formal/hol/Library/bitmatch.ml

(* ========================================================================= *)
(*           Parsing and printing for bitmatch expressions.                  *)
(*                                                                           *)
(* This supports term constructions like                                     *)
(*                                                                           *)
(*   bitmatch (word 17) with                                                 *)
(*   | [hi3:3; 0b00:2; lo3:3] -> foo hi3 lo3                                 *)
(*   | [hi3:3; 0b1:1; lo4:4]  -> bar hi3 lo4                                 *)
(*   | [0b1010101:7; b]       -> baz b                                       *)
(*                                                                           *)
(* which decomposes the number 17 = 0b00010001 into the 3 high bits          *)
(* (hi3 = 0), a 1 bit in the middle, and the 4 low bits (lo4 = 1) and calls  *)
(* `bar`.  The patterns are written with the low bits on the right.          *)
(* A pattern variable without a digit matches one bit as a bool.             *)
(*                                                                           *)
(*                (c) Copyright, Mario Carneiro 2020                         *)
(* ========================================================================= *)

needs "Library/words.ml";;

prioritize_num();;

(* Like CHOOSE, but the variable is the same as the one in the existential *)

let TRIV_CHOOSE =
  let P = `P:A->bool` and Q = `Q:bool` and an = `(/\)` in
  let pth = (* `(\x:A. Q /\ P x) = P, (?) P |- Q` *)
    let th1 = AP_THM (ASSUME `(\x:A. Q /\ P x) = P`) `x:A` in
    let th1 = TRANS (SYM th1) (BETA `(\x:A. Q /\ P x) x`) in
    let th1 = CONJUNCT1 (EQ_MP th1 (ASSUME `(P:A->bool) x`)) in
    let th1 = GEN `x:A` (DISCH `(P:A->bool) x` th1) in
    let th2 = CONV_RULE (RAND_CONV BETA_CONV) (AP_THM EXISTS_DEF P) in
    MP (SPEC `Q:bool` (EQ_MP th2 (ASSUME `(?) (P:A->bool)`))) th1 in
  fun th1 th2 ->
    try let P' = rand (concl th1) in
        let v,bod = dest_abs P' in
        let Q' = concl th2 in
        let anQ = mk_comb (an, Q') in
        let th3 = DEDUCT_ANTISYM_RULE (CONJ th2 (ASSUME bod))
          (CONJUNCT2 (ASSUME (mk_comb (anQ, bod)))) in
        let th4 = AP_TERM anQ (BETA (mk_comb (P', v))) in
        let th5 = PINST [snd(dest_var v),aty] [P',P; Q',Q] pth in
        PROVE_HYP th1 (PROVE_HYP (ABS v (TRANS th4 th3)) th5)
    with Failure _ -> failwith "TRIV_CHOOSE";;

let (WITH_GOAL:(term->tactic)->tactic) = fun tac (_,w as st) -> tac w st;;

let _BITMATCH = new_definition `_BITMATCH (s:N word) c = _MATCH (val s) c`;;
let _ELSEPATTERN = new_definition `_ELSEPATTERN (e:B) (a:A) = \x. e = x`;;

let bitpat =
  let th = prove
    (`?x:num#(num->bool). !a. SND x a ==> a < 2 EXP FST x`,
    EXISTS_TAC `0, \i:num. F` THEN REWRITE_TAC[SND]) in
  new_type_definition "bitpat" ("mk_bitpat","dest_bitpat") th;;

let pat_size = new_definition `pat_size p = FST (dest_bitpat p)`;;
let pat_set = new_definition `pat_set p = SND (dest_bitpat p)`;;

let pat_set_lt = prove (`!p s. pat_set p s ==> s < 2 EXP pat_size p`,
  REWRITE_TAC [pat_set; pat_size; bitpat]);;

let pat_thm = prove (`p = mk_bitpat (n, f) /\ (!a. f a ==> a < 2 EXP n) ==>
  pat_size p = n /\ !s. pat_set p s = f s`,
  REPEAT DISCH_TAC THEN
  SUBGOAL_THEN `dest_bitpat (mk_bitpat (n,f)) = (n,f)`
    (fun th -> ASM_REWRITE_TAC [pat_size; pat_set; th]) THEN
  ASM_REWRITE_TAC [GSYM (CONJUNCT2 bitpat)] THEN
  FIRST_X_ASSUM (ASSUME_TAC o MATCH_MP pat_set_lt));;

