Removed an old bug which made some simultaneous instantiations fail if they
were given in the "wrong" order.
Rewrote sign/infer_types.
Fixed some comments.
(* Title: Pure/drule.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Derived rules and other operations on theorems and theories
*)
infix 0 RS RSN RL RLN MRS MRL COMP;
signature DRULE =
sig
structure Thm : THM
local open Thm in
val add_defs : (string * string) list -> theory -> theory
val add_defs_i : (string * term) list -> theory -> theory
val asm_rl : thm
val assume_ax : theory -> string -> thm
val COMP : thm * thm -> thm
val compose : thm * int * thm -> thm list
val cprems_of : thm -> cterm list
val cskip_flexpairs : cterm -> cterm
val cstrip_imp_prems : cterm -> cterm list
val cterm_instantiate : (cterm*cterm)list -> thm -> thm
val cut_rl : thm
val equal_abs_elim : cterm -> thm -> thm
val equal_abs_elim_list: cterm list -> thm -> thm
val eq_thm : thm * thm -> bool
val eq_thm_sg : thm * thm -> bool
val flexpair_abs_elim_list: cterm list -> thm -> thm
val forall_intr_list : cterm list -> thm -> thm
val forall_intr_frees : thm -> thm
val forall_elim_list : cterm list -> thm -> thm
val forall_elim_var : int -> thm -> thm
val forall_elim_vars : int -> thm -> thm
val implies_elim_list : thm -> thm list -> thm
val implies_intr_list : cterm list -> thm -> thm
val MRL : thm list list * thm list -> thm list
val MRS : thm list * thm -> thm
val pprint_cterm : cterm -> pprint_args -> unit
val pprint_ctyp : ctyp -> pprint_args -> unit
val pprint_theory : theory -> pprint_args -> unit
val pprint_thm : thm -> pprint_args -> unit
val pretty_thm : thm -> Sign.Syntax.Pretty.T
val print_cterm : cterm -> unit
val print_ctyp : ctyp -> unit
val print_goals : int -> thm -> unit
val print_goals_ref : (int -> thm -> unit) ref
val print_syntax : theory -> unit
val print_theory : theory -> unit
val print_thm : thm -> unit
val prth : thm -> thm
val prthq : thm Sequence.seq -> thm Sequence.seq
val prths : thm list -> thm list
val read_instantiate : (string*string)list -> thm -> thm
val read_instantiate_sg: Sign.sg -> (string*string)list -> thm -> thm
val read_insts :
Sign.sg -> (indexname -> typ option) * (indexname -> sort option)
-> (indexname -> typ option) * (indexname -> sort option)
-> string list -> (string*string)list
-> (indexname*ctyp)list * (cterm*cterm)list
val reflexive_thm : thm
val revcut_rl : thm
val rewrite_goal_rule : bool*bool -> (meta_simpset -> thm -> thm option)
-> meta_simpset -> int -> thm -> thm
val rewrite_goals_rule: thm list -> thm -> thm
val rewrite_rule : thm list -> thm -> thm
val RS : thm * thm -> thm
val RSN : thm * (int * thm) -> thm
val RL : thm list * thm list -> thm list
val RLN : thm list * (int * thm list) -> thm list
val show_hyps : bool ref
val size_of_thm : thm -> int
val standard : thm -> thm
val string_of_cterm : cterm -> string
val string_of_ctyp : ctyp -> string
val string_of_thm : thm -> string
val symmetric_thm : thm
val thin_rl : thm
val transitive_thm : thm
val triv_forall_equality: thm
val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)
val zero_var_indexes : thm -> thm
end
end;
functor DruleFun (structure Logic: LOGIC and Thm: THM): DRULE =
struct
structure Thm = Thm;
