(* Title: Provers/eqsubst.ML
ID: $Id$
Author: Lucas Dixon, University of Edinburgh, lucas.dixon@ed.ac.uk
A proof method to perform a substiution using an equation.
*)
signature EQSUBST =
sig
val setup : theory -> theory
end;
structure EqSubst: EQSUBST =
struct
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 *)
* BasicIsaFTerm.FcTerm (* focusterm to search under *)
(* 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;
*)
(* search from top, left to right, then down *)
fun search_tlr_all_f f ft =
let
fun maux ft =
let val t' = (IsaFTerm.focus_of_fcterm ft)
(* val _ =
if !trace_subst_search then
(writeln ("Examining: " ^ (TermLib.string_of_term t'));
TermLib.writeterm t'; ())
else (); *)
in
(case t' of
(_ $ _) => Seq.append(maux (IsaFTerm.focus_left ft),
Seq.cons(f ft,
maux (IsaFTerm.focus_right ft)))
| (Abs _) => Seq.cons(f ft, maux (IsaFTerm.focus_abs ft))
| leaf => Seq.single (f ft)) end
in maux ft end;
(* search from top, left to right, then down *)
fun search_tlr_valid_f f ft =
let
fun maux ft =
let
val hereseq = if IsaFTerm.valid_match_start ft then f ft else Seq.empty
in
(case (IsaFTerm.focus_of_fcterm ft) of
(_ $ _) => Seq.append(maux (IsaFTerm.focus_left ft),
Seq.cons(hereseq,
maux (IsaFTerm.focus_right ft)))
| (Abs _) => Seq.cons(hereseq, maux (IsaFTerm.focus_abs ft))
| leaf => Seq.single (hereseq))
end
in maux ft end;
(* search all unifications *)
fun searchf_tlr_unify_all (sgn, maxidx, ft) lhs =
IsaFTerm.find_fcterm_matches
search_tlr_all_f
(IsaFTerm.clean_unify_ft sgn maxidx lhs)
ft;
(* search only for 'valid' unifiers (non abs subterms and non vars) *)
fun searchf_tlr_unify_valid (sgn, maxidx, ft) lhs =
IsaFTerm.find_fcterm_matches
search_tlr_valid_f
(IsaFTerm.clean_unify_ft sgn maxidx lhs)
ft;
(* 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.sign_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 = Term.maxidx_of_term conclterm;
val ft = ((IsaFTerm.focus_right
o IsaFTerm.focus_left
o IsaFTerm.fcterm_of_term
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;
(* Tactic.distinct_subgoals_tac -- fails to free type variables *)
(* custom version of distinct subgoals that works with term and
type variables. Asssumes th is in beta-eta normal form.
Also, does nothing if flexflex contraints are present. *)
fun distinct_subgoals th =
if List.null (Thm.tpairs_of th) then
let val (fixes,fixedthm) = IsaND.fix_vars_and_tvars th
val (fixedthconcl,cprems) = IsaND.hide_prems fixedthm
in
IsaND.unfix_frees_and_tfrees
fixes
(Drule.implies_intr_list
(Library.gen_distinct
(fn (x, y) => Thm.term_of x = Thm.term_of y)
cprems) fixedthconcl)
end
else th;
(* General substiuttion 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 (IsaPLib.skipto_seqseq occ (srchf sinfo lhs)) of
IsaPLib.skipmore _ => Seq.empty
| IsaPLib.skipseq ss => Seq.flat ss;
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_tlr_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.METHOD
(fn facts =>
HEADGOAL (Method.insert_tac facts THEN' 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 Tactic.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.sign_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 = Term.maxidx_of_term asmt;
val ft = ((IsaFTerm.focus_right
o IsaFTerm.fcterm_of_term
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)
((rev o IsaPLib.mk_num_list o length)
(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
IsaPLib.skipmore i => occ_search i moreasms
| IsaPLib.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 =
IsaPLib.skipto_seqseq 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_tlr_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.METHOD
(fn facts =>
HEADGOAL (Method.insert_tac facts THEN' 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 =
(Args.parens (Scan.repeat Args.nat))
|| (Scan.succeed [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.local_thms) src
#> (fn (ctxt, ((asmflag, occL), inthms)) =>
(if asmflag then eqsubst_asm_meth else eqsubst_meth) ctxt occL inthms);
val setup =
Method.add_method ("subst", subst_meth, "substiution with an equation");
end;