diff -r 000000000000 -r a5a9c433f639 src/Pure/thm.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/Pure/thm.ML Thu Sep 16 12:20:38 1993 +0200 @@ -0,0 +1,990 @@ +(* Title: thm + ID: $Id$ + Author: Lawrence C Paulson, Cambridge University Computer Laboratory + Copyright 1991 University of Cambridge + +The abstract types "theory" and "thm" +*) + +signature THM = + sig + structure Envir : ENVIR + structure Sequence : SEQUENCE + structure Sign : SIGN + type meta_simpset + type theory + type thm + exception THM of string * int * thm list + exception THEORY of string * theory list + exception SIMPLIFIER of string * thm + val abstract_rule: string -> Sign.cterm -> thm -> thm + val add_congs: meta_simpset * thm list -> meta_simpset + val add_prems: meta_simpset * thm list -> meta_simpset + val add_simps: meta_simpset * thm list -> meta_simpset + val assume: Sign.cterm -> thm + val assumption: int -> thm -> thm Sequence.seq + val axioms_of: theory -> (string * thm) list + val beta_conversion: Sign.cterm -> thm + val bicompose: bool -> bool * thm * int -> int -> thm -> thm Sequence.seq + val biresolution: bool -> (bool*thm)list -> int -> thm -> thm Sequence.seq + val combination: thm -> thm -> thm + val concl_of: thm -> term + val dest_state: thm * int -> (term*term)list * term list * term * term + val empty_mss: meta_simpset + val eq_assumption: int -> thm -> thm + val equal_intr: thm -> thm -> thm + val equal_elim: thm -> thm -> thm + val extend_theory: theory -> string + -> (class * class list) list * sort + * (string list * int)list + * (string list * (sort list * class))list + * (string list * string)list * Sign.Syntax.sext option + -> (string*string)list -> theory + val extensional: thm -> thm + val flexflex_rule: thm -> thm Sequence.seq + val flexpair_def: thm + val forall_elim: Sign.cterm -> thm -> thm + val forall_intr: Sign.cterm -> thm -> thm + val freezeT: thm -> thm + val get_axiom: theory -> string -> thm + val implies_elim: thm -> thm -> thm + val implies_intr: Sign.cterm -> thm -> thm + val implies_intr_hyps: thm -> thm + val instantiate: (indexname*Sign.ctyp)list * (Sign.cterm*Sign.cterm)list + -> thm -> thm + val lift_rule: (thm * int) -> thm -> thm + val merge_theories: theory * theory -> theory + val mk_rews_of_mss: meta_simpset -> thm -> thm list + val mss_of: thm list -> meta_simpset + val nprems_of: thm -> int + val parents_of: theory -> theory list + val prems_of: thm -> term list + val prems_of_mss: meta_simpset -> thm list + val pure_thy: theory + val reflexive: Sign.cterm -> thm + val rename_params_rule: string list * int -> thm -> thm + val rep_thm: thm -> {prop: term, hyps: term list, maxidx: int, sign: Sign.sg} + val rewrite_cterm: meta_simpset -> (meta_simpset -> thm -> thm option) + -> Sign.cterm -> thm + val set_mk_rews: meta_simpset * (thm -> thm list) -> meta_simpset + val sign_of: theory -> Sign.sg + val syn_of: theory -> Sign.Syntax.syntax + val stamps_of_thm: thm -> string ref list + val stamps_of_thy: theory -> string ref list + val symmetric: thm -> thm + val tpairs_of: thm -> (term*term)list + val trace_simp: bool ref + val transitive: thm -> thm -> thm + val trivial: Sign.cterm -> thm + val varifyT: thm -> thm + end; + + + +functor ThmFun (structure Logic: LOGIC and Unify: UNIFY and Pattern:PATTERN + and Net:NET + sharing type Pattern.type_sig = Unify.Sign.Type.type_sig) + : THM = +struct +structure Sequence = Unify.Sequence; +structure Envir = Unify.Envir; +structure Sign = Unify.Sign; +structure Type = Sign.Type; +structure Syntax = Sign.Syntax; +structure Symtab = Sign.Symtab; + + +(*Meta-theorems*) +datatype thm = Thm of + {sign: Sign.sg, maxidx: int, hyps: term list, prop: term}; + +fun rep_thm (Thm x) = x; + +(*Errors involving theorems*) +exception THM of string * int * thm list; + +(*maps object-rule to tpairs *) +fun tpairs_of (Thm{prop,...}) = #1 (Logic.strip_flexpairs prop); + +(*maps object-rule to premises *) +fun prems_of (Thm{prop,...}) = + Logic.strip_imp_prems (Logic.skip_flexpairs prop); + +(*counts premises in a rule*) +fun nprems_of (Thm{prop,...}) = + Logic.count_prems (Logic.skip_flexpairs prop, 0); + +(*maps object-rule to conclusion *) +fun concl_of (Thm{prop,...}) = Logic.strip_imp_concl prop; + +(*Stamps associated with a signature*) +val stamps_of_thm = #stamps o Sign.rep_sg o #sign o rep_thm; + +(*Theories. There is one pure theory. + A theory can be extended. Two theories can be merged.