Changed treatment of during type inference internally generated type
variables.
1. They are renamed to 'a, 'b, 'c etc away from a given set of used names.
2. They are either frozen (turned into TFrees) or left schematic (TVars)
depending on a parameter. In goals they are frozen, for instantiations they
are left schematic.
(* 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");
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 cv = cterm_of sign (Var(ixn,typ_subst_TVars tye2 T))
val used' = add_term_tvarnames(term_of ct,used);
in ((cv,ct)::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', 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;
in instantiate (read_insts sg ts ts [] 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;