structure Sign = Thm.Sign;
structure Type = Sign.Type;
structure Syntax = Sign.Syntax;
structure Pretty = Syntax.Pretty
structure Symtab = Sign.Symtab;
local open Thm
in
(**** Extend Theories ****)
(** add constant definitions **)
(* all_axioms_of *)
(*results may contain duplicates!*)
fun ancestry_of thy =
thy :: flat (map ancestry_of (parents_of thy));
val all_axioms_of =
flat o map (Symtab.dest o #new_axioms o rep_theory) o ancestry_of;
(* clash_types, clash_consts *)
(*check if types have common instance (ignoring sorts)*)
fun clash_types ty1 ty2 =
let
val ty1' = Type.varifyT ty1;
val ty2' = incr_tvar (maxidx_of_typ ty1' + 1) (Type.varifyT ty2);
in
Type.raw_unify (ty1', ty2')
end;
fun clash_consts (c1, ty1) (c2, ty2) =
c1 = c2 andalso clash_types ty1 ty2;
(* clash_defns *)
fun clash_defn c_ty (name, tm) =
let val (c, ty') = dest_Const (head_of (fst (Logic.dest_equals tm))) in
if clash_consts c_ty (c, ty') then Some (name, ty') else None
end handle TERM _ => None;
fun clash_defns c_ty axms =
distinct (mapfilter (clash_defn c_ty) axms);
(* dest_defn *)
fun dest_defn tm =
let
fun err msg = raise_term msg [tm];
val (lhs, rhs) = Logic.dest_equals tm
handle TERM _ => err "Not a meta-equality (==)";
val (head, args) = strip_comb lhs;
val (c, ty) = dest_Const head
handle TERM _ => err "Head of lhs not a constant";
fun occs_const (Const c_ty') = (c_ty' = (c, ty))
| occs_const (Abs (_, _, t)) = occs_const t
| occs_const (t $ u) = occs_const t orelse occs_const u
| occs_const _ = false;
val show_frees = commas_quote o map (fst o dest_Free);
val show_tfrees = commas_quote o map fst;
val lhs_dups = duplicates args;
val rhs_extras = gen_rems (op =) (term_frees rhs, args);
val rhs_extrasT = gen_rems (op =) (term_tfrees rhs, typ_tfrees ty);
in
if not (forall is_Free args) then
err "Arguments of lhs have to be variables"
else if not (null lhs_dups) then
err ("Duplicate variables on lhs: " ^ show_frees lhs_dups)
else if not (null rhs_extras) then
err ("Extra variables on rhs: " ^ show_frees rhs_extras)
else if not (null rhs_extrasT) then
err ("Extra type variables on rhs: " ^ show_tfrees rhs_extrasT)
else if occs_const rhs then
err ("Constant to be defined occurs on rhs")
else (c, ty)
end;
(* check_defn *)
fun err_in_defn name msg =
(writeln msg; error ("The error(s) above occurred in definition " ^ quote name));
fun check_defn sign (axms, (name, tm)) =
let
fun show_const (c, ty) = quote (Pretty.string_of (Pretty.block
[Pretty.str (c ^ " ::"), Pretty.brk 1, Sign.pretty_typ sign ty]));
fun show_defn c (dfn, ty') = show_const (c, ty') ^ " in " ^ dfn;
fun show_defns c = commas o map (show_defn c);
val (c, ty) = dest_defn tm
handle TERM (msg, _) => err_in_defn name msg;
val defns = clash_defns (c, ty) axms;
in
if not (null defns) then
err_in_defn name ("Definition of " ^ show_const (c, ty) ^
" clashes with " ^ show_defns c defns)
else (name, tm) :: axms
end;
(* add_defs *)
fun ext_defns prep_axm raw_axms thy =
let
val axms = map (prep_axm (sign_of thy)) raw_axms;
val all_axms = all_axioms_of thy;
in
foldl (check_defn (sign_of thy)) (all_axms, axms);
add_axioms_i axms thy
end;
val add_defs_i = ext_defns cert_axm;
val add_defs = ext_defns read_axm;
(**** More derived rules and operations on theorems ****)