*) +datatype theory = + Pure of {sign: Sign.sg} + | Extend of {sign: Sign.sg, axioms: thm Symtab.table, thy: theory} + | Merge of {sign: Sign.sg, thy1: theory, thy2: theory}; + +(*Errors involving theories*) +exception THEORY of string * theory list; + +fun sign_of (Pure {sign}) = sign + | sign_of (Extend {sign,...}) = sign + | sign_of (Merge {sign,...}) = sign; + +val syn_of = #syn o Sign.rep_sg o sign_of; + +(*return the axioms of a theory and its ancestors*) +fun axioms_of (Pure _) = [] + | axioms_of (Extend{axioms,thy,...}) = Symtab.alist_of axioms + | axioms_of (Merge{thy1,thy2,...}) = axioms_of thy1 @ axioms_of thy2; + +(*return the immediate ancestors -- also distinguishes the kinds of theories*) +fun parents_of (Pure _) = [] + | parents_of (Extend{thy,...}) = [thy] + | parents_of (Merge{thy1,thy2,...}) = [thy1,thy2]; + + +(*Merge theories of two theorems. Raise exception if incompatible. + Prefers (via Sign.merge) the signature of th1. *) +fun merge_theories(th1,th2) = + let val Thm{sign=sign1,...} = th1 and Thm{sign=sign2,...} = th2 + in Sign.merge (sign1,sign2) end + handle TERM _ => raise THM("incompatible signatures", 0, [th1,th2]); + +(*Stamps associated with a theory*) +val stamps_of_thy = #stamps o Sign.rep_sg o sign_of; + + +(**** Primitive rules ****) + +(* discharge all assumptions t from ts *) +val disch = gen_rem (op aconv); + +(*The assumption rule A|-A in a theory *) +fun assume ct : thm = + let val {sign, t=prop, T, maxidx} = Sign.rep_cterm ct + in if T<>propT then + raise THM("assume: assumptions must have type prop", 0, []) + else if maxidx <> ~1 then + raise THM("assume: assumptions may not contain scheme variables", + maxidx, []) + else Thm{sign = sign, maxidx = ~1, hyps = [prop], prop = prop} + end; + +(* Implication introduction + A |- B + ------- + A ==> B *) +fun implies_intr cA (thB as Thm{sign,maxidx,hyps,prop}) : thm = + let val {sign=signA, t=A, T, maxidx=maxidxA} = Sign.rep_cterm cA + in if T<>propT then + raise THM("implies_intr: assumptions must have type prop", 0, [thB]) + else Thm{sign= Sign.merge (sign,signA), maxidx= max[maxidxA, maxidx], + hyps= disch(hyps,A), prop= implies$A$prop} + handle TERM _ => + raise THM("implies_intr: incompatible signatures", 0, [thB]) + end; + +(* Implication elimination + A ==> B A + --------------- + B *) +fun implies_elim thAB thA : thm = + let val Thm{maxidx=maxA, hyps=hypsA, prop=propA,...} = thA + and Thm{sign, maxidx, hyps, prop,...} = thAB; + fun err(a) = raise THM("implies_elim: "^a, 0, [thAB,thA]) + in case prop of + imp$A$B => + if imp=implies andalso A aconv propA + then Thm{sign= merge_theories(thAB,thA), + maxidx= max[maxA,maxidx], + hyps= hypsA union hyps, (*dups suppressed*) + prop= B} + else err("major premise") + | _ => err("major premise") + end; + +(* Forall introduction. The Free or Var x must not be free in the hypotheses. + A + ------ + !!x.A *) +fun forall_intr cx (th as Thm{sign,maxidx,hyps,prop}) = + let val x = Sign.term_of cx; + fun result(a,T) = Thm{sign= sign, maxidx= maxidx, hyps= hyps, + prop= all(T) $ Abs(a, T, abstract_over (x,prop))} + in case x of + Free(a,T) => + if exists (apl(x, Logic.occs)) hyps + then raise THM("forall_intr: variable free in assumptions", 0, [th]) + else result(a,T) + | Var((a,_),T) => result(a,T) + | _ => raise THM("forall_intr: not a variable", 0, [th]) + end; + +(* Forall elimination + !!x.A + -------- + A[t/x] *) +fun forall_elim ct (th as Thm{sign,maxidx,hyps,prop}) : thm = + let val {sign=signt, t, T, maxidx=maxt} = Sign.rep_cterm ct + in case prop of + Const("all",Type("fun",[Type("fun",[qary,_]),_])) $ A => + if T<>qary then + raise THM("forall_elim: type mismatch", 0, [th]) + else Thm{sign= Sign.merge(sign,signt), + maxidx= max[maxidx, maxt], + hyps= hyps, prop= betapply(A,t)} + | _ => raise THM("forall_elim: not quantified", 0, [th]) + end + handle TERM _ => + raise THM("forall_elim: incompatible signatures", 0, [th]); + + +(*** Equality ***) + +(*Definition of the relation =?= *) +val flexpair_def = + Thm{sign= Sign.pure, hyps= [], maxidx= 0, + prop= Sign.term_of + (Sign.read_cterm Sign.pure + ("(?t =?