--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/eqsubst.ML Sat Feb 28 14:02:12 2009 +0100
@@ -0,0 +1,575 @@
+(* Title: Tools/eqsubst.ML
+ Author: Lucas Dixon, University of Edinburgh
+
+A proof method to perform a substiution using an equation.
+*)
+
+signature EQSUBST =
+sig
+ (* a type abbreviation for match information *)
+ type match =
+ ((indexname * (sort * typ)) list (* type instantiations *)
+ * (indexname * (typ * term)) list) (* term instantiations *)
+ * (string * typ) list (* fake named type abs env *)
+ * (string * typ) list (* type abs env *)
+ * term (* outer term *)
+
+ type searchinfo =
+ theory
+ * int (* maxidx *)
+ * Zipper.T (* focusterm to search under *)
+
+ exception eqsubst_occL_exp of
+ string * int list * Thm.thm list * int * Thm.thm
+
+ (* low level substitution functions *)
+ val apply_subst_in_asm :
+ int ->
+ Thm.thm ->
+ Thm.thm ->
+ (Thm.cterm list * int * 'a * Thm.thm) * match -> Thm.thm Seq.seq
+ val apply_subst_in_concl :
+ int ->
+ Thm.thm ->
+ Thm.cterm list * Thm.thm ->
+ Thm.thm -> match -> Thm.thm Seq.seq
+
+ (* matching/unification within zippers *)
+ val clean_match_z :
+ Context.theory -> Term.term -> Zipper.T -> match option
+ val clean_unify_z :
+ Context.theory -> int -> Term.term -> Zipper.T -> match Seq.seq
+
+ (* skipping things in seq seq's *)
+
+ (* skipping non-empty sub-sequences but when we reach the end
+ of the seq, remembering how much we have left to skip. *)
+ datatype 'a skipseq = SkipMore of int
+ | SkipSeq of 'a Seq.seq Seq.seq;
+
+ val skip_first_asm_occs_search :
+ ('a -> 'b -> 'c Seq.seq Seq.seq) ->
+ 'a -> int -> 'b -> 'c skipseq
+ val skip_first_occs_search :
+ int -> ('a -> 'b -> 'c Seq.seq Seq.seq) -> 'a -> 'b -> 'c Seq.seq
+ val skipto_skipseq : int -> 'a Seq.seq Seq.seq -> 'a skipseq
+
+ (* tactics *)
+ val eqsubst_asm_tac :
+ Proof.context ->
+ int list -> Thm.thm list -> int -> Thm.thm -> Thm.thm Seq.seq
+ val eqsubst_asm_tac' :
+ Proof.context ->
+ (searchinfo -> int -> Term.term -> match skipseq) ->
+ int -> Thm.thm -> int -> Thm.thm -> Thm.thm Seq.seq
+ val eqsubst_tac :
+ Proof.context ->
+ int list -> (* list of occurences to rewrite, use [0] for any *)
+ Thm.thm list -> int -> Thm.thm -> Thm.thm Seq.seq
+ val eqsubst_tac' :
+ Proof.context -> (* proof context *)
+ (searchinfo -> Term.term -> match Seq.seq) (* search function *)
+ -> Thm.thm (* equation theorem to rewrite with *)
+ -> int (* subgoal number in goal theorem *)
+ -> Thm.thm (* goal theorem *)
+ -> Thm.thm Seq.seq (* rewritten goal theorem *)
+
+
+ val fakefree_badbounds :
+ (string * Term.typ) list ->
+ Term.term ->
+ (string * Term.typ) list * (string * Term.typ) list * Term.term
+
+ val mk_foo_match :
+ (Term.term -> Term.term) ->
+ ('a * Term.typ) list -> Term.term -> Term.term
+
+ (* preparing substitution *)
+ val prep_meta_eq : Proof.context -> Thm.thm -> Thm.thm list
+ val prep_concl_subst :
+ int -> Thm.thm -> (Thm.cterm list * Thm.thm) * searchinfo
+ val prep_subst_in_asm :
+ int -> Thm.thm -> int ->
+ (Thm.cterm list * int * int * Thm.thm) * searchinfo