(** some cterm->cterm operations: much faster than calling cterm_of! **)
(*Discard flexflex pairs; return a cterm*)
fun cskip_flexpairs ct =
case term_of ct of
(Const("==>", _) $ (Const("=?=",_)$_$_) $ _) =>
cskip_flexpairs (#2 (dest_cimplies ct))
| _ => ct;
(* A1==>...An==>B goes to [A1,...,An], where B is not an implication *)
fun cstrip_imp_prems ct =
let val (cA,cB) = dest_cimplies ct
in cA :: cstrip_imp_prems cB end
handle TERM _ => [];
(*The premises of a theorem, as a cterm list*)
val cprems_of = cstrip_imp_prems o cskip_flexpairs o cprop_of;
(** reading of instantiations **)
fun indexname cs = case Syntax.scan_varname cs of (v,[]) => v
| _ => error("Lexical error in variable name " ^ quote (implode cs));
fun absent ixn =
error("No such variable in term: " ^ Syntax.string_of_vname ixn);
fun inst_failure ixn =
error("Instantiation of " ^ Syntax.string_of_vname ixn ^ " fails");
(* this code is a bit of a mess. add_cterm could be simplified greatly if
simultaneous instantiations were read or at least type checked
simultaneously rather than one after the other. This would make the tricky
composition of implicit type instantiations (parameter tye) superfluous.
*)
fun read_insts sign (rtypes,rsorts) (types,sorts) used insts =
let val {tsig,...} = Sign.rep_sg sign
fun split([],tvs,vs) = (tvs,vs)
| split((sv,st)::l,tvs,vs) = (case explode sv of
"'"::cs => split(l,(indexname cs,st)::tvs,vs)
| cs => split(l,tvs,(indexname cs,st)::vs));
val (tvs,vs) = split(insts,[],[]);
fun readT((a,i),st) =
let val ixn = ("'" ^ a,i);
val S = case rsorts ixn of Some S => S | None => absent ixn;
val T = Sign.read_typ (sign,sorts) st;
in if Type.typ_instance(tsig,T,TVar(ixn,S)) then (ixn,T)
else inst_failure ixn
end
val tye = map readT tvs;
fun add_cterm ((cts,tye,used), (ixn,st)) =
let val T = case rtypes ixn of
Some T => typ_subst_TVars tye T
| None => absent ixn;
val (ct,tye2) = read_def_cterm(sign,types,sorts) used false (st,T);
val cts' = (ixn,T,ct)::cts
fun inst(ixn,T,ct) = (ixn,typ_subst_TVars tye2 T,ct)
val used' = add_term_tvarnames(term_of ct,used);
in (map inst cts',tye2 @ tye,used') end
val (cterms,tye',_) = foldl add_cterm (([],tye,used), vs);
in (map (fn (ixn,T) => (ixn,ctyp_of sign T)) tye',
map (fn (ixn,T,ct) => (cterm_of sign (Var(ixn,T)), ct)) cterms)
end;
(*** Printing of theories, theorems, etc. ***)
(*If false, hypotheses are printed as dots*)
val show_hyps = ref true;
fun pretty_thm th =
let val {sign, hyps, prop,...} = rep_thm th
val hsymbs = if null hyps then []
else if !show_hyps then
[Pretty.brk 2,
Pretty.lst("[","]") (map (Sign.pretty_term sign) hyps)]
else Pretty.str" [" :: map (fn _ => Pretty.str".") hyps @
[Pretty.str"]"];
in Pretty.blk(0, Sign.pretty_term sign prop :: hsymbs) end;
val string_of_thm = Pretty.string_of o pretty_thm;
val pprint_thm = Pretty.pprint o Pretty.quote o pretty_thm;
(** Top-level commands for printing theorems **)
val print_thm = writeln o string_of_thm;
fun prth th = (print_thm th; th);
(*Print and return a sequence of theorems, separated by blank lines. *)
fun prthq thseq =
(Sequence.prints (fn _ => print_thm) 100000 thseq; thseq);
(*Print and return a list of theorems, separated by blank lines. *)
