(* 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.
*)
infix 0 RS RSN RL RLN MRS MRL OF COMP;
signature BASIC_DRULE =
sig
val mk_implies : cterm * cterm -> cterm
val list_implies : cterm list * cterm -> cterm
val dest_implies : cterm -> cterm * cterm
val dest_equals : cterm -> cterm * cterm
val skip_flexpairs : cterm -> cterm
val strip_imp_prems : cterm -> cterm list
val strip_imp_concl : cterm -> cterm
val cprems_of : thm -> cterm list
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 types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)
val strip_shyps_warning : thm -> thm
val forall_intr_list : cterm list -> thm -> thm
val forall_intr_frees : thm -> thm
val forall_intr_vars : 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 forall_elim_vars_safe : thm -> thm
val freeze_thaw : thm -> thm * (thm -> thm)
val implies_elim_list : thm -> thm list -> thm
val implies_intr_list : cterm list -> thm -> thm
val instantiate :
(indexname * ctyp) list * (cterm * cterm) list -> thm -> thm
val zero_var_indexes : thm -> thm
val standard : thm -> thm
val standard' : thm -> thm
val rotate_prems : int -> thm -> thm
val rearrange_prems : int list -> thm -> thm
val assume_ax : theory -> string -> thm
val RSN : thm * (int * thm) -> thm
val RS : thm * thm -> thm
val RLN : thm list * (int * thm list) -> thm list
val RL : thm list * thm list -> thm list
val MRS : thm list * thm -> thm
val MRL : thm list list * thm list -> thm list
val OF : thm * thm list -> thm
val compose : thm * int * thm -> thm list
val COMP : thm * thm -> thm
val read_instantiate_sg: Sign.sg -> (string*string)list -> thm -> thm
val read_instantiate : (string*string)list -> thm -> thm
val cterm_instantiate : (cterm*cterm)list -> thm -> thm
val weak_eq_thm : thm * thm -> bool
val eq_thm_sg : thm * thm -> bool
val size_of_thm : thm -> int
val reflexive_thm : thm
val symmetric_thm : thm
val transitive_thm : thm
val refl_implies : thm
val symmetric_fun : thm -> thm
val extensional : thm -> thm
val imp_cong : thm
val swap_prems_eq : thm
val equal_abs_elim : cterm -> thm -> thm
val equal_abs_elim_list: cterm list -> thm -> thm
val flexpair_abs_elim_list: cterm list -> thm -> thm
val asm_rl : thm
val cut_rl : thm
val revcut_rl : thm
val thin_rl : thm
val triv_forall_equality: thm
val swap_prems_rl : thm
val equal_intr_rule : thm
val inst : string -> string -> thm -> thm
val instantiate' : ctyp option list -> cterm option list -> thm -> thm
val incr_indexes_wrt : int list -> ctyp list -> cterm list -> thm list -> thm -> thm
end;
signature DRULE =
sig
include BASIC_DRULE
val rule_attribute: ('a -> thm -> thm) -> 'a attribute
val tag_rule: tag -> thm -> thm
val untag_rule: string -> thm -> thm
val tag: tag -> 'a attribute
val untag: string -> 'a attribute
val get_kind: thm -> string
val kind: string -> 'a attribute
val theoremK: string
val lemmaK: string
val corollaryK: string
val internalK: string
val kind_internal: 'a attribute
val has_internal: tag list -> bool
val impose_hyps: cterm list -> thm -> thm
val close_derivation: thm -> thm
val local_standard: thm -> thm
val compose_single: thm * int * thm -> thm
val add_rules: thm list -> thm list -> thm list
val del_rules: thm list -> thm list -> thm list
val merge_rules: thm list * thm list -> thm list
val norm_hhf_eq: thm
val triv_goal: thm
val rev_triv_goal: thm
val implies_intr_goals: cterm list -> thm -> thm
val freeze_all: thm -> thm
val mk_triv_goal: cterm -> thm