+ val prep_subst_in_asms :
+ int -> Thm.thm ->
+ ((Thm.cterm list * int * int * Thm.thm) * searchinfo) list
+ val prep_zipper_match :
+ Zipper.T -> Term.term * ((string * Term.typ) list * (string * Term.typ) list * Term.term)
+
+ (* search for substitutions *)
+ val valid_match_start : Zipper.T -> bool
+ val search_lr_all : Zipper.T -> Zipper.T Seq.seq
+ val search_lr_valid : (Zipper.T -> bool) -> Zipper.T -> Zipper.T Seq.seq
+ val searchf_lr_unify_all :
+ searchinfo -> Term.term -> match Seq.seq Seq.seq
+ val searchf_lr_unify_valid :
+ searchinfo -> Term.term -> match Seq.seq Seq.seq
+ val searchf_bt_unify_valid :
+ searchinfo -> Term.term -> match Seq.seq Seq.seq
+
+ (* syntax tools *)
+ val ith_syntax : Args.T list -> int list * Args.T list
+ val options_syntax : Args.T list -> bool * Args.T list
+
+ (* Isar level hooks *)
+ val eqsubst_asm_meth : Proof.context -> int list -> Thm.thm list -> Proof.method
+ val eqsubst_meth : Proof.context -> int list -> Thm.thm list -> Proof.method
+ val subst_meth : Method.src -> Proof.context -> Proof.method
+ val setup : theory -> theory
+
+end;
+
+structure EqSubst
+: EQSUBST
+= struct
+
+structure Z = Zipper;
+
+(* changes object "=" to meta "==" which prepares a given rewrite rule *)
+fun prep_meta_eq ctxt =
+ let val (_, {mk_rews = {mk, ...}, ...}) = Simplifier.rep_ss (Simplifier.local_simpset_of ctxt)
+ in mk #> map Drule.zero_var_indexes end;
+
+
+ (* a type abriviation for match information *)
+ type match =
+ ((indexname * (sort * typ)) list (* type instantiations *)
+ * (indexname * (typ * term)) list) (* term instantiations *)
+ * (string * typ) list (* fake named type abs env *)
+ * (string * typ) list (* type abs env *)
+ * term (* outer term *)
+
+ type searchinfo =
+ theory
+ * int (* maxidx *)
+ * Zipper.T (* focusterm to search under *)
+
+
+(* skipping non-empty sub-sequences but when we reach the end
+ of the seq, remembering how much we have left to skip. *)
+datatype 'a skipseq = SkipMore of int
+ | SkipSeq of 'a Seq.seq Seq.seq;
+(* given a seqseq, skip the first m non-empty seq's, note deficit *)
+fun skipto_skipseq m s =
+ let
+ fun skip_occs n sq =
+ case Seq.pull sq of
+ NONE => SkipMore n
+ | SOME (h,t) =>
+ (case Seq.pull h of NONE => skip_occs n t
+ | SOME _ => if n <= 1 then SkipSeq (Seq.cons h t)
+ else skip_occs (n - 1) t)
+ in (skip_occs m s) end;
+
+(* note: outerterm is the taget with the match replaced by a bound
+ variable : ie: "P lhs" beocmes "%x. P x"
+ insts is the types of instantiations of vars in lhs
+ and typinsts is the type instantiations of types in the lhs
+ Note: Final rule is the rule lifted into the ontext of the
+ taget thm. *)
+fun mk_foo_match mkuptermfunc Ts t =
+ let
+ val ty = Term.type_of t
+ val bigtype = (rev (map snd Ts)) ---> ty
+ fun mk_foo 0 t = t
+ | mk_foo i t = mk_foo (i - 1) (t $ (Bound (i - 1)))
+ val num_of_bnds = (length Ts)
+ (* foo_term = "fooabs y0 ... yn" where y's are local bounds *)
+ val foo_term = mk_foo num_of_bnds (Bound num_of_bnds)
+ in Abs("fooabs", bigtype, mkuptermfunc foo_term) end;
+