= ?u) == (?t == ?u::?'a::{})", propT))}; + +(*The reflexivity rule: maps t to the theorem t==t *) +fun reflexive ct = + let val {sign, t, T, maxidx} = Sign.rep_cterm ct + in Thm{sign= sign, hyps= [], maxidx= maxidx, prop= Logic.mk_equals(t,t)} + end; + +(*The symmetry rule + t==u + ---- + u==t *) +fun symmetric (th as Thm{sign,hyps,prop,maxidx}) = + case prop of + (eq as Const("==",_)) $ t $ u => + Thm{sign=sign, hyps=hyps, maxidx=maxidx, prop= eq$u$t} + | _ => raise THM("symmetric", 0, [th]); + +(*The transitive rule + t1==u u==t2 + ------------ + t1==t2 *) +fun transitive th1 th2 = + let val Thm{maxidx=max1, hyps=hyps1, prop=prop1,...} = th1 + and Thm{maxidx=max2, hyps=hyps2, prop=prop2,...} = th2; + fun err(msg) = raise THM("transitive: "^msg, 0, [th1,th2]) + in case (prop1,prop2) of + ((eq as Const("==",_)) $ t1 $ u, Const("==",_) $ u' $ t2) => + if not (u aconv u') then err"middle term" else + Thm{sign= merge_theories(th1,th2), hyps= hyps1 union hyps2, + maxidx= max[max1,max2], prop= eq$t1$t2} + | _ => err"premises" + end; + +(*Beta-conversion: maps (%(x)t)(u) to the theorem (%(x)t)(u) == t[u/x] *) +fun beta_conversion ct = + let val {sign, t, T, maxidx} = Sign.rep_cterm ct + in case t of + Abs(_,_,bodt) $ u => + Thm{sign= sign, hyps= [], + maxidx= maxidx_of_term t, + prop= Logic.mk_equals(t, subst_bounds([u],bodt))} + | _ => raise THM("beta_conversion: not a redex", 0, []) + end; + +(*The extensionality rule (proviso: x not free in f, g, or hypotheses) + f(x) == g(x) + ------------ + f == g *) +fun extensional (th as Thm{sign,maxidx,hyps,prop}) = + case prop of + (Const("==",_)) $ (f$x) $ (g$y) => + let fun err(msg) = raise THM("extensional: "^msg, 0, [th]) + in (if x<>y then err"different variables" else + case y of + Free _ => + if exists (apl(y, Logic.occs)) (f::g::hyps) + then err"variable free in hyps or functions" else () + | Var _ => + if Logic.occs(y,f) orelse Logic.occs(y,g) + then err"variable free in functions" else () + | _ => err"not a variable"); + Thm{sign=sign, hyps=hyps, maxidx=maxidx, + prop= Logic.mk_equals(f,g)} + end + | _ => raise THM("extensional: premise", 0, [th]); + +(*The abstraction rule. The Free or Var x must not be free in the hypotheses. + The bound variable will be named "a" (since x will be something like x320) + t == u + ---------------- + %(x)t == %(x)u *) +fun abstract_rule a cx (th as Thm{sign,maxidx,hyps,prop}) = + let val x = Sign.term_of cx; + val (t,u) = Logic.dest_equals prop + handle TERM _ => + raise THM("abstract_rule: premise not an equality", 0, [th]) + fun result T = + Thm{sign= sign, maxidx= maxidx, hyps= hyps, + prop= Logic.mk_equals(Abs(a, T, abstract_over (x,t)), + Abs(a, T, abstract_over (x,u)))} + in case x of + Free(_,T) => + if exists (apl(x, Logic.occs)) hyps + then raise THM("abstract_rule: variable free in assumptions", 0, [th]) + else result T + | Var(_,T) => result T + | _ => raise THM("abstract_rule: not a variable", 0, [th]) + end; + +(*The combination rule + f==g t==u + ------------ + f(t)==g(u) *) +fun combination th1 th2 = + let val Thm{maxidx=max1, hyps=hyps1, prop=prop1,...} = th1 + and Thm{maxidx=max2, hyps=hyps2, prop=prop2,...} = th2 + in case (prop1,prop2) of + (Const("==",_) $ f $ g, Const("==",_) $ t $ u) => + Thm{sign= merge_theories(th1,th2), hyps= hyps1 union hyps2, + maxidx= max[max1,max2], prop= Logic.mk_equals(f$t, g$u)} + | _ => raise THM("combination: premises", 0, [th1,th2]) + end; + + +(*The equal propositions rule + A==B A + --------- + B *) +fun equal_elim th1 th2 = + let val Thm{maxidx=max1, hyps=hyps1, prop=prop1,...} = th1 + and Thm{maxidx=max2, hyps=hyps2, prop=prop2,...} = th2; + fun err(msg) = raise THM("equal_elim: "^msg, 0, [th1,th2]) + in case prop1 of + Const("==",_) $ A $ B => + if not (prop2 aconv A) then err"not equal" else + Thm{sign= merge_theories(th1,th2), hyps= hyps1 union hyps2, + maxidx= max[max1,max2], prop= B} + | _ => err"major premise" + end; + + +(* Equality introduction + A==>B B==>A + ------------- + A==B *) +fun equal_intr th1 th2 = +let val Thm{maxidx=max1, hyps=hyps1, prop=prop1,...} = th1 + and Thm{maxidx=max2, hyps=hyps2, prop=prop2,...