val add_tvarsT: (indexname * sort) list * typ -> (indexname * sort) list
val add_tvars: (indexname * sort) list * term -> (indexname * sort) list
val add_vars: (indexname * typ) list * term -> (indexname * typ) list
val add_frees: (string * typ) list * term -> (string * typ) list
val tvars_of_terms: term list -> (indexname * sort) list
val vars_of_terms: term list -> (indexname * typ) list
val tvars_of: thm -> (indexname * sort) list
val vars_of: thm -> (indexname * typ) list
val unvarifyT: thm -> thm
val unvarify: thm -> thm
val tvars_intr_list: string list -> thm -> thm
val conj_intr: thm -> thm -> thm
val conj_intr_list: thm list -> thm
val conj_elim: thm -> thm * thm
val conj_elim_list: thm -> thm list
val conj_elim_precise: int -> thm -> thm list
val conj_intr_thm: thm
end;
structure Drule: DRULE =
struct
(** some cterm->cterm operations: much faster than calling cterm_of! **)
(** SAME NAMES as in structure Logic: use compound identifiers! **)
(*dest_implies for cterms. Note T=prop below*)
fun dest_implies ct =
case term_of ct of
(Const("==>", _) $ _ $ _) =>
let val (ct1,ct2) = Thm.dest_comb ct
in (#2 (Thm.dest_comb ct1), ct2) end
| _ => raise TERM ("dest_implies", [term_of ct]) ;
fun dest_equals ct =
case term_of ct of
(Const("==", _) $ _ $ _) =>
let val (ct1,ct2) = Thm.dest_comb ct
in (#2 (Thm.dest_comb ct1), ct2) end
| _ => raise TERM ("dest_equals", [term_of ct]) ;
(*Discard flexflex pairs; return a cterm*)
fun skip_flexpairs ct =
case term_of ct of
(Const("==>", _) $ (Const("=?=",_)$_$_) $ _) =>
skip_flexpairs (#2 (dest_implies ct))
| _ => ct;
(* A1==>...An==>B goes to [A1,...,An], where B is not an implication *)
fun strip_imp_prems ct =
let val (cA,cB) = dest_implies ct
in cA :: strip_imp_prems cB end
handle TERM _ => [];
(* A1==>...An==>B goes to B, where B is not an implication *)
fun strip_imp_concl ct =
case term_of ct of (Const("==>", _) $ _ $ _) =>
strip_imp_concl (#2 (Thm.dest_comb ct))
| _ => ct;
(*The premises of a theorem, as a cterm list*)
val cprems_of = strip_imp_prems o skip_flexpairs o cprop_of;
val proto_sign = Theory.sign_of ProtoPure.thy;
val implies = cterm_of proto_sign Term.implies;
(*cterm version of mk_implies*)
fun mk_implies(A,B) = Thm.capply (Thm.capply implies A) B;
(*cterm version of list_implies: [A1,...,An], B goes to [|A1;==>;An|]==>B *)
fun list_implies([], B) = B
| list_implies(A::AS, B) = mk_implies (A, list_implies(AS,B));
(** reading of instantiations **)
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
fun split([],tvs,vs) = (tvs,vs)
| split((sv,st)::l,tvs,vs) = (case Symbol.explode sv of
"'"::cs => split(l,(Syntax.indexname cs,st)::tvs,vs)
| cs => split(l,tvs,(Syntax.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 Sign.typ_instance sign (T, TVar(ixn,S)) then (ixn,T)
else inst_failure ixn
end
val tye = map readT tvs;
fun mkty(ixn,st) = (case rtypes ixn of
Some T => (ixn,(st,typ_subst_TVars tye T))
| None => absent ixn);
val ixnsTs = map mkty vs;
val ixns = map fst ixnsTs
and sTs = map snd ixnsTs
val (cts,tye2) = read_def_cterms(sign,types,sorts) used false sTs;
fun mkcVar(ixn,T) =
let val U = typ_subst_TVars tye2 T
in cterm_of sign (Var(ixn,U)) end
val ixnTs = ListPair.zip(ixns, map snd sTs)
in (map (fn (ixn,T) => (ixn,ctyp_of sign T)) (tye2 @ tye),
ListPair.zip(map mkcVar ixnTs,cts))
end;
(*** 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;
(** basic attributes **)
(* dependent rules *)
fun rule_attribute f (x, thm) = (x, (f x thm));