+(* T is outer bound vars, n is number of locally bound vars *)
+(* THINK: is order of Ts correct...? or reversed? *)
+fun fakefree_badbounds Ts t =
+ let val (FakeTs,Ts,newnames) =
+ List.foldr (fn ((n,ty),(FakeTs,Ts,usednames)) =>
+ let val newname = Name.variant usednames n
+ in ((RWTools.mk_fake_bound_name newname,ty)::FakeTs,
+ (newname,ty)::Ts,
+ newname::usednames) end)
+ ([],[],[])
+ Ts
+ in (FakeTs, Ts, Term.subst_bounds (map Free FakeTs, t)) end;
+
+(* before matching we need to fake the bound vars that are missing an
+abstraction. In this function we additionally construct the
+abstraction environment, and an outer context term (with the focus
+abstracted out) for use in rewriting with RWInst.rw *)
+fun prep_zipper_match z =
+ let
+ val t = Z.trm z
+ val c = Z.ctxt z
+ val Ts = Z.C.nty_ctxt c
+ val (FakeTs', Ts', t') = fakefree_badbounds Ts t
+ val absterm = mk_foo_match (Z.C.apply c) Ts' t'
+ in
+ (t', (FakeTs', Ts', absterm))
+ end;
+
+(* Matching and Unification with exception handled *)
+fun clean_match thy (a as (pat, t)) =
+ let val (tyenv, tenv) = Pattern.match thy a (Vartab.empty, Vartab.empty)
+ in SOME (Vartab.dest tyenv, Vartab.dest tenv)
+ end handle Pattern.MATCH => NONE;
+
+(* given theory, max var index, pat, tgt; returns Seq of instantiations *)
+fun clean_unify thry ix (a as (pat, tgt)) =
+ let
+ (* type info will be re-derived, maybe this can be cached
+ for efficiency? *)
+ val pat_ty = Term.type_of pat;
+ val tgt_ty = Term.type_of tgt;
+ (* is it OK to ignore the type instantiation info?
+ or should I be using it? *)
+ val typs_unify =
+ SOME (Sign.typ_unify thry (pat_ty, tgt_ty) (Vartab.empty, ix))
+ handle Type.TUNIFY => NONE;
+ in
+ case typs_unify of
+ SOME (typinsttab, ix2) =>
+ let
+ (* is it right to throw away the flexes?
+ or should I be using them somehow? *)
+ fun mk_insts env =
+ (Vartab.dest (Envir.type_env env),
+ Envir.alist_of env);
+ val initenv = Envir.Envir {asol = Vartab.empty,
+ iTs = typinsttab, maxidx = ix2};
+ val useq = Unify.smash_unifiers thry [a] initenv
+ handle UnequalLengths => Seq.empty
+ | Term.TERM _ => Seq.empty;
+ fun clean_unify' useq () =
+ (case (Seq.pull useq) of
+ NONE => NONE
+ | SOME (h,t) => SOME (mk_insts h, Seq.make (clean_unify' t)))
+ handle UnequalLengths => NONE
+ | Term.TERM _ => NONE
+ in
+ (Seq.make (clean_unify' useq))
+ end
+ | NONE => Seq.empty
+ end;
+
+(* Matching and Unification for zippers *)
+(* Note: Ts is a modified version of the original names of the outer
+bound variables. New names have been introduced to make sure they are
+unique w.r.t all names in the term and each other. usednames' is
+oldnames + new names. *)
+fun clean_match_z thy pat z =
+ let val (t, (FakeTs,Ts,absterm)) = prep_zipper_match z in
+ case clean_match thy (pat, t) of
+ NONE => NONE
+ | SOME insts => SOME (insts, FakeTs, Ts, absterm) end;
+(* ix = max var index *)
+fun clean_unify_z sgn ix pat z =
+ let val (t, (FakeTs, Ts,absterm)) = prep_zipper_match z in
+ Seq.map (fn insts => (insts, FakeTs, Ts, absterm))
+ (clean_unify sgn ix (t, pat)) end;
+
+
+(* FOR DEBUGGING...
