src/Tools/eqsubst.ML
changeset 30160 5f7b17941730
parent 29269 5c25a2012975
child 30318 3d03190d2864
--- /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;