(* add / delete tags *)
fun map_tags f thm =
Thm.put_name_tags (Thm.name_of_thm thm, f (#2 (Thm.get_name_tags thm))) thm;
fun tag_rule tg = map_tags (fn tgs => if tg mem tgs then tgs else tgs @ [tg]);
fun untag_rule s = map_tags (filter_out (equal s o #1));
fun tag tg x = rule_attribute (K (tag_rule tg)) x;
fun untag s x = rule_attribute (K (untag_rule s)) x;
fun simple_tag name x = tag (name, []) x;
(* theorem kinds *)
val theoremK = "theorem";
val lemmaK = "lemma";
val corollaryK = "corollary";
val internalK = "internal";
fun get_kind thm =
(case Library.assoc (#2 (Thm.get_name_tags thm), "kind") of
Some (k :: _) => k
| _ => "unknown");
fun kind_rule k = tag_rule ("kind", [k]) o untag_rule "kind";
fun kind k x = rule_attribute (K (kind_rule k)) x;
fun kind_internal x = kind internalK x;
fun has_internal tags = exists (equal internalK o fst) tags;
(** Standardization of rules **)
(*Strip extraneous shyps as far as possible*)
fun strip_shyps_warning thm =
let
val str_of_sort = Sign.str_of_sort (Thm.sign_of_thm thm);
val thm' = Thm.strip_shyps thm;
val xshyps = Thm.extra_shyps thm';
in
if null xshyps then ()
else warning ("Pending sort hypotheses: " ^ commas (map str_of_sort xshyps));
thm'
end;
(*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 (make_ord atless) (term_frees prop)))
th
end;
val forall_elim_var = PureThy.forall_elim_var;
val forall_elim_vars = PureThy.forall_elim_vars;
fun forall_elim_vars_safe th =
forall_elim_vars_safe (forall_elim_var (#maxidx (Thm.rep_thm th) + 1) 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);
(* maps |- B to A1,...,An |- B *)
fun impose_hyps chyps th =
let val chyps' = gen_rems (op aconv o apfst Thm.term_of) (chyps, #hyps (Thm.rep_thm th))
in implies_elim_list (implies_intr_list chyps' th) (map Thm.assume chyps') end;
(*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 = ListPair.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 Thm.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 close_derivation thm =
if Thm.get_name_tags thm = ("", []) then Thm.name_thm ("", thm)
else thm;
fun standard' th =
let val {maxidx,...} = rep_thm th in
th
|> implies_intr_hyps
|> forall_intr_frees |> forall_elim_vars (maxidx + 1)
|> strip_shyps_warning
|> zero_var_indexes |> Thm.varifyT |> Thm.compress
end;
val standard = close_derivation o standard';
fun local_standard th =
th |> strip_shyps_warning |> zero_var_indexes
|> Thm.compress |> close_derivation;
(*Convert all Vars in a theorem to Frees. Also return a function for
reversing that operation. DOES NOT WORK FOR TYPE VARIABLES.
Similar code in type/freeze_thaw*)
fun freeze_thaw th =
let val fth = freezeT th
val {prop,sign,...} = rep_thm fth
in
case term_vars prop of
[] => (fth, fn x => x)
| vars =>
let fun newName (Var(ix,_), (pairs,used)) =
let val v = variant used (string_of_indexname ix)
in ((ix,v)::pairs, v::used) end;
val (alist, _) = foldr newName
(vars, ([], add_term_names (prop, [])))
fun mk_inst (Var(v,T)) =
(cterm_of sign (Var(v,T)),
cterm_of sign (Free(the (assoc(alist,v)), T)))
val insts = map mk_inst vars
fun thaw th' =
th' |> forall_intr_list (map #2 insts)
|> forall_elim_list (map #1 insts)
in (Thm.instantiate ([],insts) fth, thaw) end
end;
(*Rotates a rule's premises to the left by k*)
val rotate_prems = permute_prems 0;
(* permute prems, where the i-th position in the argument list (counting from 0)
gives the position within the original thm to be transferred to position i.