fun prths ths = (print_list_ln print_thm ths; ths);
(* other printing commands *)
fun pprint_ctyp cT =
let val {sign, T} = rep_ctyp cT in Sign.pprint_typ sign T end;
fun string_of_ctyp cT =
let val {sign, T} = rep_ctyp cT in Sign.string_of_typ sign T end;
val print_ctyp = writeln o string_of_ctyp;
fun pprint_cterm ct =
let val {sign, t, ...} = rep_cterm ct in Sign.pprint_term sign t end;
fun string_of_cterm ct =
let val {sign, t, ...} = rep_cterm ct in Sign.string_of_term sign t end;
val print_cterm = writeln o string_of_cterm;
(* print theory *)
val pprint_theory = Sign.pprint_sg o sign_of;
val print_syntax = Syntax.print_syntax o syn_of;
fun print_axioms thy =
let
val {sign, new_axioms, ...} = rep_theory thy;
val axioms = Symtab.dest new_axioms;
fun prt_axm (a, t) = Pretty.block [Pretty.str (a ^ ":"), Pretty.brk 1,
Pretty.quote (Sign.pretty_term sign t)];
in
Pretty.writeln (Pretty.big_list "additional axioms:" (map prt_axm axioms))
end;
fun print_theory thy = (Sign.print_sg (sign_of thy); print_axioms thy);
(** Print thm A1,...,An/B in "goal style" -- premises as numbered subgoals **)
(* get type_env, sort_env of term *)
local
open Syntax;
fun ins_entry (x, y) [] = [(x, [y])]
| ins_entry (x, y) ((pair as (x', ys')) :: pairs) =
if x = x' then (x', y ins ys') :: pairs
else pair :: ins_entry (x, y) pairs;
fun add_type_env (Free (x, T), env) = ins_entry (T, x) env
| add_type_env (Var (xi, T), env) = ins_entry (T, string_of_vname xi) env
| add_type_env (Abs (_, _, t), env) = add_type_env (t, env)
| add_type_env (t $ u, env) = add_type_env (u, add_type_env (t, env))
| add_type_env (_, env) = env;
fun add_sort_env (Type (_, Ts), env) = foldr add_sort_env (Ts, env)
| add_sort_env (TFree (x, S), env) = ins_entry (S, x) env
| add_sort_env (TVar (xi, S), env) = ins_entry (S, string_of_vname xi) env;
val sort = map (apsnd sort_strings);
in
fun type_env t = sort (add_type_env (t, []));
fun sort_env t = rev (sort (it_term_types add_sort_env (t, [])));
end;
(* print_goals *)
fun print_goals maxgoals state =
let
open Syntax;
val {sign, prop, ...} = rep_thm state;
val pretty_term = Sign.pretty_term sign;
val pretty_typ = Sign.pretty_typ sign;
val pretty_sort = Sign.pretty_sort;
fun pretty_vars prtf (X, vs) = Pretty.block
[Pretty.block (Pretty.commas (map Pretty.str vs)),
Pretty.str " ::", Pretty.brk 1, prtf X];
fun print_list _ _ [] = ()
| print_list name prtf lst =
(writeln ""; Pretty.writeln (Pretty.big_list name (map prtf lst)));
fun print_goals (_, []) = ()
| print_goals (n, A :: As) = (Pretty.writeln (Pretty.blk (0,
[Pretty.str (" " ^ string_of_int n ^ ". "), pretty_term A]));
print_goals (n + 1, As));
val print_ffpairs =
print_list "Flex-flex pairs:" (pretty_term o Logic.mk_flexpair);
val print_types = print_list "Types:" (pretty_vars pretty_typ) o type_env;
val print_sorts = print_list "Sorts:" (pretty_vars pretty_sort) o sort_env;
val (tpairs, As, B) = Logic.strip_horn prop;
val ngoals = length As;
val orig_no_freeTs = ! show_no_free_types;
val orig_sorts = ! show_sorts;
fun restore () =
(show_no_free_types := orig_no_freeTs; show_sorts := orig_sorts);
in
(show_no_free_types := true; show_sorts := false;
Pretty.writeln (pretty_term B);
if ngoals = 0 then writeln "No subgoals!"
