(* Title: Pure/drule.ML ID: $Id: drule.ML,v 1.168 2005/09/29 10:30:30 berghofe Exp $ 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 strip_imp_prems : cterm -> cterm list val strip_imp_concl : cterm -> cterm val cprems_of : thm -> cterm list val cterm_fun : (term -> term) -> (cterm -> cterm) val ctyp_fun : (typ -> typ) -> (ctyp -> ctyp) val read_insts : theory -> (indexname -> typ option) * (indexname -> sort option) -> (indexname -> typ option) * (indexname -> sort option) -> string list -> (indexname * string) list -> (ctyp * 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 gen_all : thm -> thm val freeze_thaw : thm -> thm * (thm -> thm) val freeze_thaw_robust: thm -> thm * (int -> thm -> thm) val implies_elim_list : thm -> thm list -> thm val implies_intr_list : cterm list -> thm -> thm val instantiate : (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm val zero_var_indexes : thm -> thm val implies_intr_hyps : 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: theory -> (string*string)list -> thm -> thm val read_instantiate : (string*string)list -> thm -> thm val cterm_instantiate : (cterm*cterm)list -> thm -> thm val eq_thm_thy : thm * thm -> bool val eq_thm_prop : thm * thm -> bool val weak_eq_thm : thm * thm -> bool val size_of_thm : thm -> int val reflexive_thm : thm val symmetric_thm : thm val transitive_thm : 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 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 equal_elim_rule1 : 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 list_comb: cterm * cterm list -> cterm val strip_comb: cterm -> cterm * cterm list val strip_type: ctyp -> ctyp list * ctyp val beta_conv: cterm -> cterm -> cterm val plain_prop_of: thm -> term val add_used: thm -> string list -> string list 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 satisfy_hyps: thm list -> thm -> thm val flexflex_unique: thm -> thm val close_derivation: thm -> thm val local_standard: thm -> thm val compose_single: thm * int * thm -> thm val add_rule: thm -> thm list -> thm list val del_rule: thm -> thm list -> thm list 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 imp_cong' : thm -> thm -> thm val beta_eta_conversion: cterm -> thm val eta_long_conversion: cterm -> thm val goals_conv : (int -> bool) -> (cterm -> thm) -> cterm -> thm val forall_conv : (cterm -> thm) -> cterm -> thm val fconv_rule : (cterm -> thm) -> thm -> thm val norm_hhf_eq: thm val is_norm_hhf: term -> bool val norm_hhf: theory -> term -> term 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 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 rename_bvars: (string * string) list -> thm -> thm val rename_bvars': string option list -> thm -> thm val unvarifyT: thm -> thm val unvarify: thm -> thm val tvars_intr_list: string list -> thm -> thm * (string * (indexname * sort)) list val remdups_rl: 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 val abs_def: thm -> thm val read_instantiate_sg': theory -> (indexname * string) list -> thm -> thm val read_instantiate': (indexname * string) list -> thm -> thm end; structure Drule: DRULE = struct (** some cterm->cterm operations: faster than calling cterm_of! **) (* FIXME exception CTERM (!?) *) fun dest_implies ct = (case Thm.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 Thm.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])); (* 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 cprop_of; fun cterm_fun f ct = let val {t, thy, ...} = Thm.rep_cterm ct in Thm.cterm_of thy (f t) end; fun ctyp_fun f cT = let val {T, thy, ...