let NILPAT = new_definition `NILPAT = mk_bitpat(0,\s:num. s = 0)`;;

let NILPAT_pat_size,NILPAT_pat_set = (CONJ_PAIR o prove)
 (`pat_size NILPAT = 0 /\ !s. pat_set NILPAT s <=> s = 0`,
  MATCH_MP_TAC pat_thm THEN REWRITE_TAC [NILPAT] THEN
  STRIP_TAC THEN DISCH_THEN SUBST1_TAC THEN ARITH_TAC);;

let CONSPAT = new_definition `CONSPAT p (a:N word) =
  let n = dimindex(:N) in
  mk_bitpat(n + pat_size p,
    \s:num. (?t. pat_set p t /\ s = val a + 2 EXP n * t))`;;

let CONSPAT_pat_size,CONSPAT_pat_set = (CONJ_PAIR o prove)
 (`pat_size (CONSPAT p (a:N word)) = dimindex(:N) + pat_size p /\
   !s. pat_set (CONSPAT p (a:N word)) s <=>
     ?t. pat_set p t /\ s = val a + 2 EXP dimindex(:N) * t`,
  MATCH_MP_TAC pat_thm THEN
  REWRITE_TAC [CONV_RULE (ONCE_DEPTH_CONV let_CONV) CONSPAT] THEN
  REPEAT STRIP_TAC THEN POP_ASSUM SUBST1_TAC THEN
  MATCH_MP_TAC LTE_TRANS THEN EXISTS_TAC `2 EXP dimindex (:N) * SUC t` THEN
  POP_ASSUM (fun th -> REWRITE_TAC [EXP_ADD; LE_MULT_LCANCEL; LE_SUC_LT;
    MATCH_MP pat_set_lt th]) THEN
  REWRITE_TAC [MULT_SUC; LT_ADD_RCANCEL; VAL_BOUND]);;

reserve_words["BITPAT"; "bitmatch"];;

let word1 = new_definition `word1 p = (word (bitval p)):1 word`;;

let BIT0_WORD1 = prove (`bit 0 (word1 b) = b`,
  REWRITE_TAC [word1; BIT_LSB; bitval; VAL_WORD] THEN DIMINDEX_TAC THEN
  BOOL_CASES_TAC `b:bool` THEN REWRITE_TAC [] THEN ARITH_TAC);;

let VAL_WORD1 = prove (`val (word1 b) = bitval b`,
  REWRITE_TAC [word1; VAL_WORD_BITVAL]);;

let WORD1_ODD = prove (`word1 (ODD n) = word n`,
  CONV_TAC SYM_CONV THEN REWRITE_TAC [word1; WORD_EQ; DIMINDEX_CLAUSES] THEN
  ASM_CASES_TAC `ODD n` THEN
  ASM_REWRITE_TAC[ODD_MOD; BITVAL_CLAUSES; EXP_1; CONG] THEN
  CONV_TAC NUM_REDUCE_CONV THEN
  ASM_REWRITE_TAC[GSYM EVEN_MOD; GSYM ODD_MOD; GSYM NOT_ODD]);;

