reft,maxt)
(mss as Mss{rules, procs, termless, prems, congs, ...})
(t:term,etc as (shypst,hypst,ders)) =
let
val signt = Sign.deref sign_reft;
val tsigt = Sign.tsig_of signt;
fun rew{thm as Thm{sign_ref,der,shyps,hyps,prop,maxidx,...},
lhs, elhs, fo, perm} =
let
val _ = if Sign.subsig (Sign.deref sign_ref, signt) then ()
else (prthm true "Rewrite rule from different theory:" thm;
raise Pattern.MATCH);
val rprop = if maxt = ~1 then prop
else Logic.incr_indexes([],maxt+1) prop;
val insts = if fo then Pattern.first_order_match tsigt (elhs,t)
else Pattern.match tsigt (elhs,t);
val insts = if maxt = ~1 then insts else incr_insts (maxt+1) insts
val prop' = ren_inst(insts,rprop,lhs,t);
val hyps' = union_term(hyps,hypst);
val shyps' = add_insts_sorts (insts, union_sort(shyps,shypst));
val unconditional = (Logic.count_prems(prop',0) = 0);
val maxidx' = if unconditional then maxt else maxidx+maxt+1
val ct' = Cterm{sign_ref = sign_reft, (*used for deriv only*)
t = prop', T = propT, maxidx = maxidx'}
val der' = infer_derivs (RewriteC ct', [der]);
val thm' = Thm{sign_ref = sign_reft, 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
(trace_thm false "Applying instance of rewrite rule:" thm;
if unconditional
then
(trace_thm false "Rewriting:" thm';
let val lr = Logic.dest_equals prop
val trec' = (rhs', (shyps', hyps', der'::ders))
in Some(trec',uncond_skel(congs,lr)) end)
else
(trace_thm false "Trying to rewrite:" thm';
case prover mss thm' of
None => (trace_thm false "FAILED" thm'; None)
| Some(thm2) =>
(case check_conv(thm2,prop',ders) of
None => None |
Some trec =>
let val concl = Logic.strip_imp_concl prop
val lr = Logic.dest_equals concl
in Some(trec,cond_skel(congs,lr)) end)))
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,...}, ...}:rrule) = 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
fun proc_rews _ ([]:simproc list) = None
| proc_rews eta_t ({name, proc, lhs = Cterm {t = plhs, ...}, ...} :: ps) =
if Pattern.matches tsigt (plhs, t) then
(trace_term false ("Trying procedure " ^ quote name ^ " on:") signt eta_t;
case proc signt prems eta_t of
None => (trace false "FAILED"; proc_rews eta_t ps)
| Some raw_thm =>
(trace_thm false ("Procedure " ^ quote name ^ " produced rewrite rule:") raw_thm;
(case rews (mk_procrule raw_thm) of
None => (trace false "IGNORED"; proc_rews eta_t ps)
| some => some)))
else proc_rews eta_t ps;
in case t of
Abs (_, _, body) $ u => Some ((subst_bound (u, body), etc),skel0)
| _ => (case rews (sort_rrules (Net.match_term rules t)) of
None => proc_rews (Pattern.eta_contract t)
(Net.match_term procs t)
| some => some)
end;
(* conversion to apply a congruence rule to a term *)
fun congc (prover,sign_reft,maxt) {thm=cong,lhs=lhs} (t,(shypst,hypst,ders)) =
let val signt = Sign.deref sign_reft;
val tsig = Sign.tsig_of signt;
val Thm{sign_ref,der,shyps,hyps,maxidx,prop,...} = cong
val _ = if Sign.subsig(Sign.deref sign_ref,signt) then ()
else error("Congruence rule from different theory")
val rprop = if maxt = ~1 then prop
else Logic.incr_indexes([],maxt+1) prop;