} = th2; + fun err(msg) = raise THM("equal_intr: "^msg, 0, [th1,th2]) +in case (prop1,prop2) of + (Const("==>",_) $ A $ B, Const("==>",_) $ B' $ A') => + if A aconv A' andalso B aconv B' + then Thm{sign= merge_theories(th1,th2), hyps= hyps1 union hyps2, + maxidx= max[max1,max2], prop= Logic.mk_equals(A,B)} + else err"not equal" + | _ => err"premises" +end; + +(**** Derived rules ****) + +(*Discharge all hypotheses (need not verify cterms) + Repeated hypotheses are discharged only once; fold cannot do this*) +fun implies_intr_hyps (Thm{sign, maxidx, hyps=A::As, prop}) = + implies_intr_hyps + (Thm{sign=sign, maxidx=maxidx, + hyps= disch(As,A), prop= implies$A$prop}) + | implies_intr_hyps th = th; + +(*Smash" unifies the list of term pairs leaving no flex-flex pairs. + Instantiates the theorem and deletes trivial tpairs. + Resulting sequence may contain multiple elements if the tpairs are + not all flex-flex. *) +fun flexflex_rule (Thm{sign,maxidx,hyps,prop}) = + let fun newthm env = + let val (tpairs,horn) = + Logic.strip_flexpairs (Envir.norm_term env prop) + (*Remove trivial tpairs, of the form t=t*) + val distpairs = filter (not o op aconv) tpairs + val newprop = Logic.list_flexpairs(distpairs, horn) + in Thm{sign= sign, hyps= hyps, + maxidx= maxidx_of_term newprop, prop= newprop} + end; + val (tpairs,_) = Logic.strip_flexpairs prop + in Sequence.maps newthm + (Unify.smash_unifiers(sign, Envir.empty maxidx, tpairs)) + end; + + +(*Instantiation of Vars + A + -------------------- + A[t1/v1,....,tn/vn] *) + +(*Check that all the terms are Vars and are distinct*) +fun instl_ok ts = forall is_Var ts andalso null(findrep ts); + +(*For instantiate: process pair of cterms, merge theories*) +fun add_ctpair ((ct,cu), (sign,tpairs)) = + let val {sign=signt, t=t, T= T, ...} = Sign.rep_cterm ct + and {sign=signu, t=u, T= U, ...} = Sign.rep_cterm cu + in if T=U then (Sign.merge(sign, Sign.merge(signt, signu)), (t,u)::tpairs) + else raise TYPE("add_ctpair", [T,U], [t,u]) + end; + +fun add_ctyp ((v,ctyp), (sign',vTs)) = + let val {T,sign} = Sign.rep_ctyp ctyp + in (Sign.merge(sign,sign'), (v,T)::vTs) end; + +fun duplicates t = findrep (map (#1 o dest_Var) (term_vars t)); + +(*Left-to-right replacements: ctpairs = [...,(vi,ti),...]. + Instantiates distinct Vars by terms of same type. + Normalizes the new theorem! *) +fun instantiate (vcTs,ctpairs) (th as Thm{sign,maxidx,hyps,prop}) = + let val (newsign,tpairs) = foldr add_ctpair (ctpairs, (sign,[])); + val (newsign,vTs) = foldr add_ctyp (vcTs, (newsign,[])); + val prop = Type.inst_term_tvars(#tsig(Sign.rep_sg newsign),vTs) prop; + val newprop = Envir.norm_term (Envir.empty 0) (subst_atomic tpairs prop) + val newth = Thm{sign= newsign, hyps= hyps, + maxidx= maxidx_of_term newprop, prop= newprop} + in if not(instl_ok(map #1 tpairs)) orelse not(null(findrep(map #1 vTs))) + then raise THM("instantiate: not distinct Vars", 0, [th]) + else case duplicates newprop of + [] => newth + | ix::_ => raise THM("instantiate: conflicting types for " ^ + Syntax.string_of_vname ix ^ "\n", 0, [newth]) + end + handle TERM _ => + raise THM("instantiate: incompatible signatures",0,[th]) + | TYPE _ => raise THM("instantiate: types", 0, [th]); + + +(*The trivial implication A==>A, justified by assume and forall rules. + A can contain Vars, not so for assume! *) +fun trivial ct : thm = + let val {sign, t=A, T, maxidx} = Sign.rep_cterm ct + in if T<>propT then + raise THM("trivial: the term must have type prop", 0, []) + else Thm{sign= sign, maxidx= maxidx, hyps= [], prop= implies$A$A} + end; + +(* Replace all TFrees not in the hyps by new TVars *) +fun varifyT(Thm{sign,maxidx,hyps,prop}) = + let val tfrees = foldr add_term_tfree_names (hyps,[]) + in Thm{sign=sign, maxidx=max[0,maxidx], hyps=hyps, + prop= Type.varify(prop,tfrees)} + end; + +(* Replace all TVars by new TFrees *) +fun freezeT(Thm{sign,maxidx,hyps,prop}) = + let val prop' = Type.freeze (K true) prop + in Thm{sign=sign, maxidx=maxidx_of_term prop', hyps=hyps, prop=prop'} end; + + +(*** Inference rules for tactics ***) + +(*Destruct proof state into constraints, other goals, goal(i), rest *) +fun dest_state (state as Thm{prop,...}, i) = + let val (tpairs,horn) = Logic.strip_flexpairs prop + in case Logic.strip_prems(i, [], horn) of + (B::rBs, C) => (tpairs, rev rBs, B, C) + | _ => raise THM("dest_state", i, [state]) + end + handle TERM _ => raise THM("dest_state", i, [state]); + +(*Increment variables and parameters of rule as required for + resolution with goal i of state. *) +fun lift_rule (state, i) orule = + let val Thm{prop=sprop,maxidx=smax,...