} = Thm.rep_ctyp cT in Thm.ctyp_of thy (f T) end; val implies = cterm_of ProtoPure.thy 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)); (*cterm version of list_comb: maps (f, [t1,...,tn]) to f(t1,...,tn) *) fun list_comb (f, []) = f | list_comb (f, t::ts) = list_comb (Thm.capply f t, ts); (*cterm version of strip_comb: maps f(t1,...,tn) to (f, [t1,...,tn]) *) fun strip_comb ct = let fun stripc (p as (ct, cts)) = let val (ct1, ct2) = Thm.dest_comb ct in stripc (ct1, ct2 :: cts) end handle CTERM _ => p in stripc (ct, []) end; (* cterm version of strip_type: maps [T1,...,Tn]--->T to ([T1,T2,...,Tn], T) *) fun strip_type cT = (case Thm.typ_of cT of Type ("fun", _) => let val [cT1, cT2] = Thm.dest_ctyp cT; val (cTs, cT') = strip_type cT2 in (cT1 :: cTs, cT') end | _ => ([], cT)); (*Beta-conversion for cterms, where x is an abstraction. Simply returns the rhs of the meta-equality returned by the beta_conversion rule.*) fun beta_conv x y = #2 (Thm.dest_comb (cprop_of (Thm.beta_conversion false (Thm.capply x y)))); fun plain_prop_of raw_thm = let val thm = Thm.strip_shyps raw_thm; fun err msg = raise THM ("plain_prop_of: " ^ msg, 0, [thm]); val {hyps, prop, tpairs, ...} = Thm.rep_thm thm; in if not (null hyps) then err "theorem may not contain hypotheses" else if not (null (Thm.extra_shyps thm)) then err "theorem may not contain sort hypotheses" else if not (null tpairs) then err "theorem may not contain flex-flex pairs" else prop end; (** 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 thy (rtypes,rsorts) (types,sorts) used insts = let fun is_tv ((a, _), _) = (case Symbol.explode a of "'" :: _ => true | _ => false); val (tvs, vs) = List.partition is_tv insts; fun sort_of ixn = case rsorts ixn of SOME S => S | NONE => absent ixn; fun readT (ixn, st) = let val S = sort_of ixn; val T = Sign.read_typ (thy,sorts) st; in if Sign.typ_instance thy (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(thy,types,sorts) used false sTs; fun mkcVar(ixn,T) = let val U = typ_subst_TVars tye2 T in cterm_of thy (Var(ixn,U)) end val ixnTs = ListPair.zip(ixns, map snd sTs) in (map (fn (ixn, T) => (ctyp_of thy (TVar (ixn, sort_of ixn)), ctyp_of thy 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, tpairs, ...} = Thm.rep_thm thm; (* bogus term! *) val big = Term.list_comb (Term.list_comb (prop, hyps), Thm.terms_of_tpairs tpairs); 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 AList.lookup (op =) frees a else AList.lookup (op =) vars (a,i); fun sort(a,i) = if i<0 then AList.lookup (op =) tfrees a else AList.lookup (op =) tvars (a,i); in (typ,sort) end; fun add_used thm used = let val {prop, hyps, tpairs, ...} = Thm.rep_thm thm in add_term_tvarnames (prop, used) |> fold (curry add_term_tvarnames) hyps |> fold (curry add_term_tvarnames) (Thm.terms_of_tpairs tpairs) 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 AList.lookup (op =) ((#2 o 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 = if k = "" then x else 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 = Pretty.str_of o Sign.pretty_sort (Thm.theory_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,thy,...} = rep_thm th in forall_intr_list (map (cterm_of thy) (sort Term.term_ord (term_frees prop))) th end; val forall_elim_var = PureThy.forall_elim_var; val forall_elim_vars = PureThy.forall_elim_vars; fun gen_all thm = let val {thy, prop, maxidx, ...} = Thm.rep_thm thm; fun elim (x, T) = Thm.forall_elim (Thm.cterm_of thy (Var ((x, maxidx + 1), T))); val vs = Term.strip_all_vars prop; in fold elim (Term.variantlist (map #1 vs, []) ~~ map #2 vs) thm end; (*specialization over a list of cterms*) val forall_elim_list = fold forall_elim; (*maps A1,...,An |- B to [| A1;...;An |] ==> B*) val implies_intr_list = fold_rev implies_intr; (*maps [| A1;...;An |] ==> B and [A1,...,An] to B*) fun implies_elim_list impth ths = Library.foldl (uncurry implies_elim) (impth,ths); (*maps |- B to A1,...,An |- B*) val impose_hyps = fold Thm.weaken; (* maps A1,...,An and A1,...,An |- B to |- B *) fun satisfy_hyps ths th = implies_elim_list (implies_intr_list (map (#prop o Thm.crep_thm) ths) th) ths; (*Reset Var indexes to zero, renaming to preserve distinctness*) fun zero_var_indexes th = let val thy = Thm.theory_of_thm th; val certT = Thm.ctyp_of thy and cert = Thm.cterm_of thy; val (instT, inst) = Term.zero_var_indexes_inst (Thm.full_prop_of th); val cinstT = map (fn (v, T) => (certT (TVar v), certT T)) instT; val cinst = map (fn (v, t) => (cert (Var v), cert t)) inst; in Thm.adjust_maxidx_thm (Thm.instantiate (cinstT, cinst) th) end; (** Standard form of object-rule: no hypotheses, flexflex constraints, Frees, or outer quantifiers; all generality expressed by Vars of index 0.