let preparse_bitpat,preparse_bitmatch =
  let bitmatch_ptm = Varp("_BITMATCH",dpty)
  and NILPAT_ptm = Varp("NILPAT",dpty)
  and CONSPAT_ptm = Varp("CONSPAT",dpty)
  and word_ptm = Varp("word",dpty)
  and word1_ptm = Varp("word1",dpty)
  and arb_ptm = Varp("ARB",dpty)
  and ung_ptm = Varp("_UNGUARDED_PATTERN",dpty)
  and seqp_ptm = Varp("_SEQPATTERN",dpty)
  and else_ptm = Varp("_ELSEPATTERN",dpty) in
  let rec pfrees ptm = match ptm with
  | Varp(v,pty) ->
      if v = "" && pty = dpty then []
      else if can get_const_type v || can num_of_string v
              || exists (fun (w,_) -> v = w) (!the_interface) then []
      else [ptm]
  | Constp(_,_) -> []
  | Combp(p1,p2) -> union (pfrees p1) (pfrees p2)
  | Absp(p1,p2) -> subtract (pfrees p2) (pfrees p1)
  | Typing(p,_) -> pfrees p in
  let pgenvar =
    let gcounter = ref 0 in
    fun () -> let count = !gcounter in
              (gcounter := count + 1;
               Varp("BM%PVAR%"^(string_of_int count),dpty)) in
  let pmk_exists v ptm = Combp(Varp("?",dpty),Absp(v,ptm)) in
  let pmk_bitmatch (((_,e),_),cs) =
    let x = pgenvar() and y = pgenvar() in
    let rec pmk_clauses cs = match pmk_clauses_opt cs with
    | Some t -> t
    | None -> Combp(else_ptm, arb_ptm)
    and pmk_clauses_opt cs = match cs with
    | [] -> None
    | (None,res)::cs -> Some (Combp(else_ptm, res))
    | (Some pat,res)::cs ->
      let tx = Combp(Combp(Varp("pat_set",dpty),pat),x)
      and ty = Combp(Combp(Varp("=",dpty),res),y) in
      let t = itlist pmk_exists (pfrees pat) (Combp(Combp(ung_ptm,tx),ty)) in
      Some(Combp(Combp(seqp_ptm, Absp(x, Absp(y, t))), pmk_clauses cs)) in
    Combp(Combp(bitmatch_ptm,e), pmk_clauses cs) in
  let rec to_bitpat p e = match e with
  | Varp("NIL",_) -> p
  | Combp(Combp(Varp("CONS",_),a),e) ->
    let e2 = match a with
    | Typing(a,i) ->
        let a = match a with
        | Varp(s,_) when can num_of_string s -> Combp(word_ptm,a)
        | _ -> a in
        Typing(a,Ptycon("word",[i]))
    | a -> Combp(word1_ptm,a) in
    to_bitpat (Combp(Combp(CONSPAT_ptm,p),e2)) e
  | _ -> raise Noparse in
  let bitpat = parse_preterm >> to_bitpat NILPAT_ptm in
  let pattern = (a (Ident "_") >> (fun _ -> None))
    ||| (bitpat >> fun x -> Some(x)) in
  let clause = pattern ++ (a (Resword "->") ++ parse_preterm >> snd) in
  let clauses =
    possibly (a (Resword "|")) ++
      listof clause (a (Resword "|")) "pattern-match clause" >> snd in
  (a (Resword "BITPAT") ++ bitpat >> snd),
  (a (Resword "bitmatch") ++ parse_preterm ++ a (Resword "with") ++ clauses
    >> pmk_bitmatch);;

install_parser("bitpat",preparse_bitpat);;
install_parser("bitmatch",preparse_bitmatch);;