} = state; + val (Bi::_, _) = Logic.strip_prems(i, [], Logic.skip_flexpairs sprop) + handle TERM _ => raise THM("lift_rule", i, [orule,state]); + val (lift_abs,lift_all) = Logic.lift_fns(Bi,smax+1); + val (Thm{sign,maxidx,hyps,prop}) = orule + val (tpairs,As,B) = Logic.strip_horn prop + in Thm{hyps=hyps, sign= merge_theories(state,orule), + maxidx= maxidx+smax+1, + prop= Logic.rule_of(map (pairself lift_abs) tpairs, + map lift_all As, lift_all B)} + end; + +(*Solve subgoal Bi of proof state B1...Bn/C by assumption. *) +fun assumption i state = + let val Thm{sign,maxidx,hyps,prop} = state; + val (tpairs, Bs, Bi, C) = dest_state(state,i) + fun newth (env as Envir.Envir{maxidx,asol,iTs}, tpairs) = + Thm{sign=sign, hyps=hyps, maxidx=maxidx, prop= + case (Envir.alist_of_olist asol, iTs) of + (*avoid wasted normalizations*) + ([],[]) => Logic.rule_of(tpairs, Bs, C) + | _ => (*normalize the new rule fully*) + Envir.norm_term env (Logic.rule_of(tpairs, Bs, C))}; + fun addprfs [] = Sequence.null + | addprfs ((t,u)::apairs) = Sequence.seqof (fn()=> Sequence.pull + (Sequence.mapp newth + (Unify.unifiers(sign,Envir.empty maxidx, (t,u)::tpairs)) + (addprfs apairs))) + in addprfs (Logic.assum_pairs Bi) end; + +(*Solve subgoal Bi of proof state B1...Bn/C by assumption. + Checks if Bi's conclusion is alpha-convertible to one of its assumptions*) +fun eq_assumption i state = + let val Thm{sign,maxidx,hyps,prop} = state; + val (tpairs, Bs, Bi, C) = dest_state(state,i) + in if exists (op aconv) (Logic.assum_pairs Bi) + then Thm{sign=sign, hyps=hyps, maxidx=maxidx, + prop=Logic.rule_of(tpairs, Bs, C)} + else raise THM("eq_assumption", 0, [state]) + end; + + +(** User renaming of parameters in a subgoal **) + +(*Calls error rather than raising an exception because it is intended + for top-level use -- exception handling would not make sense here. + The names in cs, if distinct, are used for the innermost parameters; + preceding parameters may be renamed to make all params distinct.*) +fun rename_params_rule (cs, i) state = + let val Thm{sign,maxidx,hyps,prop} = state + val (tpairs, Bs, Bi, C) = dest_state(state,i) + val iparams = map #1 (Logic.strip_params Bi) + val short = length iparams - length cs + val newnames = + if short<0 then error"More names than abstractions!" + else variantlist(take (short,iparams), cs) @ cs + val freenames = map (#1 o dest_Free) (term_frees prop) + val newBi = Logic.list_rename_params (newnames, Bi) + in + case findrep cs of + c::_ => error ("Bound variables not distinct: " ^ c) + | [] => (case cs inter freenames of + a::_ => error ("Bound/Free variable clash: " ^ a) + | [] => Thm{sign=sign, hyps=hyps, maxidx=maxidx, prop= + Logic.rule_of(tpairs, Bs@[newBi], C)}) + end; + +(*** Preservation of bound variable names ***) + +(*Scan a pair of terms; while they are similar, + accumulate corresponding bound vars in "al"*) +fun match_bvs(Abs(x,_,s),Abs(y,_,t), al) = match_bvs(s,t,(x,y)::al) + | match_bvs(f$s, g$t, al) = match_bvs(f,g,match_bvs(s,t,al)) + | match_bvs(_,_,al) = al; + +(* strip abstractions created by parameters *) +fun match_bvars((s,t),al) = match_bvs(strip_abs_body s, strip_abs_body t, al); + + +(* strip_apply f A(,B) strips off all assumptions/parameters from A + introduced by lifting over B, and applies f to remaining part of A*) +fun strip_apply f = + let fun strip(Const("==>",_)$ A1 $ B1, + Const("==>",_)$ _ $ B2) = implies $ A1 $ strip(B1,B2) + | strip((c as Const("all",_)) $ Abs(a,T,t1), + Const("all",_) $ Abs(_,_,t2)) = c$Abs(a,T,strip(t1,t2)) + | strip(A,_) = f A + in strip end; + +(*Use the alist to rename all bound variables and some unknowns in a term + dpairs = current disagreement pairs; tpairs = permanent ones (flexflex); + Preserves unknowns in tpairs and on lhs of dpairs. *) +fun rename_bvs([],_,_,_) = I + | rename_bvs(al,dpairs,tpairs,B) = + let val vars = foldr add_term_vars + (map fst dpairs @ map fst tpairs @ map snd tpairs, []) + (*unknowns appearing elsewhere be preserved!*) + val vids = map (#1 o #1 o dest_Var) vars; + fun rename(t as Var((x,i),T)) = + (case assoc(al,x) of + Some(y) => if x mem vids orelse y mem vids then t + else Var((y,i),T) + | None=> t) + | rename(Abs(x,T,t)) = + Abs(case assoc(al,x) of Some(y) => y | None => x, + T, rename t) + | rename(f$t) = rename f $ rename t + | rename(t) = t; + fun strip_ren Ai = strip_apply rename (Ai,B) + in strip_ren end; + +(*Function to rename bounds/unknowns in the argument, lifted over B*) +fun rename_bvars(dpairs, tpairs, B) = + rename_bvs(foldr match_bvars (dpairs,[]), dpairs, tpairs, B); + + +(*** RESOLUTION ***) + +(*strip off pairs of assumptions/parameters in parallel -- they are + identical because of lifting*) +fun strip_assums2 (Const("==>", _) $ _ $ B1, + Const("==>", _) $ _ $ B2) = strip_assums2 (B1,B2) + | strip_assums2 (Const("all",_)$Abs(a,T,t1), + Const("all",_)$Abs(_,_,t2)) = + let val (B1,B2) = strip_assums2 (t1,t2) + in (Abs(a,T,B1), Abs(a,T,B2)) end + | strip_assums2 BB = BB; + + +(*Composition of object rule r=(A1...Am/B) with proof state s=(B1...Bn/C) + Unifies B with Bi, replacing subgoal i (1 <= i <= n) + If match then forbid instantiations in proof state + If lifted then shorten the dpair using strip_assums2. + If eres_flg then simultaneously proves A1 by assumption. + nsubgoal is the number of new subgoals (written m above). + Curried so that resolution calls dest_state only once. +*) +local open Sequence; exception Bicompose +in +fun bicompose_aux match (state, (stpairs, Bs, Bi, C), lifted) + (eres_flg, orule, nsubgoal) = + let val Thm{maxidx=smax, hyps=shyps, ...} = state + and Thm{maxidx=rmax, hyps=rhyps, prop=rprop,...} = orule; + val sign = merge_theories(state,orule); + (** Add new theorem with prop = '[| Bs; As |] ==> C' to thq **) + fun addth As ((env as Envir.Envir{maxidx,asol,iTs}, tpairs), thq) = + let val minenv = case Envir.alist_of_olist asol of + [] => ~1 | ((_,i),_) :: _ => i; + val minx = min (minenv :: map (fn ((_,i),_) => i) iTs); + val normt = Envir.norm_term env; + (*Perform minimal copying here by examining env*) + val normp = if minx = ~1 then (tpairs, Bs@As, C) + else + let val ntps = map (pairself normt) tpairs + in if minx>smax then (*no assignments in state*) + (ntps, Bs @ map normt As, C) + else if match then raise Bicompose + else (*normalize the new rule fully*) + (ntps, map normt (Bs @ As), normt C) + end + val th = Thm{sign=sign, hyps=rhyps union shyps, maxidx=maxidx, + prop= Logic.rule_of normp} + in cons(th, thq) end handle Bicompose => thq + val (rtpairs,rhorn) = Logic.strip_flexpairs(rprop); + val (rAs,B) = Logic.strip_prems(nsubgoal, [], rhorn) + handle TERM _ => raise THM("bicompose: rule", 0, [orule,state]); + (*Modify assumptions, deleting n-th if n>0 for e-resolution*) + fun newAs(As0, n, dpairs, tpairs) = + let val As1 = if !Logic.auto_rename orelse not lifted then As0 + else map (rename_bvars(dpairs,tpairs,B)) As0 + in (map (Logic.flatten_params n) As1) + handle TERM _ => + raise THM("bicompose: 1st premise", 0, [orule]) + end; + val env = Envir.empty(max[rmax,smax]); + val BBi = if lifted then strip_assums2(B,Bi) else (B,Bi); + val dpairs = BBi :: (rtpairs@stpairs); + (*elim-resolution: try each assumption in turn. Initially n=1*) + fun tryasms (_, _, []) = null + | tryasms (As, n, (t,u)::apairs) = + (case pull(Unify.unifiers(sign, env, (t,u)::dpairs)) of + None => tryasms (As, n+1, apairs) + | cell as Some((_,tpairs),_) => + its_right (addth (newAs(As, n, [BBi,(u,t)], tpairs))) + (seqof (fn()=> cell), + seqof (fn()=> pull (tryasms (As, n+1, apairs))))); + fun eres [] = raise THM("bicompose: no premises", 0, [orule,state]) + | eres (A1::As) = tryasms (As, 1, Logic.assum_pairs A1); + (*ordinary resolution*) + fun res(None) = null + | res(cell as Some((_,tpairs),_)) = + its_right (addth(newAs(rev rAs, 0, [BBi], tpairs))) + (seqof (fn()=> cell), null) + in if eres_flg then eres(rev rAs) + else res(pull(Unify.unifiers(sign, env, dpairs))) + end; +end; (*open Sequence*) + + +fun bicompose match arg i state = + bicompose_aux match (state, dest_state(state,i), false) arg; + +(*Quick test whether rule is resolvable with the subgoal with hyps Hs + and conclusion B. If eres_flg then checks 1st premise of rule also*) +fun could_bires (Hs, B, eres_flg, rule) = + let fun could_reshyp (A1::_) = exists (apl(A1,could_unify)) Hs + | could_reshyp [] = false; (*no premise -- illegal*) + in could_unify(concl_of rule, B) andalso + (not eres_flg orelse could_reshyp (prems_of rule)) + end; + +(*Bi-resolution of a state with a list of (flag,rule) pairs. + Puts the rule above: rule/state. Renames vars in the rules. *) +fun biresolution match brules i state = + let val lift = lift_rule(state, i); + val (stpairs, Bs, Bi, C) = dest_state(state,i) + val B = Logic.strip_assums_concl Bi; + val Hs = Logic.strip_assums_hyp Bi; + val comp = bicompose_aux match (state, (stpairs, Bs, Bi, C), true); + fun res [] = Sequence.null + | res ((eres_flg, rule)::brules) = + if could_bires (Hs, B, eres_flg, rule) + then Sequence.seqof (*delay processing remainder til needed*) + (fn()=> Some(comp (eres_flg, lift rule, nprems_of rule), + res brules)) + else res brules + in Sequence.flats (res brules) end; + + +(**** THEORIES ****) + +val pure_thy = Pure{sign = Sign.pure}; + +(*Look up the named axiom in the theory*) +fun get_axiom thy axname = + let fun get (Pure _) = raise Match + | get (Extend{axioms,thy,...