val rlhs = if maxt = ~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, union_sort(shyps,shypst))
val maxidx' = maxidx_of_term prop'
val ct' = Cterm{sign_ref = sign_reft, (*used for deriv only*)
t = prop',
T = propT,
maxidx = maxidx'}
val thm' = Thm{sign_ref = sign_reft,
der = infer_derivs (CongC ct', [der]),
shyps = shyps',
hyps = union_term(hyps,hypst),
prop = prop',
maxidx = maxidx'};
val unit = trace_thm false "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,mutsimp),prover,sign_ref,maxidx) =
let
fun botc fail skel mss trec =
if is_Var skel then if fail then None else Some(trec)
else
(case subc skel mss trec of
some as Some(trec1) =>
(case rewritec (prover,sign_ref,maxidx) mss trec1 of
Some(trec2,skel2) => botc false skel2 mss trec2
| None => some)
| None =>
(case rewritec (prover,sign_ref,maxidx) mss trec of
Some(trec2,skel2) => botc false skel2 mss trec2
| None => if fail then None else Some(trec)))
and try_botc mss trec =
(case botc true skel0 mss trec of
Some(trec1) => trec1 | None => trec)
and subc skel
(mss as Mss{rules,congs,procs,bounds,prems,mk_rews,termless})
(trec as (t0:term,etc:sort list*term list * rule mtree list)) =
(case t0 of
Abs(a,T,t) =>
let val b = variant bounds a
val v = Free("." ^ b,T)
val mss' = mk_mss (rules, congs, procs, b :: bounds, prems, mk_rews, termless)
val skel' = case skel of Abs(_,_,sk) => sk | _ => skel0
in case botc true skel' mss' (subst_bound(v,t),etc) of
Some(t',etc') => Some(Abs(a, T, abstract_over(v,t')), etc')
| None => None
end
| t$u => (case t of
Const("==>",_)$s => Some(impc(s,u,mss,etc))
| Abs(_,_,body) =>
let val trec = (subst_bound(u,body), etc)
in case subc skel0 mss trec of
None => Some(trec)
| trec => trec
end
| _ =>
let fun appc() =
let val (tskel,uskel) =
case skel of tskel$uskel => (tskel,uskel)
| _ => (skel0,skel0)
in
(case botc true tskel mss (t,etc) of
Some(t1,etc1) =>
(case botc true uskel mss (u,etc1) of
Some(u1,etc2) => Some(t1$u1, etc2)
| None => Some(t1$u, etc1))
| None =>
(case botc true uskel mss (u,etc) of
Some(u1,etc1) => Some(t$u1, etc1)
| None => None))
end
val (h,ts) = strip_comb t
in case h of
Const(a,_) =>
(case assoc_string(fst congs,a) of
None => appc()
| Some(cong) =>
(congc (prover mss,sign_ref,maxidx) cong trec
handle Pattern.MATCH => appc() ) )
| _ => appc()
end)
| _ => None)
and impc args =
if mutsimp
then let val (prem, conc, mss, etc) = args
in snd(mut_impc([], prem, conc, mss, etc)) end
else nonmut_impc args
and mut_impc (prems, prem, conc, mss, etc) =
let val (prem1,etc1) = try_botc mss (prem,etc)
in mut_impc1(prems, prem1, conc, mss, etc1) end
and mut_impc1(prems, prem1, conc, mss, etc1 as (_,hyps1,_)) =
let
fun uncond({thm,lhs,perm}) =
if nprems_of thm = 0 then Some lhs else None
val (lhss1,mss1) =
if maxidx_of_term prem1 <> ~1
then (trace_term true "Cannot add premise as rewrite rule because it contains (type) unknowns:"
(Sign.deref sign_ref) prem1;
([],mss))
else let val thm = assume (Cterm{sign_ref=sign_ref, t=prem1,
T=propT, maxidx= ~1})
val rrules1 = extract_safe_rrules(mss,thm)
val lhss1 = mapfilter uncond rrules1