+type trace_subst_errT = int (* subgoal *)
+ * thm (* thm with all goals *)
+ * (Thm.cterm list (* certified free var placeholders for vars *)
+ * thm) (* trivial thm of goal concl *)
+ (* possible matches/unifiers *)
+ * thm (* rule *)
+ * (((indexname * typ) list (* type instantiations *)
+ * (indexname * term) list ) (* term instantiations *)
+ * (string * typ) list (* Type abs env *)
+ * term) (* outer term *);
+
+val trace_subst_err = (ref NONE : trace_subst_errT option ref);
+val trace_subst_search = ref false;
+exception trace_subst_exp of trace_subst_errT;
+*)
+
+
+fun bot_left_leaf_of (l $ r) = bot_left_leaf_of l
+ | bot_left_leaf_of (Abs(s,ty,t)) = bot_left_leaf_of t
+ | bot_left_leaf_of x = x;
+
+(* Avoid considering replacing terms which have a var at the head as
+ they always succeed trivially, and uninterestingly. *)
+fun valid_match_start z =
+ (case bot_left_leaf_of (Z.trm z) of
+ Var _ => false
+ | _ => true);
+
+(* search from top, left to right, then down *)
+val search_lr_all = ZipperSearch.all_bl_ur;
+
+(* search from top, left to right, then down *)
+fun search_lr_valid validf =
+ let
+ fun sf_valid_td_lr z =
+ let val here = if validf z then [Z.Here z] else [] in
+ case Z.trm z
+ of _ $ _ => [Z.LookIn (Z.move_down_left z)]
+ @ here
+ @ [Z.LookIn (Z.move_down_right z)]
+ | Abs _ => here @ [Z.LookIn (Z.move_down_abs z)]
+ | _ => here
+ end;
+ in Z.lzy_search sf_valid_td_lr end;
+
+(* search from bottom to top, left to right *)
+
+fun search_bt_valid validf =
+ let
+ fun sf_valid_td_lr z =
+ let val here = if validf z then [Z.Here z] else [] in
+ case Z.trm z
+ of _ $ _ => [Z.LookIn (Z.move_down_left z),
+ Z.LookIn (Z.move_down_right z)] @ here
+ | Abs _ => [Z.LookIn (Z.move_down_abs z)] @ here
+ | _ => here
+ end;
+ in Z.lzy_search sf_valid_td_lr end;
+
+fun searchf_unify_gen f (sgn, maxidx, z) lhs =
+ Seq.map (clean_unify_z sgn maxidx lhs)
+ (Z.limit_apply f z);
+
+(* search all unifications *)
+val searchf_lr_unify_all =
+ searchf_unify_gen search_lr_all;
+
+(* search only for 'valid' unifiers (non abs subterms and non vars) *)
+val searchf_lr_unify_valid =
+ searchf_unify_gen (search_lr_valid valid_match_start);
+
+val searchf_bt_unify_valid =
+ searchf_unify_gen (search_bt_valid valid_match_start);
+
+(* apply a substitution in the conclusion of the theorem th *)
+(* cfvs are certified free var placeholders for goal params *)
+(* conclthm is a theorem of for just the conclusion *)
+(* m is instantiation/match information *)
+(* rule is the equation for substitution *)
+fun apply_subst_in_concl i th (cfvs, conclthm) rule m =
+ (RWInst.rw m rule conclthm)
+ |> IsaND.unfix_frees cfvs
+ |> RWInst.beta_eta_contract
+ |> (fn r => Tactic.rtac r i th);
+
+(* substitute within the conclusion of goal i of gth, using a meta
+equation rule. Note that we assume rule has var indicies zero'd *)
+fun prep_concl_subst i gth =
+ let
+ val th = Thm.incr_indexes 1 gth;
+ val tgt_term = Thm.prop_of th;
+
+ val sgn = Thm.theory_of_thm th;
+ val ctermify = Thm.cterm_of sgn;
+ val trivify = Thm.trivial o ctermify;
+
+ val (fixedbody, fvs) = IsaND.fix_alls_term i tgt_term;
+ val cfvs = rev (map ctermify fvs);
+
+ val conclterm = Logic.strip_imp_concl fixedbody;
+ val conclthm = trivify conclterm;
+ val maxidx = Thm.maxidx_of th;