}) = + (case Symtab.lookup(axioms,axname) of + Some th => th + | None => get thy) + | get (Merge{thy1,thy2,...}) = + get thy1 handle Match => get thy2 + in get thy + handle Match => raise THEORY("get_axiom: No axiom "^axname, [thy]) + end; + +(*Converts Frees to Vars: axioms can be written without question marks*) +fun prepare_axiom sign sP = + Logic.varify (Sign.term_of (Sign.read_cterm sign (sP,propT))); + +(*Read an axiom for a new theory*) +fun read_ax sign (a, sP) : string*thm = + let val prop = prepare_axiom sign sP + in (a, Thm{sign=sign, hyps=[], maxidx= maxidx_of_term prop, prop= prop}) + end + handle ERROR => + error("extend_theory: The error above occurred in axiom " ^ a); + +fun mk_axioms sign axpairs = + Symtab.st_of_alist(map (read_ax sign) axpairs, Symtab.null) + handle Symtab.DUPLICATE(a) => error("Two axioms named " ^ a); + +(*Extension of a theory with given classes, types, constants and syntax. + Reads the axioms from strings: axpairs have the form (axname, axiom). *) +fun extend_theory thy thyname ext axpairs = + let val sign = Sign.extend (sign_of thy) thyname ext; + val axioms= mk_axioms sign axpairs + in Extend{sign=sign, axioms= axioms, thy = thy} end; + +(*The union of two theories*) +fun merge_theories (thy1,thy2) = + Merge{sign = Sign.merge(sign_of thy1, sign_of thy2), + thy1 = thy1, thy2 = thy2}; + + +(*** Meta simp sets ***) + +type rrule = {thm:thm, lhs:term}; +datatype meta_simpset = + Mss of {net:rrule Net.net, congs:(string * rrule)list, primes:string, + prems: thm list, mk_rews: thm -> thm list}; + +(*A "mss" contains data needed during conversion: + net: discrimination net of rewrite rules + congs: association list of congruence rules + primes: offset for generating unique new names + for rewriting under lambda abstractions + mk_rews: used when local assumptions are added +*) + +val empty_mss = Mss{net= Net.empty, congs= [], primes="", prems= [], + mk_rews = K[]}; + +exception SIMPLIFIER of string * thm; + +fun prtm a sg t = (writeln a; writeln(Sign.string_of_term sg t)); + +(*simple test for looping rewrite*) +fun loops sign prems (lhs,rhs) = + null(prems) andalso + Pattern.eta_matches (#tsig(Sign.rep_sg sign)) (lhs,rhs); + +fun mk_rrule (thm as Thm{hyps,sign,prop,maxidx,...}) = + let val prems = Logic.strip_imp_prems prop + val concl = Pattern.eta_contract (Logic.strip_imp_concl prop) + val (lhs,rhs) = Logic.dest_equals concl handle TERM _ => + raise SIMPLIFIER("Rewrite rule not a meta-equality",thm) + in if loops sign prems (lhs,rhs) + then (prtm "Warning: ignoring looping rewrite rule" sign prop; None) + else Some{thm=thm,lhs=lhs} + end; + +fun add_simp(mss as Mss{net,congs,primes,prems,mk_rews}, + thm as Thm{sign,prop,...}) = + let fun eq({thm=Thm{prop=p1,...},...}:rrule, + {thm=Thm{prop=p2,...},...}:rrule) = p1 aconv p2 + in case mk_rrule thm of + None => mss + | Some(rrule as {lhs,...}) => + Mss{net= (Net.insert_term((lhs,rrule),net,eq) + handle Net.INSERT => + (prtm "Warning: ignoring duplicate rewrite rule" sign prop; + net)), + congs=congs, primes=primes, prems=prems,mk_rews=mk_rews} + end; + +val add_simps = foldl add_simp; + +fun mss_of thms = add_simps(empty_mss,thms); + +fun add_cong(Mss{net,congs,primes,prems,mk_rews},thm) = + let val (lhs,_) = Logic.dest_equals(concl_of thm) handle TERM _ => + raise SIMPLIFIER("Congruence not a meta-equality",thm) + val lhs = Pattern.eta_contract lhs + val (a,_) = dest_Const (head_of lhs) handle TERM _ => + raise SIMPLIFIER("Congruence must start with a constant",thm) + in Mss{net=net, congs=(a,{lhs=lhs,thm=thm})::congs, primes=primes, + prems=prems, mk_rews=mk_rews} + end; + +val (op add_congs) = foldl add_cong; + +fun add_prems(Mss{net,congs,primes,prems,mk_rews},thms) = + Mss{net=net, congs=congs, primes=primes, prems=thms@prems, mk_rews=mk_rews}; + +fun prems_of_mss(Mss{prems,...}) = prems; + +fun set_mk_rews(Mss{net,congs,primes,prems,...},mk_rews) = + Mss{net=net, congs=congs, primes=primes, prems=prems, mk_rews=mk_rews}; +fun mk_rews_of_mss(Mss{mk_rews,...}) = mk_rews; + + +(*** Meta-level rewriting + uses conversions, omitting proofs for efficiency. See + L C Paulson, A higher-order implementation of rewriting, + Science of Computer Programming 3 (1983), pages 119-149. ***) + +type prover = meta_simpset -> thm -> thm option; +type termrec = (Sign.sg * term list) * term; +type conv = meta_simpset -> termrec -> termrec; + +val trace_simp = ref false; + +fun trace_term a sg t = if !trace_simp then prtm a sg t else (); + +fun trace_thm a (Thm{sign,prop,...