(* 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 del_simps: meta_simpset * thm list -> meta_simpset
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 * indexname list * string) 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:
bool*bool -> 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*) = (* FIXME *)
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;
(*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 newprop =
Envir.norm_term (Envir.empty 0)
(subst_atomic tpairs
(Type.inst_term_tvars(#tsig(Sign.rep_sg newsign),vTs) prop))
val newth = Thm{sign= newsign, hyps= hyps,
maxidx= maxidx_of_term newprop, prop= newprop}
in if not(instl_ok(map #1 tpairs))
then raise THM("instantiate: variables not distinct", 0, [th])
else if not(null(findrep(map #1 vTs)))
then raise THM("instantiate: type variables not distinct", 0, [th])
else (*Check types of Vars for agreement*)
case findrep (map (#1 o dest_Var) (term_vars newprop)) of
ix::_ => raise THM("instantiate: conflicting types for variable " ^
Syntax.string_of_vname ix ^ "\n", 0, [newth])
| [] =>
case findrep (map #1 (term_tvars newprop)) of
ix::_ => raise THM
("instantiate: conflicting sorts for type variable " ^
Syntax.string_of_vname ix ^ "\n", 0, [newth])
| [] => newth
end
handle TERM _ =>
raise THM("instantiate: incompatible signatures",0,[th])
| TYPE _ => raise THM("instantiate: type conflict", 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));
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;
(*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;
local
fun eq({thm=Thm{prop=p1,...},...}:rrule,
{thm=Thm{prop=p2,...},...}:rrule) = p1 aconv p2
in
fun add_simp(mss as Mss{net,congs,primes,prems,mk_rews},
thm as Thm{sign,prop,...}) =
case mk_rrule thm of
None => mss
| Some(rrule as {lhs,...}) =>
(trace_thm "Adding rewrite rule:" thm;
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});
fun del_simp(mss as Mss{net,congs,primes,prems,mk_rews},
thm as Thm{sign,prop,...}) =
case mk_rrule thm of
None => mss
| Some(rrule as {lhs,...}) =>
Mss{net= (Net.delete_term((lhs,rrule),net,eq)
handle Net.INSERT =>
(prtm "Warning: rewrite rule not in simpset" sign prop;
net)),
congs=congs, primes=primes, prems=prems,mk_rews=mk_rews}
end;
val add_simps = foldl add_simp;
val del_simps = foldl del_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;
fun check_conv(thm as Thm{hyps,prop,...}, prop0) =
let fun err() = (trace_thm "Proved wrong thm (Check subgoaler?)" thm; None)
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; Some(hyps,rhs))
else err()
| _ => err()
end;
(*Conversion to apply the meta simpset to a term*)
fun rewritec (prover,signt) (mss as Mss{net,...}) (hypst,t) =
let fun rew (t, {thm as Thm{sign,hyps,maxidx,prop,...}, lhs}) =
let val unit = if Sign.subsig(sign,signt) then ()
else (writeln"Warning: rewrite rule from different theory";
raise Pattern.MATCH)
val insts = Pattern.match (#tsig(Sign.rep_sg signt)) (lhs,t)
val prop' = subst_vars insts prop;
val hyps' = hyps union hypst;
val thm' = Thm{sign=signt, 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(hyps',rhs) end
else (trace_thm "Trying to rewrite:" thm';
case prover mss thm' of
None => (trace_thm "FAILED" thm'; None)
| Some(thm2) => check_conv(thm2,prop'))
end
fun rews t =
let fun rews1 [] = None
| rews1 (rrule::rrules) =
let val opt = rew(t,rrule) handle Pattern.MATCH => None
in case opt of None => rews1 rrules | some => some end;
in rews1 end
fun eta_rews([]) = None
| eta_rews(rrules) = rews (Pattern.eta_contract t) rrules
in case t of
Abs(_,_,body) $ u => Some(hypst,subst_bounds([u], body))
| _ => eta_rews(Net.match_term net t)
end;
(*Conversion to apply a congruence rule to a term*)
fun congc (prover,signt) {thm=cong,lhs=lhs} (hypst,t) =
let val Thm{sign,hyps,maxidx,prop,...} = cong
val unit = if Sign.subsig(sign,signt) then ()
else error("Congruence rule from different theory")
val tsig = #tsig(Sign.rep_sg signt)
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=signt, hyps=hyps union hypst,
prop=prop', maxidx=maxidx}
val unit = trace_thm "Applying congruence rule" thm';
fun err() = error("Failed congruence proof!")
in case prover thm' of
None => err()
| Some(thm2) => (case check_conv(thm2,prop') of
None => err() | Some(x) => x)
end;
fun bottomc ((simprem,useprem),prover,sign) =
let fun botc mss trec = let val trec1 = subc mss trec
in case rewritec (prover,sign) mss trec1 of
None => trec1
| Some(trec2) => botc mss trec2
end
and subc (mss as Mss{net,congs,primes,prems,mk_rews})
(trec as (hyps,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 (hyps',t') = botc mss' (hyps,subst_bounds([v],t))
in (hyps', Abs(a, T, abstract_over(v,t'))) end
| t$u => (case t of
Const("==>",_)$s => impc(hyps,s,u,mss)
| Abs(_,_,body) => subc mss (hyps,subst_bounds([u], body))
| _ =>
let fun appc() = let val (hyps1,t1) = botc mss (hyps,t)
val (hyps2,u1) = botc mss (hyps1,u)
in (hyps2,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,sign) cong trec)
| _ => appc()
end)
| _ => trec)
and impc(hyps,s,u,mss as Mss{mk_rews,...}) =
let val (hyps1,s') = if simprem then botc mss (hyps,s) else (hyps,s)
val mss' =
if not useprem orelse maxidx_of_term s' <> ~1 then mss
else let val thm = Thm{sign=sign,hyps=[s'],prop=s',maxidx= ~1}
in add_simps(add_prems(mss,[thm]), mk_rews thm) end
val (hyps2,u') = botc mss' (hyps1,u)
val hyps2' = if s' mem hyps1 then hyps2 else hyps2\s'
in (hyps2', Logic.mk_implies(s',u')) end
in botc end;
(*** Meta-rewriting: rewrites t to u and returns the theorem t==u ***)
(* Parameters:
mode = (simplify A, use A in simplifying B) when simplifying A ==> B
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 mode mss prover ct =
let val {sign, t, T, maxidx} = Sign.rep_cterm ct;
val (hyps,u) = bottomc (mode,prover,sign) mss ([],t);
val prop = Logic.mk_equals(t,u)
in Thm{sign= sign, hyps= hyps, maxidx= maxidx_of_term prop, prop= prop}
end
end;