**) (*Discharge all hypotheses.*) fun implies_intr_hyps th = fold Thm.implies_intr (#hyps (Thm.crep_thm th)) th; (*Squash a theorem's flexflex constraints provided it can be done uniquely. This step can lose information.*) fun flexflex_unique th = if null (tpairs_of th) then th else case Seq.chop (2, flexflex_rule th) of ([th],_) => th | ([],_) => raise THM("flexflex_unique: impossible constraints", 0, [th]) | _ => raise THM("flexflex_unique: multiple unifiers", 0, [th]); fun close_derivation thm = if Thm.get_name_tags thm = ("", []) then Thm.name_thm ("", thm) else thm; val standard' = implies_intr_hyps #> forall_intr_frees #> `(#maxidx o Thm.rep_thm) #-> (fn maxidx => forall_elim_vars (maxidx + 1) #> strip_shyps_warning #> zero_var_indexes #> Thm.varifyT #> Thm.compress); val standard = flexflex_unique #> standard' #> close_derivation; val local_standard = strip_shyps #> 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_robust th = let val fth = freezeT th val {prop, tpairs, thy, ...} = rep_thm fth in case foldr add_term_vars [] (prop :: Thm.terms_of_tpairs tpairs) of [] => (fth, fn i => fn x => x) (*No vars: nothing to do!*) | vars => let fun newName (Var(ix,_), pairs) = let val v = gensym (string_of_indexname ix) in ((ix,v)::pairs) end; val alist = foldr newName [] vars fun mk_inst (Var(v,T)) = (cterm_of thy (Var(v,T)), cterm_of thy (Free(((the o AList.lookup (op =) alist) v), T))) val insts = map mk_inst vars fun thaw i th' = (*i is non-negative increment for Var indexes*) th' |> forall_intr_list (map #2 insts) |> forall_elim_list (map (Thm.cterm_incr_indexes i o #1) insts) in (Thm.instantiate ([],insts) fth, thaw) end end; (*Basic version of the function above. No option to rename Vars apart in thaw. The Frees created from Vars have nice names.*) fun freeze_thaw th = let val fth = freezeT th val {prop, tpairs, thy, ...} = rep_thm fth in case foldr add_term_vars [] (prop :: Thm.terms_of_tpairs tpairs) 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 ([], Library.foldr add_term_names (prop :: Thm.terms_of_tpairs tpairs, [])) vars fun mk_inst (Var(v,T)) = (cterm_of thy (Var(v,T)), cterm_of thy (Free(((the o AList.lookup (op =) 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(?a) |] [ !(A,P,a)[| ALL x:A. P(x) |] ==> [| P(a) |] ] *) fun assume_ax thy sP = let val prop = Logic.close_form (term_of (read_cterm thy (sP, propT))) in forall_elim_vars 0 (Thm.assume (cterm_of thy 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 **) (*True if the two theorems have the same theory.*) val eq_thm_thy = eq_thy o pairself Thm.theory_of_thm; (*True if the two theorems have the same prop field, ignoring hyps, der, etc.*) val eq_thm_prop = op aconv o pairself Thm.full_prop_of; (*Useful "distance" function for BEST_FIRST*) val size_of_thm = size_of_term o Thm.full_prop_of; (*maintain lists of theorems --- preserving canonical order*) fun del_rules rs rules = Library.gen_rems eq_thm_prop (rules, rs); fun add_rules rs rules = rs @ del_rules rs rules; val del_rule = del_rules o single; val add_rule = add_rules o single; fun merge_rules (rules1, rules2) = gen_merge_lists' eq_thm_prop 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,thy,...