}) = trace_term a sign prop; + +fun check_conv(thm as Thm{sign,hyps,prop,...}, prop0) = + let fun err() = (trace_term "Proved wrong thm" sign prop; + error "Check your prover") + val (lhs0,_) = Logic.dest_equals(Logic.strip_imp_concl prop0) + in case prop of + Const("==",_) $ lhs $ rhs => + if (lhs = lhs0) orelse + (lhs aconv (Envir.norm_term (Envir.empty 0) lhs0)) + then (trace_thm "SUCCEEDED" thm; ((sign,hyps),rhs)) + else err() + | _ => err() + end; + +(*Conversion to apply the meta simpset to a term*) +fun rewritec prover (mss as Mss{net,...}) (sghyt as (sgt,hypst),t) = + let val t = Pattern.eta_contract t + fun rew {thm as Thm{sign,hyps,maxidx,prop,...}, lhs} = + let val sign' = Sign.merge(sgt,sign) + val tsig = #tsig(Sign.rep_sg sign') + val insts = Pattern.match tsig (lhs,t) + val prop' = subst_vars insts prop; + val hyps' = hyps union hypst; + val thm' = Thm{sign=sign', hyps=hyps', prop=prop', maxidx=maxidx} + in if nprems_of thm' = 0 + then let val (_,rhs) = Logic.dest_equals prop' + in trace_thm "Rewriting:" thm'; Some((sign',hyps'),rhs) end + else (trace_thm "Trying to rewrite:" thm'; + case prover mss thm' of + None => (trace_thm "FAILED" thm'; None) + | Some(thm2) => Some(check_conv(thm2,prop'))) + end + + fun rewl [] = None + | rewl (rrule::rrules) = + let val opt = rew rrule handle Pattern.MATCH => None + in case opt of None => rewl rrules | some => some end; + + in case t of + Abs(_,_,body) $ u => Some(sghyt,subst_bounds([u], body)) + | _ => rewl (Net.match_term net t) + end; + +(*Conversion to apply a congruence rule to a term*) +fun congc prover {thm=cong,lhs=lhs} (sghyt as (sgt,hypst),t) = + let val Thm{sign,hyps,maxidx,prop,...} = cong + val sign' = Sign.merge(sgt,sign) + val tsig = #tsig(Sign.rep_sg sign') + val insts = Pattern.match tsig (lhs,t) handle Pattern.MATCH => + error("Congruence rule did not match") + val prop' = subst_vars insts prop; + val thm' = Thm{sign=sign', hyps=hyps union hypst, + prop=prop', maxidx=maxidx} + val unit = trace_thm "Applying congruence rule" thm'; + + in case prover thm' of + None => error("Failed congruence proof!") + | Some(thm2) => check_conv(thm2,prop') + end; + + +fun bottomc prover = + let fun botc mss trec = let val trec1 = subc mss trec + in case rewritec prover mss trec1 of + None => trec1 + | Some(trec2) => botc mss trec2 + end + + and subc (mss as Mss{net,congs,primes,prems,mk_rews}) + (trec as (sghy,t)) = + (case t of + Abs(a,T,t) => + let val v = Free(".subc." ^ primes,T) + val mss' = Mss{net=net, congs=congs, primes=primes^"'", + prems=prems,mk_rews=mk_rews} + val (sghy',t') = botc mss' (sghy,subst_bounds([v],t)) + in (sghy', Abs(a, T, abstract_over(v,t'))) end + | t$u => (case t of + Const("==>",_)$s => impc(sghy,s,u,mss) + | Abs(_,_,body) => subc mss (sghy,subst_bounds([u], body)) + | _ => + let fun appc() = let val (sghy1,t1) = botc mss (sghy,t) + val (sghy2,u1) = botc mss (sghy1,u) + in (sghy2,t1$u1) end + val (h,ts) = strip_comb t + in case h of + Const(a,_) => + (case assoc(congs,a) of + None => appc() + | Some(cong) => congc (prover mss) cong trec) + | _ => appc() + end) + | _ => trec) + + and impc(sghy,s,u,mss as Mss{mk_rews,...}) = + let val (sghy1 as (sg1,hyps1),s') = botc mss (sghy,s) + val (rthms,mss) = + if maxidx_of_term s' <> ~1 then ([],mss) + else let val thm = Thm{sign=sg1,hyps=[s'],prop=s',maxidx= ~1} + in (mk_rews thm, add_prems(mss,[thm])) end + val unit = seq (trace_thm "Adding rewrite rule:") rthms + val mss' = add_simps(mss,rthms) + val ((sg2,hyps2),u') = botc mss' (sghy1,u) + in ((sg2,hyps2\s'), Logic.mk_implies(s',u')) end + + in botc end; + + +(*** Meta-rewriting: rewrites t to u and returns the theorem t==u ***) +(* Parameters: + mss: contains equality theorems of the form [|p1,...|] ==> t==u + prover: how to solve premises in conditional rewrites and congruences +*) + +(*** FIXME: check that #primes(mss) does not "occur" in ct alread ***) +fun rewrite_cterm mss prover ct = + let val {sign, t, T, maxidx} = Sign.rep_cterm ct; + val ((sign',hyps),u) = bottomc prover mss ((sign,[]),t); + val prop = Logic.mk_equals(t,u) + in Thm{sign= sign', hyps= hyps, maxidx= maxidx_of_term prop, prop= prop} + end + +end;