let pp_print_bitpat,pp_print_bitmatch =
  let rec dest_pat_rev tm =
    match tm with
    | Comb(Comb(Const("CONSPAT",_),p),a) -> a::dest_pat_rev p
    | Const("NILPAT",_) -> []
    | _ -> failwith "dest_pat" in
  let dest_pat = rev o dest_pat_rev in
  let dest_clause tm =
    match snd(strip_exists(body(body tm))) with
    | Comb(Comb(Const("_UNGUARDED_PATTERN",_),lhs),
        Comb(Comb(Const("=",_),res),_)) ->
      (match lhs with
      | Comb(Comb(Const("pat_set",_),pat),_) -> dest_pat pat, res
      | _ -> failwith "dest_clause")
    | _ -> failwith "dest_clause" in
  let rec dest_clauses tm =
    match tm with
    | Comb(Comb(Const("_SEQPATTERN",_),c),cs) ->
        let c = dest_clause c and l,r = dest_clauses cs in c::l,r
    | Comb(Const("_ELSEPATTERN",_),res) -> [],res
    | _ -> failwith "dest_clauses" in
  let f fmt =
    let unword i = function
    | Comb(Const("word",_),a) when is_numeral a ->
      if dest_numeral a < power_num (Int 2) (dest_finty i)
      then a
      else failwith "numeral out of range"
    | a -> a in
    let print_pat = function
    | Comb(Const("word1",_),a) -> pp_print_term fmt a
    | a -> match type_of a with
      | Tyapp("word",[i]) ->
          (pp_print_term fmt (unword i a);
          pp_print_string fmt ":";
          pp_print_type fmt i)
      | _ -> failwith "print_pat" in
    let print_opat = function
    | None -> pp_print_string fmt "_"
    | Some [] -> pp_print_string fmt "[]"
    | Some (x::xs) ->
       (pp_print_string fmt "[";
        print_pat x;
        List.iter (fun x ->
          pp_print_string fmt "; ";
          print_pat x) xs;
        pp_print_string fmt "]") in
    let print_clause p r =
      (print_opat p;
       pp_print_string fmt " -> ";
       pp_print_term fmt r) in
    let rec print_clauses cls r = match cls,r with
    | [p,res],Const("ARB",_) -> print_clause (Some p) res
    | (p,res)::cs,_ -> (
      print_clause (Some p) res;
      pp_print_break fmt 1 0;
      pp_print_string fmt "| ";
      print_clauses cs r)
    | [],r -> print_clause None r in
    let pp_print_bitpat tm =
      let pat = dest_pat tm in
      (pp_open_hvbox fmt 0;
       pp_print_string fmt "(BITPAT";
       pp_print_space fmt ();
       print_opat (Some pat);
       pp_print_string fmt ")";
       pp_close_box fmt ()) in
    let pp_print_bitmatch = function
    | Comb(Comb(Const("_BITMATCH",_),e),c) ->
        let cls,r = dest_clauses c in
        (pp_open_hvbox fmt 0;
         pp_print_string fmt "(bitmatch ";
         pp_print_term fmt e;
         pp_print_string fmt " with";
         pp_print_break fmt 1 2;
         print_clauses cls r;
         pp_close_box fmt ();
         pp_print_string fmt ")")
    | _ -> failwith "print_bitmatch" in
    pp_print_bitpat,pp_print_bitmatch in
  fst o f, snd o f;;

let print_bitpat = pp_print_bitpat std_formatter
and string_of_bitpat = print_to_string pp_print_bitpat;;

let print_bitmatch = pp_print_bitmatch std_formatter
and string_of_bitmatch = print_to_string pp_print_bitmatch;;

install_user_printer("bitpat",pp_print_bitpat);;
install_user_printer("bitmatch",pp_print_bitmatch);;

(* ------------------------------------------------------------------------- *)
(* Some tactics for dealing with bitmatch                                    *)
(* ------------------------------------------------------------------------- *)

let dest_word_ty = function
| Tyapp("word",[i]) -> i
| _ -> failwith "dest_word_ty";;

let SPLIT_IF =
  let th = (UNDISCH_ALL o TAUT)
    `(p ==> a:A = t1) ==> (~p ==> a = t2) ==> a = if p then t1 else t2` in
  let [Var(_,A) as a; p; t1; t2] = frees (concl th) and tnot = `~` in
  fun p' th1 th2 ->
    let a',t1' = dest_eq (concl th1) and
    t2' = snd (dest_eq (concl th2)) in
    (PROVE_HYP (DISCH p' th1) o PROVE_HYP (DISCH (mk_comb(tnot, p')) th2))
    (PINST [type_of a',A] [p',p; a',a; t1',t1; t2',t2] th);;

let MATCH_SEQPATTERN = prove
 (`_MATCH (e:A) (_SEQPATTERN c cs) =
   if ?y:B. c e y then _MATCH e c else _MATCH e cs`,
  COND_CASES_TAC THEN ASM_REWRITE_TAC [_MATCH; _SEQPATTERN]);;

let BITMATCH_SEQPATTERN = prove
 (`_BITMATCH (e:N word) (_SEQPATTERN c cs) =
   if ?y:B. c (val e) y then _BITMATCH e c else _BITMATCH e cs`,
  REWRITE_TAC [_BITMATCH; MATCH_SEQPATTERN]);;