val mss1 = foldl insert_rrule (add_prems(mss,[thm]),rrules1)
in (lhss1, mss1) end
fun disch1(conc2,(shyps2,hyps2,ders2)) =
let val hyps2' = if gen_mem (op aconv) (prem1, hyps1)
then hyps2 else hyps2\prem1
in (Logic.mk_implies(prem1,conc2),(shyps2,hyps2',ders2)) end
fun rebuild trec2 =
let val trec = disch1 trec2
in case rewritec (prover,sign_ref,maxidx) mss trec of
None => (None,trec)
| Some((Const("==>",_)$prem$conc,etc),_) =>
mut_impc(prems,prem,conc,mss,etc)
| Some(trec',_) => (None,trec')
end
fun simpconc() =
case conc of
Const("==>",_)$s$t =>
(case mut_impc(prems@[prem1],s,t,mss1,etc1) of
(Some(i,prem),trec2) =>
let val trec2' = disch1 trec2
in if i=0 then mut_impc1(prems,prem,fst trec2',mss,snd trec2')
else (Some(i-1,prem),trec2')
end
| (None,trec) => rebuild(trec))
| _ => rebuild(try_botc mss1 (conc,etc1))
in let val sg = Sign.deref sign_ref
val tsig = #tsig(Sign.rep_sg sg)
fun reducible t =
exists (fn lhs => Pattern.matches_subterm tsig (lhs,t))
lhss1;
in case dropwhile (not o reducible) prems of
[] => simpconc()
| red::rest => (trace_term false "Can now reduce premise:" sg
red;
(Some(length rest,prem1),(conc,etc1)))
end
end
(* legacy code - only for backwards compatibility *)
and nonmut_impc(prem, conc, mss, etc as (_,hyps1,_)) =
let val (prem1,etc1) = if simprem then try_botc mss (prem,etc)
else (prem,etc)
val maxidx1 = maxidx_of_term prem1
val mss1 =
if not useprem then mss else
if maxidx1 <> ~1
then (trace_term true "Cannot add premise as rewrite rule because it contains (type) unknowns:"
(Sign.deref sign_ref) prem1;
mss)
else let val thm = assume (Cterm{sign_ref=sign_ref, t=prem1,
T=propT, maxidx= ~1})
in add_safe_simp(add_prems(mss,[thm]), thm) end
val (conc2,(shyps2,hyps2,ders2)) = try_botc mss1 (conc,etc1)
val hyps2' = if prem1 mem hyps1 then hyps2 else hyps2\prem1
in (Logic.mk_implies(prem1,conc2), (shyps2, hyps2', 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,
use prems of B (if B is again a meta-impl.) to simplify A)
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 Cterm {sign_ref, t, T, maxidx} = ct;
val (u,(shyps,hyps,ders)) = bottomc (mode,prover, sign_ref, maxidx) mss
(t, (add_term_sorts(t,[]), [], []));
val prop = Logic.mk_equals(t,u)
in
Thm{sign_ref = sign_ref,
der = infer_derivs (Rewrite_cterm ct, ders),
maxidx = maxidx,
shyps = shyps,
hyps = hyps,
prop = prop}
end;
(*** Oracles ***)
fun invoke_oracle thy raw_name =
let
val {sign = sg, oracles, ...} = rep_theory thy;
val name = Sign.intern sg Theory.oracleK raw_name;
val oracle =
(case Symtab.lookup (oracles, name) of
None => raise THM ("Unknown oracle: " ^ name, 0, [])
| Some (f, _) => f);
in
fn (sign, exn) =>
let
val sign_ref' = Sign.merge_refs (Sign.self_ref sg, Sign.self_ref sign);
val sign' = Sign.deref sign_ref';
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: " ^ name, 0, [])
else fix_shyps [] []
(Thm {sign_ref = sign_ref',
der = Join (Oracle (name, sign, exn), []),
maxidx = maxidx,
shyps = [],
hyps = [],
prop = prop})
end
end;
end;
structure BasicThm: BASIC_THM = Thm;
open BasicThm;