} = rep_thm th; val vars = distinct (term_vars' prop); in forall_intr_list (map (cterm_of thy) vars) th end; val weak_eq_thm = Thm.eq_thm o pairself (forall_intr_vars o freezeT); (*** Meta-Rewriting Rules ***) fun read_prop s = read_cterm ProtoPure.thy (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 ProtoPure.thy (Var(("x",0),TVar(("'a",0),[]))) in store_standard_thm_open "reflexive" (Thm.reflexive cx) end; val symmetric_thm = let val xy = read_prop "x == y" in store_standard_thm_open "symmetric" (Thm.implies_intr xy (Thm.symmetric (Thm.assume xy))) end; val transitive_thm = let val xy = read_prop "x == y" val yz = read_prop "y == 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 imp_cong' = combination o combination (reflexive implies) fun abs_def thm = let val (_, cvs) = strip_comb (fst (dest_equals (cprop_of thm))); val thm' = foldr (fn (ct, thm) => Thm.abstract_rule (case term_of ct of Var ((a, _), _) => a | Free (a, _) => a | _ => "x") ct thm) thm cvs in transitive (symmetric (eta_conversion (fst (dest_equals (cprop_of thm'))))) thm' end; local val dest_eq = dest_equals o cprop_of val rhs_of = snd o dest_eq in fun beta_eta_conversion t = let val thm = beta_conversion true t in transitive thm (eta_conversion (rhs_of thm)) end end; fun eta_long_conversion ct = transitive (beta_eta_conversion ct) (symmetric (beta_eta_conversion (cterm_fun (Pattern.eta_long []) ct))); (*In [A1,...,An]==>B, rewrite the selected A's only -- for rewrite_goals_tac*) fun goals_conv pred cv = let fun gconv i ct = let val (A,B) = dest_implies ct in imp_cong' (if pred i then cv A else reflexive A) (gconv (i+1) B) end handle TERM _ => reflexive ct in gconv 1 end (* Rewrite A in !!x1,...,xn. A *) fun forall_conv cv ct = let val p as (ct1, ct2) = Thm.dest_comb ct in (case pairself term_of p of (Const ("all", _), Abs (s, _, _)) => let val (v, ct') = Thm.dest_abs (SOME (gensym "all_")) ct2; in Thm.combination (Thm.reflexive ct1) (Thm.abstract_rule s v (forall_conv cv ct')) end | _ => cv ct) end handle TERM _ => cv ct; (*Use a conversion to transform a theorem*) fun fconv_rule cv th = equal_elim (cv (cprop_of th)) th; (*** 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 ProtoPure.thy ("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 == PROP ?psi; PROP ?phi |] ==> PROP ?psi *) val equal_elim_rule1 = let val eq = read_prop "PROP phi == PROP psi" and P = read_prop "PROP phi" in store_standard_thm_open "equal_elim_rule1" (Thm.equal_elim (assume eq) (assume P) |> implies_intr_list [eq, P]) end; (* "[| PROP ?phi; PROP ?phi; PROP ?psi |] ==> PROP ?psi" *) val remdups_rl = let val P = read_prop "PROP phi" and Q = read_prop "PROP psi"; in store_standard_thm_open "remdups_rl" (implies_intr_list [P, P, Q] (Thm.assume Q)) 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 ProtoPure.thy; val aT = TFree ("'a", []); 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; fun is_norm_hhf tm = let fun is_norm (Const ("==>", _) $ _ $ (Const ("all", _) $ _)) = false | is_norm (t $ u) = is_norm t andalso is_norm u | is_norm (Abs (_, _, t)) = is_norm t | is_norm _ = true; in is_norm (Pattern.beta_eta_contract tm) end; fun norm_hhf thy t = if is_norm_hhf t then t else Pattern.rewrite_term thy [Logic.dest_equals (prop_of norm_hhf_eq)] [] t; (*** Instantiate theorem th, reading instantiations in theory thy ****) (*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' thy sinsts th = let val ts = types_sorts th; val used = add_used th []; in instantiate (read_insts thy ts ts used sinsts) th end; fun read_instantiate_sg thy sinsts th = read_instantiate_sg' thy (map (apfst Syntax.indexname) sinsts) th; (*Instantiate theorem th, reading instantiations under theory of th*) fun read_instantiate sinsts th = read_instantiate_sg (Thm.theory_of_thm th) sinsts th; fun read_instantiate' sinsts th = read_instantiate_sg' (Thm.theory_of_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), (thy,tye,maxidx)) = let val {thy=thyt, t=t, T= T, maxidx=maxt,...} = rep_cterm ct and {thy=thyu, t=u, T= U, maxidx=maxu,...