let BITMATCH_SEQPATTERN2 = prove
 (`_BITMATCH (e:N word) (_SEQPATTERN c cs) =
   if ?y:B. c (val e) y then
     _BITMATCH e (_SEQPATTERN c (_ELSEPATTERN ARB))
   else _BITMATCH e cs`,
  REWRITE_TAC [BITMATCH_SEQPATTERN] THEN COND_CASES_TAC THEN REWRITE_TAC []);;

let MATCH_ELSEPATTERN = prove
 (`_MATCH (e:B) (_ELSEPATTERN (a:A)) = a`,
  REWRITE_TAC [_MATCH; _ELSEPATTERN] THEN METIS_TAC[]);;

let BITMATCH_ELSEPATTERN = prove
 (`_BITMATCH (e:N word) (_ELSEPATTERN (a:A)) = a`,
  REWRITE_TAC [_BITMATCH; MATCH_ELSEPATTERN]);;

let pat_extract = new_definition `pat_extract p i b <=>
  (!n. pat_set p n ==> numbit i n = b)`;;

let PAT_EXTRACT_THM =
  let N,n,m,i,p,b = `:N`,`n:num`,`m:num`,`i:num`,`p:bitpat`,`b:bool`
  and a,lt,pl,nb,di = `a:N word`,`(<)`,`(+)`,`numbit`,`dimindex(:N)` in
  let word1_eq = prove
   (`pat_extract (CONSPAT p (word1 b)) 0 b`,
    REWRITE_TAC [pat_extract; numbit; CONSPAT_pat_set; word1; VAL_WORD] THEN
    REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST1_TAC THEN
    DIMINDEX_TAC THEN NUM_REDUCE_TAC THEN
    REWRITE_TAC [DIV_1; EVEN_ADD; GSYM NOT_EVEN; EVEN_DOUBLE] THEN
    BOOL_CASES_TAC `b:bool` THEN REWRITE_TAC [bitval] THEN ARITH_TAC) in

  let word_lt = (UNDISCH_ALL o prove)
   (`dimindex(:N) = n ==> numbit i m = b ==> (i < n <=> T) ==>
     pat_extract (CONSPAT p (word m:N word)) i b`,
    REWRITE_TAC[] THEN REPEAT (DISCH_THEN (SUBST1_TAC o SYM)) THEN
    REWRITE_TAC [pat_extract; CONSPAT_pat_set; numbit] THEN
    REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST1_TAC THEN
    REWRITE_TAC [VAL_WORD; ODD_MOD; DIV_MOD] THEN
    AP_THM_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
    ONCE_REWRITE_TAC [MULT_SYM] THEN
    REWRITE_TAC [GSYM (CONJUNCT2 EXP)] THEN
    ONCE_REWRITE_TAC [GSYM MOD_ADD_MOD] THEN
    FIRST_X_ASSUM (SUBST1_TAC o GSYM o
      MATCH_MP (ARITH_RULE `m <= n ==> (m + (n - m) = n:num)`) o
      REWRITE_RULE [GSYM LE_SUC_LT]) THEN
    REWRITE_TAC [EXP_ADD; MOD_MOD] THEN ONCE_REWRITE_TAC [MULT_AC] THEN
    REWRITE_TAC [MOD_MULT; ADD_0; MOD_MOD_REFL]) in

  let word_skip = (UNDISCH_ALL o prove)
   (`dimindex(:N) = n ==> m + n = i ==> pat_extract p m b ==>
     pat_extract (CONSPAT p (a:N word)) i b`,
    REPEAT (DISCH_THEN (SUBST1_TAC o SYM)) THEN
    REWRITE_TAC [pat_extract; CONSPAT_pat_set; numbit] THEN
    REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST1_TAC THEN
    POP_ASSUM (fun th -> POP_ASSUM (fun f ->
      REWRITE_TAC [GSYM (MATCH_MP f th)])) THEN
    REWRITE_TAC [SPECL [m;`dimindex(:N)`] ADD_SYM] THEN
    REWRITE_TAC [EXP_ADD; GSYM DIV_DIV] THEN
    AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
    IMP_REWRITE_TAC [DIV_MULT_ADD; DIV_LT; ADD; VAL_BOUND; EXP_2_NE_0]) in