Any remaining trailing positions are left unchanged. *)
val rearrange_prems = let
fun rearr new [] thm = thm
| rearr new (p::ps) thm = rearr (new+1)
(map (fn q => if new<=q andalso q<p then q+1 else q) ps)
(permute_prems (new+1) (new-p) (permute_prems new (p-new) thm))
in rearr 0 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 = Theory.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 Seq.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 = Seq.list_of (resolve thb) handle THM _ => []
in List.concat (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;
(*A version of MRS with more appropriate argument order*)
fun bottom_rl OF rls = rls MRS bottom_rl;
(*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) =
Seq.list_of (bicompose false (false,tha,0) i thb);
fun compose_single (tha,i,thb) =
(case compose (tha,i,thb) of
[th] => th
| _ => raise THM ("compose: unique result expected", i, [tha,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]);
(** theorem equality **)
(*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;
(*maintain lists of theorems --- preserving canonical order*)
fun del_rules rs rules = Library.gen_rems Thm.eq_thm (rules, rs);
fun add_rules rs rules = rs @ del_rules rs rules;
fun merge_rules (rules1, rules2) = Library.generic_merge Thm.eq_thm I I rules1 rules2;
(** Mark Staples's weaker version of eq_thm: ignores variable renaming and
(some) type variable renaming **)
(* Can't use term_vars, because it sorts the resulting list of variable names.
We instead need the unique list noramlised by the order of appearance
in the term. *)
fun term_vars' (t as Var(v,T)) = [t]
| term_vars' (Abs(_,_,b)) = term_vars' b
| term_vars' (f$a) = (term_vars' f) @ (term_vars' a)
| term_vars' _ = [];
fun forall_intr_vars th =
let val {prop,sign,...} = rep_thm th;
val vars = distinct (term_vars' prop);
in forall_intr_list (map (cterm_of sign) vars) th end;
fun weak_eq_thm (tha,thb) =
eq_thm(forall_intr_vars (freezeT tha), forall_intr_vars (freezeT thb));
(*** Meta-Rewriting Rules ***)
fun read_prop s = read_cterm proto_sign (s, propT);
fun store_thm name thm = hd (PureThy.smart_store_thms (name, [thm]));
fun store_standard_thm name thm = store_thm name (standard thm);
fun store_thm_open name thm = hd (PureThy.smart_store_thms_open (name, [thm]));
fun store_standard_thm_open name thm = store_thm_open name (standard' thm);
val reflexive_thm =
let val cx = cterm_of proto_sign (Var(("x",0),TVar(("'a",0),logicS)))
in store_standard_thm_open "reflexive" (Thm.reflexive cx) end;
val symmetric_thm =
let val xy = read_prop "x::'a::logic == y"
in store_standard_thm_open "symmetric" (Thm.implies_intr_hyps (Thm.symmetric (Thm.assume xy))) end;
val transitive_thm =
let val xy = read_prop "x::'a::logic == y"
val yz = read_prop "y::'a::logic == z"
val xythm = Thm.assume xy and yzthm = Thm.assume yz
in store_standard_thm_open "transitive" (Thm.implies_intr yz (Thm.transitive xythm yzthm)) end;
fun symmetric_fun thm = thm RS symmetric_thm;
fun extensional eq =
let val eq' =
abstract_rule "x" (snd (Thm.dest_comb (fst (dest_equals (cprop_of eq))))) eq
in equal_elim (eta_conversion (cprop_of eq')) eq' end;
val imp_cong =
let
val ABC = read_prop "PROP A ==> PROP B == PROP C"
val AB = read_prop "PROP A ==> PROP B"
val AC = read_prop "PROP A ==> PROP C"
val A = read_prop "PROP A"
in
store_standard_thm_open "imp_cong" (implies_intr ABC (equal_intr
(implies_intr AB (implies_intr A