else if ngoals > maxgoals then
(print_goals (1, take (maxgoals, As));
writeln ("A total of " ^ string_of_int ngoals ^ " subgoals..."))
else print_goals (1, As);
print_ffpairs tpairs;
if orig_sorts then
(print_types prop; print_sorts prop)
else if ! show_types then
print_types prop
else ())
handle exn => (restore (); raise exn);
restore ()
end;
(*"hook" for user interfaces: allows print_goals to be replaced*)
val print_goals_ref = ref print_goals;
(*** Find the type (sort) associated with a (T)Var or (T)Free in a term
Used for establishing default types (of variables) and sorts (of
type variables) when reading another term.
Index -1 indicates that a (T)Free rather than a (T)Var is wanted.
***)
fun types_sorts thm =
let val {prop,hyps,...} = rep_thm thm;
val big = list_comb(prop,hyps); (* bogus term! *)
val vars = map dest_Var (term_vars big);
val frees = map dest_Free (term_frees big);
val tvars = term_tvars big;
val tfrees = term_tfrees big;
fun typ(a,i) = if i<0 then assoc(frees,a) else assoc(vars,(a,i));
fun sort(a,i) = if i<0 then assoc(tfrees,a) else assoc(tvars,(a,i));
in (typ,sort) end;
(** Standardization of rules **)
(*Generalization over a list of variables, IGNORING bad ones*)
fun forall_intr_list [] th = th
| forall_intr_list (y::ys) th =
let val gth = forall_intr_list ys th
in forall_intr y gth handle THM _ => gth end;
(*Generalization over all suitable Free variables*)
fun forall_intr_frees th =
let val {prop,sign,...} = rep_thm th
in forall_intr_list
(map (cterm_of sign) (sort atless (term_frees prop)))
th
end;
(*Replace outermost quantified variable by Var of given index.
Could clash with Vars already present.*)
fun forall_elim_var i th =
let val {prop,sign,...} = rep_thm th
in case prop of
Const("all",_) $ Abs(a,T,_) =>
forall_elim (cterm_of sign (Var((a,i), T))) th
| _ => raise THM("forall_elim_var", i, [th])
end;
(*Repeat forall_elim_var until all outer quantifiers are removed*)
fun forall_elim_vars i th =
forall_elim_vars i (forall_elim_var i th)
handle THM _ => th;
(*Specialization over a list of cterms*)
fun forall_elim_list cts th = foldr (uncurry forall_elim) (rev cts, th);
(* maps [A1,...,An], B to [| A1;...;An |] ==> B *)
fun implies_intr_list cAs th = foldr (uncurry implies_intr) (cAs,th);
(* maps [| A1;...;An |] ==> B and [A1,...,An] to B *)
fun implies_elim_list impth ths = foldl (uncurry implies_elim) (impth,ths);
(*Reset Var indexes to zero, renaming to preserve distinctness*)
fun zero_var_indexes th =
let val {prop,sign,...} = rep_thm th;
val vars = term_vars prop
val bs = foldl add_new_id ([], map (fn Var((a,_),_)=>a) vars)
val inrs = add_term_tvars(prop,[]);
val nms' = rev(foldl add_new_id ([], map (#1 o #1) inrs));
val tye = map (fn ((v,rs),a) => (v, TVar((a,0),rs))) (inrs ~~ nms')
val ctye = map (fn (v,T) => (v,ctyp_of sign T)) tye;
fun varpairs([],[]) = []
| varpairs((var as Var(v,T)) :: vars, b::bs) =
let val T' = typ_subst_TVars tye T
in (cterm_of sign (Var(v,T')),
cterm_of sign (Var((b,0),T'))) :: varpairs(vars,bs)
end
| varpairs _ = raise TERM("varpairs", []);
in instantiate (ctye, varpairs(vars,rev bs)) th end;
(*Standard form of object-rule: no hypotheses, Frees, or outer quantifiers;
all generality expressed by Vars having index 0.*)
fun standard th =
let val {maxidx,...} = rep_thm th
in varifyT (zero_var_indexes (forall_elim_vars(maxidx+1)
(forall_intr_frees(implies_intr_hyps th))))
end;
(*Assume a new formula, read following the same conventions as axioms.