+ val ft = ((Z.move_down_right (* ==> *)
+ o Z.move_down_left (* Trueprop *)
+ o Z.mktop
+ o Thm.prop_of) conclthm)
+ in
+ ((cfvs, conclthm), (sgn, maxidx, ft))
+ end;
+
+(* substitute using an object or meta level equality *)
+fun eqsubst_tac' ctxt searchf instepthm i th =
+ let
+ val (cvfsconclthm, searchinfo) = prep_concl_subst i th;
+ val stepthms = Seq.of_list (prep_meta_eq ctxt instepthm);
+ fun rewrite_with_thm r =
+ let val (lhs,_) = Logic.dest_equals (Thm.concl_of r);
+ in searchf searchinfo lhs
+ |> Seq.maps (apply_subst_in_concl i th cvfsconclthm r) end;
+ in stepthms |> Seq.maps rewrite_with_thm end;
+
+
+(* distinct subgoals *)
+fun distinct_subgoals th =
+ the_default th (SINGLE distinct_subgoals_tac th);
+
+(* General substitution of multiple occurances using one of
+ the given theorems*)
+
+
+exception eqsubst_occL_exp of
+ string * (int list) * (thm list) * int * thm;
+fun skip_first_occs_search occ srchf sinfo lhs =
+ case (skipto_skipseq occ (srchf sinfo lhs)) of
+ SkipMore _ => Seq.empty
+ | SkipSeq ss => Seq.flat ss;
+
+(* The occL is a list of integers indicating which occurence
+w.r.t. the search order, to rewrite. Backtracking will also find later
+occurences, but all earlier ones are skipped. Thus you can use [0] to
+just find all rewrites. *)
+
+fun eqsubst_tac ctxt occL thms i th =
+ let val nprems = Thm.nprems_of th in
+ if nprems < i then Seq.empty else
+ let val thmseq = (Seq.of_list thms)
+ fun apply_occ occ th =
+ thmseq |> Seq.maps
+ (fn r => eqsubst_tac'
+ ctxt
+ (skip_first_occs_search
+ occ searchf_lr_unify_valid) r
+ (i + ((Thm.nprems_of th) - nprems))
+ th);
+ val sortedoccL =
+ Library.sort (Library.rev_order o Library.int_ord) occL;
+ in
+ Seq.map distinct_subgoals (Seq.EVERY (map apply_occ sortedoccL) th)
+ end
+ end
+ handle THM _ => raise eqsubst_occL_exp ("THM",occL,thms,i,th);
+
+
+(* inthms are the given arguments in Isar, and treated as eqstep with
+ the first one, then the second etc *)
+fun eqsubst_meth ctxt occL inthms =
+ Method.SIMPLE_METHOD' (eqsubst_tac ctxt occL inthms);
+
+(* apply a substitution inside assumption j, keeps asm in the same place *)
+fun apply_subst_in_asm i th rule ((cfvs, j, ngoalprems, pth),m) =
+ let
+ val th2 = Thm.rotate_rule (j - 1) i th; (* put premice first *)
+ val preelimrule =
+ (RWInst.rw m rule pth)
+ |> (Seq.hd o prune_params_tac)
+ |> Thm.permute_prems 0 ~1 (* put old asm first *)
+ |> IsaND.unfix_frees cfvs (* unfix any global params *)
+ |> RWInst.beta_eta_contract; (* normal form *)
+ (* val elimrule =
+ preelimrule
+ |> Tactic.make_elim (* make into elim rule *)
+ |> Thm.lift_rule (th2, i); (* lift into context *)
+ *)
+ in
+ (* ~j because new asm starts at back, thus we subtract 1 *)
+ Seq.map (Thm.rotate_rule (~j) ((Thm.nprems_of rule) + i))
+ (Tactic.dtac preelimrule i th2)
+
+ (* (Thm.bicompose
+ false (* use unification *)
+ (true, (* elim resolution *)
+ elimrule, (2 + (Thm.nprems_of rule)) - ngoalprems)
+ i th2) *)
+ end;
+
+
+(* prepare to substitute within the j'th premise of subgoal i of gth,
+using a meta-level equation. Note that we assume rule has var indicies
+zero'd. Note that we also assume that premt is the j'th premice of
+subgoal i of gth. Note the repetition of work done for each
+assumption, i.e. this can be made more efficient for search over