(equal_elim (implies_elim (assume ABC) (assume A))
(implies_elim (assume AB) (assume A)))))
(implies_intr AC (implies_intr A
(equal_elim (symmetric (implies_elim (assume ABC) (assume A)))
(implies_elim (assume AC) (assume A)))))))
end;
val swap_prems_eq =
let
val ABC = read_prop "PROP A ==> PROP B ==> PROP C"
val BAC = read_prop "PROP B ==> PROP A ==> PROP C"
val A = read_prop "PROP A"
val B = read_prop "PROP B"
in
store_standard_thm_open "swap_prems_eq" (equal_intr
(implies_intr ABC (implies_intr B (implies_intr A
(implies_elim (implies_elim (assume ABC) (assume A)) (assume B)))))
(implies_intr BAC (implies_intr A (implies_intr B
(implies_elim (implies_elim (assume BAC) (assume B)) (assume A))))))
end;
val refl_implies = reflexive implies;
(*** Some useful meta-theorems ***)
(*The rule V/V, obtains assumption solving for eresolve_tac*)
val asm_rl = store_standard_thm_open "asm_rl" (Thm.trivial (read_prop "PROP ?psi"));
val _ = store_thm "_" asm_rl;
(*Meta-level cut rule: [| V==>W; V |] ==> W *)
val cut_rl =
store_standard_thm_open "cut_rl"
(Thm.trivial (read_prop "PROP ?psi ==> PROP ?theta"));
(*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_prop "PROP V"
and VW = read_prop "PROP V ==> PROP W";
in
store_standard_thm_open "revcut_rl"
(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_prop "PROP V"
and W = read_prop "PROP W";
in store_standard_thm_open "thin_rl" (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_prop "PROP V"
and QV = read_prop "!!x::'a. PROP V"
and x = read_cterm proto_sign ("x", TypeInfer.logicT);
in
store_standard_thm_open "triv_forall_equality"
(equal_intr (implies_intr QV (forall_elim x (assume QV)))
(implies_intr V (forall_intr x (assume V))))
end;
(* (PROP ?PhiA ==> PROP ?PhiB ==> PROP ?Psi) ==>
(PROP ?PhiB ==> PROP ?PhiA ==> PROP ?Psi)
`thm COMP swap_prems_rl' swaps the first two premises of `thm'
*)
val swap_prems_rl =
let val cmajor = read_prop "PROP PhiA ==> PROP PhiB ==> PROP Psi";
val major = assume cmajor;
val cminor1 = read_prop "PROP PhiA";
val minor1 = assume cminor1;
val cminor2 = read_prop "PROP PhiB";
val minor2 = assume cminor2;
in store_standard_thm_open "swap_prems_rl"
(implies_intr cmajor (implies_intr cminor2 (implies_intr cminor1
(implies_elim (implies_elim major minor1) minor2))))
end;
(* [| PROP ?phi ==> PROP ?psi; PROP ?psi ==> PROP ?phi |]
==> PROP ?phi == PROP ?psi
Introduction rule for == as a meta-theorem.
*)
val equal_intr_rule =
let val PQ = read_prop "PROP phi ==> PROP psi"
and QP = read_prop "PROP psi ==> PROP phi"
in
store_standard_thm_open "equal_intr_rule"
(implies_intr PQ (implies_intr QP (equal_intr (assume PQ) (assume QP))))
end;
(*(PROP ?phi ==> (!!x. PROP ?psi(x))) == (!!x. PROP ?phi ==> PROP ?psi(x))
Rewrite rule for HHF normalization.
*)
val norm_hhf_eq =
let
val cert = Thm.cterm_of proto_sign;
val aT = TFree ("'a", Term.logicS);
val all = Term.all aT;
val x = Free ("x", aT);
val phi = Free ("phi", propT);
val psi = Free ("psi", aT --> propT);
val cx = cert x;
val cphi = cert phi;
val lhs = cert (Logic.mk_implies (phi, all $ Abs ("x", aT, psi $ Bound 0)));
val rhs = cert (all $ Abs ("x", aT, Logic.mk_implies (phi, psi $ Bound 0)));
in
Thm.equal_intr
(Thm.implies_elim (Thm.assume lhs) (Thm.assume cphi)
|> Thm.forall_elim cx
|> Thm.implies_intr cphi
|> Thm.forall_intr cx
|> Thm.implies_intr lhs)
(Thm.implies_elim
(Thm.assume rhs |> Thm.forall_elim cx) (Thm.assume cphi)