Generalizes over Free variables,
creates the assumption, and then strips quantifiers.
Example is [| ALL x:?A. ?P(x) |] ==> [| ?P(?a) |]
[ !(A,P,a)[| ALL x:A. P(x) |] ==> [| P(a) |] ] *)
fun assume_ax thy sP =
let val sign = sign_of thy
val prop = Logic.close_form (term_of (read_cterm sign
(sP, propT)))
in forall_elim_vars 0 (assume (cterm_of sign prop)) end;
(*Resolution: exactly one resolvent must be produced.*)
fun tha RSN (i,thb) =
case Sequence.chop (2, biresolution false [(false,tha)] i thb) of
([th],_) => th
| ([],_) => raise THM("RSN: no unifiers", i, [tha,thb])
| _ => raise THM("RSN: multiple unifiers", i, [tha,thb]);
(*resolution: P==>Q, Q==>R gives P==>R. *)
fun tha RS thb = tha RSN (1,thb);
(*For joining lists of rules*)
fun thas RLN (i,thbs) =
let val resolve = biresolution false (map (pair false) thas) i
fun resb thb = Sequence.list_of_s (resolve thb) handle THM _ => []
in flat (map resb thbs) end;
fun thas RL thbs = thas RLN (1,thbs);
(*Resolve a list of rules against bottom_rl from right to left;
makes proof trees*)
fun rls MRS bottom_rl =
let fun rs_aux i [] = bottom_rl
| rs_aux i (rl::rls) = rl RSN (i, rs_aux (i+1) rls)
in rs_aux 1 rls end;
(*As above, but for rule lists*)
fun rlss MRL bottom_rls =
let fun rs_aux i [] = bottom_rls
| rs_aux i (rls::rlss) = rls RLN (i, rs_aux (i+1) rlss)
in rs_aux 1 rlss end;
(*compose Q and [...,Qi,Q(i+1),...]==>R to [...,Q(i+1),...]==>R
with no lifting or renaming! Q may contain ==> or meta-quants
ALWAYS deletes premise i *)
fun compose(tha,i,thb) =
Sequence.list_of_s (bicompose false (false,tha,0) i thb);
(*compose Q and [Q1,Q2,...,Qk]==>R to [Q2,...,Qk]==>R getting unique result*)
fun tha COMP thb =
case compose(tha,1,thb) of
[th] => th
| _ => raise THM("COMP", 1, [tha,thb]);
(*Instantiate theorem th, reading instantiations under signature sg*)
fun read_instantiate_sg sg sinsts th =
let val ts = types_sorts th;
val used = add_term_tvarnames(#prop(rep_thm th),[]);
in instantiate (read_insts sg ts ts used sinsts) th end;
(*Instantiate theorem th, reading instantiations under theory of th*)
fun read_instantiate sinsts th =
read_instantiate_sg (#sign (rep_thm th)) sinsts th;
(*Left-to-right replacements: tpairs = [...,(vi,ti),...].