  let rec go = function
  | Comb(Comb(Const("CONSPAT",_),p'),a'),i' ->
    let i'' = dest_numeral i' in
    let aty = type_of a' in
    let Tyapp(_,[N']) = aty in
    let n'' = dest_finty N' in
    if i'' < n'' then
      match a' with
      | Comb(Const("word1",_),b') -> INST [p',p; b',b] word1_eq
      | Comb(Const("word",_),m') when is_numeral m' ->
        let th = INST_TYPE [N',N] word_lt in
        let th1 = DIMINDEX_CONV (inst [N',N] di) in
        let n' = rhs (concl th1) in
        let th2 = NUMBIT_CONV (mk_comb (mk_comb (nb,i'),m')) in
        let b' = rhs (concl th2) in
        let th3 = NUM_LT_CONV (mk_comb (mk_comb (lt,i'),n')) in
        (PROVE_HYP th3 o PROVE_HYP th2 o PROVE_HYP th1)
          (INST [i',i; m',m; n',n; p',p; b',b] th)
      | _ -> failwith "PAT_EXTRACT_THM: not a constant at this position"
    else
      let th = INST_TYPE [N',N] word_skip in
      let th1 = DIMINDEX_CONV (inst [N',N] di) in
      let n' = rhs (concl th1) in
      let m' = mk_numeral (sub_num i'' n'') in
      let th2 = NUM_ADD_CONV (mk_comb (mk_comb (pl,m'),n')) in
      let th3 = go (p', m') in
      let b' = rand (concl th3) in
      (PROVE_HYP th3 o PROVE_HYP th2 o PROVE_HYP th1)
        (INST [i',i; m',m; n',n; p',p; b',b; a',mk_var("a",aty)] th)
  | _ -> failwith "PAT_EXTRACT_THM: out of range"
  in go;;

(* (pat_to_bit false `i` `pat_set p (val e)`) proves
    `bit i e |- ~pat_set p (val e)` or
    `~bit i e |- ~pat_set p (val e)`
    (pat_to_bit true `i` `pat_set p (val e)`) proves
    `pat_set p (val e) |- ~bit i e` or
    `pat_set p (val e) |- bit i e` *)
let pat_to_bit =
  (* thT := ~bit i e, pat_extract p i T |- ~pat_set p (val e)
      thF := bit i e, pat_extract p i F |- ~pat_set p (val e) *)
  let thT_pos,thF_pos =
    (* TODO: I got frustrated with tactics so this is just a direct proof. *)
    let a = PURE_REWRITE_RULE [pat_extract] (ASSUME `pat_extract p i b`) in
    let a = UNDISCH (SPEC `val (e:N word)` a) in
    let th = TRANS (SYM NUMBIT_VAL) a in
    EQT_ELIM (INST [`T`,`b:bool`] th), EQF_ELIM (INST [`F`,`b:bool`] th) in
  let thT_neg,thF_neg =
    let f x y = NOT_INTRO (DISCH `pat_set p (val (e:N word))`
      (MP (NOT_ELIM x) y)) in
    f (ASSUME `~bit i (e:N word)`) thT_pos,
    f thF_pos (ASSUME `bit i (e:N word)`) in
  let N,i,e,p = `:N`,`i:num`,`e:N word`,`p:bitpat` in
  fun pos i' -> function
  | Comb(Comb(Const("pat_set",_),p'),
      Comb(Const("val",Tyapp(_,[Tyapp(_, [N']); _])),e')) ->
    let thp = PAT_EXTRACT_THM (p', i') in
    let th = match rand (concl thp) with
    | Const("T",_) -> if pos then thT_pos else thT_neg
    | Const("F",_) -> if pos then thF_pos else thF_neg
    | _ -> failwith "pat_to_bit" in
    PROVE_HYP thp (PINST [N',N] [i',i; e',e; p',p] th)
  | _ -> failwith "pat_to_bit";;