|> Thm.forall_intr cx
|> Thm.implies_intr cphi
|> Thm.implies_intr rhs)
|> store_standard_thm_open "norm_hhf_eq"
end;
(*** Instantiate theorem th, reading instantiations under signature sg ****)
(*Version that normalizes the result: Thm.instantiate no longer does that*)
fun instantiate instpair th = Thm.instantiate instpair th COMP asm_rl;
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,maxidx)) =
let val {sign=signt, t=t, T= T, maxidx=maxt,...} = rep_cterm ct
and {sign=signu, t=u, T= U, maxidx=maxu,...} = rep_cterm cu;
val maxi = Int.max(maxidx, Int.max(maxt, maxu));
val sign' = Sign.merge(sign, Sign.merge(signt, signu))
val (tye',maxi') = Type.unify (#tsig(Sign.rep_sg sign')) maxi tye (T,U)
handle Type.TUNIFY => raise TYPE("Ill-typed instantiation", [T,U], [t,u])
in (sign', tye', maxi') end;
in
fun cterm_instantiate ctpairs0 th =
let val (sign,tye,_) = foldr add_types (ctpairs0, (#sign(rep_thm th), Vartab.empty, 0))
fun instT(ct,cu) = let val inst = subst_TVars_Vartab tye
in (cterm_fun inst ct, cterm_fun inst cu) end
fun ctyp2 (ix,T) = (ix, ctyp_of sign T)
in instantiate (map ctyp2 (Vartab.dest tye), map instT ctpairs0) th end
handle TERM _ =>
raise THM("cterm_instantiate: incompatible signatures",0,[th])
| TYPE (msg, _, _) => raise THM(msg, 0, [th])
end;
(** 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 false (cterm (abst$a))))
(transitive combth (beta_conversion false (cterm (absu$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
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 ProtoPure.flexpair_def)
and flexpair_elim = flexpair_inst ProtoPure.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;
(*** Goal (PROP A) <==> PROP A ***)
local
val cert = Thm.cterm_of proto_sign;
val A = Free ("A", propT);
val G = Logic.mk_goal A;
val (G_def, _) = freeze_thaw ProtoPure.Goal_def;
in
val triv_goal = store_thm "triv_goal" (kind_rule internalK (standard
(Thm.equal_elim (Thm.symmetric G_def) (Thm.assume (cert A)))));
val rev_triv_goal = store_thm "rev_triv_goal" (kind_rule internalK (standard
(Thm.equal_elim G_def (Thm.assume (cert G)))));
end;
val mk_cgoal = Thm.capply (Thm.cterm_of proto_sign Logic.goal_const);
fun assume_goal ct = Thm.assume (mk_cgoal ct) RS rev_triv_goal;
fun implies_intr_goals cprops thm =
implies_elim_list (implies_intr_list cprops thm) (map assume_goal cprops)
|> implies_intr_list (map mk_cgoal cprops);
(** variations on instantiate **)
(*shorthand for instantiating just one variable in the current theory*)
fun inst x t = read_instantiate_sg (sign_of (the_context())) [(x,t)];
(* collect vars *)
val add_tvarsT = foldl_atyps (fn (vs, TVar v) => v ins vs | (vs, _) => vs);
val add_tvars = foldl_types add_tvarsT;
val add_vars = foldl_aterms (fn (vs, Var v) => v ins vs | (vs, _) => vs);
val add_frees = foldl_aterms (fn (vs, Free v) => v ins vs | (vs, _) => vs);
fun tvars_of_terms ts = rev (foldl add_tvars ([], ts));
fun vars_of_terms ts = rev (foldl add_vars ([], ts));
fun tvars_of thm = tvars_of_terms [#prop (Thm.rep_thm thm)];
fun vars_of thm = vars_of_terms [#prop (Thm.rep_thm thm)];
(* instantiate by left-to-right occurrence of variables *)
fun instantiate' cTs cts thm =
let
fun err msg =
raise TYPE ("instantiate': " ^ msg,
mapfilter (apsome Thm.typ_of) cTs,
mapfilter (apsome Thm.term_of) cts);
fun inst_of (v, ct) =
(Thm.cterm_of (#sign (Thm.rep_cterm ct)) (Var v), ct)
handle TYPE (msg, _, _) => err msg;
fun zip_vars _ [] = []