} = rep_cterm cu; val maxi = Int.max(maxidx, Int.max(maxt, maxu)); val thy' = Theory.merge(thy, Theory.merge(thyt, thyu)) val (tye',maxi') = Sign.typ_unify thy' (T, U) (tye, maxi) handle Type.TUNIFY => raise TYPE("Ill-typed instantiation", [T,U], [t,u]) in (thy', tye', maxi') end; in fun cterm_instantiate ctpairs0 th = let val (thy,tye,_) = foldr add_types (Thm.theory_of_thm th, Vartab.empty, 0) ctpairs0 fun instT(ct,cu) = let val inst = cterm_of thy o Envir.subst_TVars tye o term_of in (inst ct, inst cu) end fun ctyp2 (ixn, (S, T)) = (ctyp_of thy (TVar (ixn, S)), ctyp_of thy T) in instantiate (map ctyp2 (Vartab.dest tye), map instT ctpairs0) th end handle TERM _ => raise THM("cterm_instantiate: incompatible theories",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 {thy=thya, t=a, ...} = rep_cterm ca and combth = combination eqth (reflexive ca) val {thy,prop,...} = rep_thm eqth val (abst,absu) = Logic.dest_equals prop val cterm = cterm_of (Theory.merge (thy,thya)) 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) th (rev cts); (*** Goal (PROP A) <==> PROP A ***) local val cert = Thm.cterm_of ProtoPure.thy; 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 ProtoPure.thy 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 (the_context()) [(x,t)]; (* vars in left-to-right order *) fun tvars_of_terms ts = rev (fold Term.add_tvars ts []); fun vars_of_terms ts = rev (fold Term.add_vars ts []); fun tvars_of thm = tvars_of_terms [Thm.full_prop_of thm]; fun vars_of thm = vars_of_terms [Thm.full_prop_of thm]; (* instantiate by left-to-right occurrence of variables *) fun instantiate' cTs cts thm = let fun err msg = raise TYPE ("instantiate': " ^ msg, List.mapPartial (Option.map Thm.typ_of) cTs, List.mapPartial (Option.map Thm.term_of) cts); fun inst_of (v, ct) = (Thm.cterm_of (Thm.theory_of_cterm ct) (Var v), ct) handle TYPE (msg, _, _) => err msg; fun tyinst_of (v, cT) = (Thm.ctyp_of (Thm.theory_of_ctyp cT) (TVar 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 (map tyinst_of (zip_vars (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; (** renaming of bound variables **) (* replace bound variables x_i in thm by y_i *) (* where vs = [(x_1, y_1), ..., (x_n, y_n)] *) fun rename_bvars [] thm = thm | rename_bvars vs thm = let val {thy, prop, ...} = rep_thm thm; fun ren (Abs (x, T, t)) = Abs (AList.lookup (op =) vs x |> the_default x, T, ren t) | ren (t $ u) = ren t $ ren u | ren t = t; in equal_elim (reflexive (cterm_of thy (ren prop))) thm end; (* renaming in left-to-right order *) fun rename_bvars' xs thm = let val {thy, prop, ...} = rep_thm thm; fun rename [] t = ([], t) | rename (x' :: xs) (Abs (x, T, t)) = let val (xs', t') = rename xs t in (xs', Abs (getOpt (x',x), T, t')) end | rename xs (t $ u) = let val (xs', t') = rename xs t; val (xs'', u') = rename xs' u in (xs'', t' $ u') end | rename xs t = (xs, t); in case rename xs prop of ([], prop') => equal_elim (reflexive (cterm_of thy prop')) thm | _ => error "More names than abstractions in theorem" 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.theory_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.theory_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_tfrees [] (prop :: hyps) end; fun tvars_intr_list tfrees thm = apsnd (map (fn ((s, S), ixn) => (s, (ixn, S)))) (Thm.varifyT' (gen_rems (op = o apfst fst) (tfrees_of thm, tfrees)) thm); (* increment var indexes *) fun incr_indexes_wrt is cTs cts thms = let val maxidx = Library.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.theory_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.theory_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); fun conj_intr_list [] = asm_rl | conj_intr_list ths = foldr1 (uncurry conj_intr) ths; 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 0 _ = [] | 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;