+multiple assumptions. *)
+fun prep_subst_in_asm i gth j =
+ let
+ val th = Thm.incr_indexes 1 gth;
+ val tgt_term = Thm.prop_of th;
+
+ val sgn = Thm.theory_of_thm th;
+ val ctermify = Thm.cterm_of sgn;
+ val trivify = Thm.trivial o ctermify;
+
+ val (fixedbody, fvs) = IsaND.fix_alls_term i tgt_term;
+ val cfvs = rev (map ctermify fvs);
+
+ val asmt = nth (Logic.strip_imp_prems fixedbody) (j - 1);
+ val asm_nprems = length (Logic.strip_imp_prems asmt);
+
+ val pth = trivify asmt;
+ val maxidx = Thm.maxidx_of th;
+
+ val ft = ((Z.move_down_right (* trueprop *)
+ o Z.mktop
+ o Thm.prop_of) pth)
+ in ((cfvs, j, asm_nprems, pth), (sgn, maxidx, ft)) end;
+
+(* prepare subst in every possible assumption *)
+fun prep_subst_in_asms i gth =
+ map (prep_subst_in_asm i gth)
+ ((fn l => Library.upto (1, length l))
+ (Logic.prems_of_goal (Thm.prop_of gth) i));
+
+
+(* substitute in an assumption using an object or meta level equality *)
+fun eqsubst_asm_tac' ctxt searchf skipocc instepthm i th =
+ let
+ val asmpreps = prep_subst_in_asms i th;
+ val stepthms = Seq.of_list (prep_meta_eq ctxt instepthm);
+ fun rewrite_with_thm r =
+ let val (lhs,_) = Logic.dest_equals (Thm.concl_of r)
+ fun occ_search occ [] = Seq.empty
+ | occ_search occ ((asminfo, searchinfo)::moreasms) =
+ (case searchf searchinfo occ lhs of
+ SkipMore i => occ_search i moreasms
+ | SkipSeq ss =>
+ Seq.append (Seq.map (Library.pair asminfo) (Seq.flat ss))
+ (occ_search 1 moreasms))
+ (* find later substs also *)
+ in
+ occ_search skipocc asmpreps |> Seq.maps (apply_subst_in_asm i th r)
+ end;
+ in stepthms |> Seq.maps rewrite_with_thm end;
+
+
+fun skip_first_asm_occs_search searchf sinfo occ lhs =
+ skipto_skipseq occ (searchf sinfo lhs);
+
+fun eqsubst_asm_tac ctxt occL thms i th =
+ let val nprems = Thm.nprems_of th
+ in
+ if nprems < i then Seq.empty else
+ let val thmseq = (Seq.of_list thms)
+ fun apply_occ occK th =
+ thmseq |> Seq.maps
+ (fn r =>
+ eqsubst_asm_tac' ctxt (skip_first_asm_occs_search
+ searchf_lr_unify_valid) occK r
+ (i + ((Thm.nprems_of th) - nprems))
+ th);
+ val sortedoccs =
+ Library.sort (Library.rev_order o Library.int_ord) occL
+ in
+ Seq.map distinct_subgoals
+ (Seq.EVERY (map apply_occ sortedoccs) th)
+ end
+ end
+ handle THM _ => raise eqsubst_occL_exp ("THM",occL,thms,i,th);
+
+(* inthms are the given arguments in Isar, and treated as eqstep with
+ the first one, then the second etc *)
+fun eqsubst_asm_meth ctxt occL inthms =
+ Method.SIMPLE_METHOD' (eqsubst_asm_tac ctxt occL inthms);
+
+(* syntax for options, given "(asm)" will give back true, without
+ gives back false *)
+val options_syntax =
+ (Args.parens (Args.$$$ "asm") >> (K true)) ||
+ (Scan.succeed false);
+
+val ith_syntax =
+ Scan.optional (Args.parens (Scan.repeat OuterParse.nat)) [0];
+
+(* combination method that takes a flag (true indicates that subst
+should be done to an assumption, false = apply to the conclusion of
+the goal) as well as the theorems to use *)
+fun subst_meth src =
+ Method.syntax ((Scan.lift options_syntax) -- (Scan.lift ith_syntax) -- Attrib.thms) src
+ #> (fn (((asmflag, occL), inthms), ctxt) =>
+ (if asmflag then eqsubst_asm_meth else eqsubst_meth) ctxt occL inthms);
+
+
+val setup =
+ Method.add_method ("subst", subst_meth, "single-step substitution");
+
+end;