| zip_vars (_ :: vs) (None :: opt_ts) = zip_vars vs opt_ts
| zip_vars (v :: vs) (Some t :: opt_ts) = (v, t) :: zip_vars vs opt_ts
| zip_vars [] _ = err "more instantiations than variables in thm";
(*instantiate types first!*)
val thm' =
if forall is_none cTs then thm
else Thm.instantiate (zip_vars (map fst (tvars_of thm)) cTs, []) thm;
in
if forall is_none cts then thm'
else Thm.instantiate ([], map inst_of (zip_vars (vars_of thm') cts)) thm'
end;
(* unvarify(T) *)
(*assume thm in standard form, i.e. no frees, 0 var indexes*)
fun unvarifyT thm =
let
val cT = Thm.ctyp_of (Thm.sign_of_thm thm);
val tfrees = map (fn ((x, _), S) => Some (cT (TFree (x, S)))) (tvars_of thm);
in instantiate' tfrees [] thm end;
fun unvarify raw_thm =
let
val thm = unvarifyT raw_thm;
val ct = Thm.cterm_of (Thm.sign_of_thm thm);
val frees = map (fn ((x, _), T) => Some (ct (Free (x, T)))) (vars_of thm);
in instantiate' [] frees thm end;
(* tvars_intr_list *)
fun tfrees_of thm =
let val {hyps, prop, ...} = Thm.rep_thm thm
in foldr Term.add_term_tfree_names (prop :: hyps, []) end;
fun tvars_intr_list tfrees thm =
Thm.varifyT' (tfrees_of thm \\ tfrees) thm;
(* increment var indexes *)
fun incr_indexes_wrt is cTs cts thms =
let
val maxidx =
foldl Int.max (~1, is @
map (maxidx_of_typ o #T o Thm.rep_ctyp) cTs @
map (#maxidx o Thm.rep_cterm) cts @
map (#maxidx o Thm.rep_thm) thms);
in Thm.incr_indexes (maxidx + 1) end;
(* freeze_all *)
(*freeze all (T)Vars; assumes thm in standard form*)
fun freeze_all_TVars thm =
(case tvars_of thm of
[] => thm
| tvars =>
let val cert = Thm.ctyp_of (Thm.sign_of_thm thm)
in instantiate' (map (fn ((x, _), S) => Some (cert (TFree (x, S)))) tvars) [] thm end);
fun freeze_all_Vars thm =
(case vars_of thm of
[] => thm
| vars =>
let val cert = Thm.cterm_of (Thm.sign_of_thm thm)
in instantiate' [] (map (fn ((x, _), T) => Some (cert (Free (x, T)))) vars) thm end);
val freeze_all = freeze_all_Vars o freeze_all_TVars;
(* mk_triv_goal *)
(*make an initial proof state, "PROP A ==> (PROP A)" *)
fun mk_triv_goal ct = instantiate' [] [Some ct] triv_goal;
(** meta-level conjunction **)
local
val A = read_prop "PROP A";
val B = read_prop "PROP B";
val C = read_prop "PROP C";
val ABC = read_prop "PROP A ==> PROP B ==> PROP C";
val proj1 =
forall_intr_list [A, B] (implies_intr_list [A, B] (Thm.assume A))
|> forall_elim_vars 0;
val proj2 =
forall_intr_list [A, B] (implies_intr_list [A, B] (Thm.assume B))
|> forall_elim_vars 0;
val conj_intr_rule =
forall_intr_list [A, B] (implies_intr_list [A, B]
(Thm.forall_intr C (Thm.implies_intr ABC
(implies_elim_list (Thm.assume ABC) [Thm.assume A, Thm.assume B]))))
|> forall_elim_vars 0;
val incr = incr_indexes_wrt [] [] [];
in
fun conj_intr tha thb = thb COMP (tha COMP incr [tha, thb] conj_intr_rule);
val conj_intr_list = foldr1 (uncurry conj_intr);
fun conj_elim th =
let val th' = forall_elim_var (#maxidx (Thm.rep_thm th) + 1) th
in (incr [th'] proj1 COMP th', incr [th'] proj2 COMP th') end;
fun conj_elim_list th =
let val (th1, th2) = conj_elim th
in conj_elim_list th1 @ conj_elim_list th2 end handle THM _ => [th];
fun conj_elim_precise 1 th = [th]
| conj_elim_precise n th =
let val (th1, th2) = conj_elim th
in th1 :: conj_elim_precise (n - 1) th2 end;
val conj_intr_thm = store_standard_thm_open "conjunctionI"
(implies_intr_list [A, B] (conj_intr (Thm.assume A) (Thm.assume B)));
end;
end;
structure BasicDrule: BASIC_DRULE = Drule;
open BasicDrule;