Instantiates distinct Vars by terms, inferring type instantiations. *)
local
fun add_types ((ct,cu), (sign,tye)) =
let val {sign=signt, t=t, T= T, ...} = rep_cterm ct
and {sign=signu, t=u, T= U, ...} = rep_cterm cu
val sign' = Sign.merge(sign, Sign.merge(signt, signu))
val tye' = Type.unify (#tsig(Sign.rep_sg sign')) ((T,U), tye)
handle Type.TUNIFY => raise TYPE("add_types", [T,U], [t,u])
in (sign', tye') end;
in
fun cterm_instantiate ctpairs0 th =
let val (sign,tye) = foldr add_types (ctpairs0, (#sign(rep_thm th),[]))
val tsig = #tsig(Sign.rep_sg sign);
fun instT(ct,cu) = let val inst = subst_TVars tye
in (cterm_fun inst ct, cterm_fun inst cu) end
fun ctyp2 (ix,T) = (ix, ctyp_of sign T)
in instantiate (map ctyp2 tye, map instT ctpairs0) th end
handle TERM _ =>
raise THM("cterm_instantiate: incompatible signatures",0,[th])
| TYPE _ => raise THM("cterm_instantiate: types", 0, [th])
end;
(** theorem equality test is exported and used by BEST_FIRST **)
(*equality of theorems uses equality of signatures and
the a-convertible test for terms*)
fun eq_thm (th1,th2) =
let val {sign=sg1, hyps=hyps1, prop=prop1, ...} = rep_thm th1
and {sign=sg2, hyps=hyps2, prop=prop2, ...} = rep_thm th2
in Sign.eq_sg (sg1,sg2) andalso
aconvs(hyps1,hyps2) andalso
prop1 aconv prop2
end;
(*Do the two theorems have the same signature?*)
fun eq_thm_sg (th1,th2) = Sign.eq_sg(#sign(rep_thm th1), #sign(rep_thm th2));
(*Useful "distance" function for BEST_FIRST*)
val size_of_thm = size_of_term o #prop o rep_thm;
(*** Meta-Rewriting Rules ***)
val reflexive_thm =
let val cx = cterm_of Sign.proto_pure (Var(("x",0),TVar(("'a",0),logicS)))
in Thm.reflexive cx end;
val symmetric_thm =
let val xy = read_cterm Sign.proto_pure ("x::'a::logic == y",propT)
in standard(Thm.implies_intr_hyps(Thm.symmetric(Thm.assume xy))) end;
val transitive_thm =
let val xy = read_cterm Sign.proto_pure ("x::'a::logic == y",propT)
val yz = read_cterm Sign.proto_pure ("y::'a::logic == z",propT)
val xythm = Thm.assume xy and yzthm = Thm.assume yz
in standard(Thm.implies_intr yz (Thm.transitive xythm yzthm)) end;
(** Below, a "conversion" has type cterm -> thm **)
val refl_cimplies = reflexive (cterm_of Sign.proto_pure implies);
(*In [A1,...,An]==>B, rewrite the selected A's only -- for rewrite_goals_tac*)
(*Do not rewrite flex-flex pairs*)
fun goals_conv pred cv =
let fun gconv i ct =
let val (A,B) = Thm.dest_cimplies ct
val (thA,j) = case term_of A of
Const("=?=",_)$_$_ => (reflexive A, i)
| _ => (if pred i then cv A else reflexive A, i+1)
in combination (combination refl_cimplies thA) (gconv j B) end
handle TERM _ => reflexive ct
in gconv 1 end;
(*Use a conversion to transform a theorem*)
fun fconv_rule cv th = equal_elim (cv (cprop_of th)) th;
(*rewriting conversion*)
fun rew_conv mode prover mss = rewrite_cterm mode mss prover;
(*Rewrite a theorem*)
fun rewrite_rule thms =
fconv_rule (rew_conv (true,false) (K(K None)) (Thm.mss_of thms));
(*Rewrite the subgoals of a proof state (represented by a theorem) *)
fun rewrite_goals_rule thms =
fconv_rule (goals_conv (K true) (rew_conv (true,false) (K(K None))
(Thm.mss_of thms)));
(*Rewrite the subgoal of a proof state (represented by a theorem) *)
fun rewrite_goal_rule mode prover mss i thm =
if 0 < i andalso i <= nprems_of thm
then fconv_rule (goals_conv (fn j => j=i) (rew_conv mode prover mss)) thm
else raise THM("rewrite_goal_rule",i,[thm]);
(** Derived rules mainly for METAHYPS **)
(*Given the term "a", takes (%x.t)==(%x.u) to t[a/x]==u[a/x]*)
fun equal_abs_elim ca eqth =
let val {sign=signa, t=a, ...} = rep_cterm ca
and combth = combination eqth (reflexive ca)
val {sign,prop,...} = rep_thm eqth
val (abst,absu) = Logic.dest_equals prop
val cterm = cterm_of (Sign.merge (sign,signa))
in transitive (symmetric (beta_conversion (cterm (abst$a))))
(transitive combth (beta_conversion (cterm (absu$a))))
end
handle THM _ => raise THM("equal_abs_elim", 0, [eqth]);
(*Calling equal_abs_elim with multiple terms*)
fun equal_abs_elim_list cts th = foldr (uncurry equal_abs_elim) (rev cts, th);
local
open Logic
val alpha = TVar(("'a",0), []) (* type ?'a::{} *)
fun err th = raise THM("flexpair_inst: ", 0, [th])
fun flexpair_inst def th =
let val {prop = Const _ $ t $ u, sign,...} = rep_thm th
val cterm = cterm_of sign
fun cvar a = cterm(Var((a,0),alpha))
val def' = cterm_instantiate [(cvar"t", cterm t), (cvar"u", cterm u)]
def
in equal_elim def' th
end
handle THM _ => err th | bind => err th
in
val flexpair_intr = flexpair_inst (symmetric flexpair_def)
and flexpair_elim = flexpair_inst flexpair_def
end;
(*Version for flexflex pairs -- this supports lifting.*)
fun flexpair_abs_elim_list cts =
flexpair_intr o equal_abs_elim_list cts o flexpair_elim;
(*** Some useful meta-theorems ***)
(*The rule V/V, obtains assumption solving for eresolve_tac*)
val asm_rl = trivial(read_cterm Sign.proto_pure ("PROP ?psi",propT));
(*Meta-level cut rule: [| V==>W; V |] ==> W *)
val cut_rl = trivial(read_cterm Sign.proto_pure
("PROP ?psi ==> PROP ?theta", propT));
(*Generalized elim rule for one conclusion; cut_rl with reversed premises:
[| PROP V; PROP V ==> PROP W |] ==> PROP W *)
val revcut_rl =
let val V = read_cterm Sign.proto_pure ("PROP V", propT)
and VW = read_cterm Sign.proto_pure ("PROP V ==> PROP W", propT);
in standard (implies_intr V
(implies_intr VW
(implies_elim (assume VW) (assume V))))
end;
(*for deleting an unwanted assumption*)
val thin_rl =
let val V = read_cterm Sign.proto_pure ("PROP V", propT)
and W = read_cterm Sign.proto_pure ("PROP W", propT);
in standard (implies_intr V (implies_intr W (assume W)))
end;
(* (!!x. PROP ?V) == PROP ?V Allows removal of redundant parameters*)
val triv_forall_equality =
let val V = read_cterm Sign.proto_pure ("PROP V", propT)
and QV = read_cterm Sign.proto_pure ("!!x::'a. PROP V", propT)
and x = read_cterm Sign.proto_pure ("x", TFree("'a",logicS));
in standard (equal_intr (implies_intr QV (forall_elim x (assume QV)))
(implies_intr V (forall_intr x (assume V))))
end;
end
end;