let bm_analyze_pat sz =
  let A = Array.make sz None in
  let rec go i = function
  | Comb(Comb(Const("CONSPAT",_),p),a) ->
    let n = Num.int_of_num (dest_finty (dest_word_ty (type_of a))) in
    if i + n > sz then
      raise (Invalid_argument "incorrect bit length") else
    let () = match a with
    | Comb(Const("word1",_),a) ->
      A.(i) <- (match a with
        | Const("T",_) -> Some true
        | Const("F",_) -> Some false
        | Var(_,_) -> None
        | _ -> failwith "bm_analyze_pat")
    | Comb(Const("word",_),Comb(Const("NUMERAL",_),a)) ->
      let rec analyze_num n a = match a with
      | Comb(Const("BIT0",_),a) ->
        (A.(n) <- Some false; analyze_num (n+1) a)
      | Comb(Const("BIT1",_),a) ->
        (A.(n) <- Some true; analyze_num (n+1) a)
      | Const("_0",_) -> Array.fill A n (sz - n) (Some false)
      | _ -> failwith "bm_analyze_pat" in
      analyze_num i a
    | Var(_,_) -> Array.fill A i n None
    | _ -> failwith "bm_analyze_pat" in
    go (i + n) p
  | Const("NILPAT",_) ->
    if i = sz then () else
    raise (Invalid_argument "incorrect bit length")
  | Abs(_,c) -> go i c
  | Comb(Const("?",_),c) -> go i c
  | Comb(Comb(Const("_UNGUARDED_PATTERN",_),c),_) -> go i c
  | Comb(Comb(Const("pat_set",_),c),_) -> go i c
  | _ -> failwith "bm_analyze_pat" in
  fun pat ->
    try go 0 pat; A
    with Invalid_argument _ -> failwith (sprintf
      "bm_analyze_pat: pattern %s has incorrect bit length"
      (string_of_term pat));;

let rec bm_analyze_clauses sz = function
| Comb(Comb(Const("_SEQPATTERN",_),c),cs) ->
  let pat = (lhand o lhand o snd o strip_exists o body o body) c in
  bm_analyze_pat sz pat :: bm_analyze_clauses sz cs
| _ -> [];;

(* (bm_skip_clause f `_BITMATCH e (_SEQPATTERN r rs)`) returns
   `A |- _BITMATCH e (_SEQPATTERN r rs) = _BITMATCH e rs` if
   (f `pat_set p (val e)`) proves `A |- ~pat_set p (val e)`
   (probably via pat_to_bit false) *)
let bm_skip_clause =
  let th = (UNDISCH o prove)
    (`(?) (r (val e)) = F ==>
      (_BITMATCH e (_SEQPATTERN r rs):B) = _BITMATCH (e:N word) rs`,
    REPEAT DISCH_TAC THEN REWRITE_TAC [BITMATCH_SEQPATTERN] THEN
    CONV_TAC (ONCE_DEPTH_CONV ETA_CONV) THEN ASM_REWRITE_TAC []) in
  let N,B,e,r,rs =
    `:N`,`:B`,`e:N word`,`r:num->B->bool`,`rs:num->B->bool` in

  (* (strip_ex `x` `A |- P[x] = F`) proves `A |- (?x. P[x]) = F` *)
  let strip_ex =
    let th1 = (UNDISCH o prove)
      (`(P = \x:A. F) ==> (?) P = F`, DISCH_TAC THEN ASM_REWRITE_TAC[]) in
    let A = `:A` and P = `P:A->bool` in
    fun x th ->
      let th' = ABS x th in
      PROVE_HYP th' (PINST [type_of x,A] [lhs (concl th'),P] th1) in

  (* (skip_guard `A` `B`) proves `~A |- _UNGUARDED_PATTERN A B = F` *)
  let skip_guard =
    let th = (UNDISCH o prove)
      (`~A ==> _UNGUARDED_PATTERN A B = F`,
      DISCH_TAC THEN ASM_REWRITE_TAC [_UNGUARDED_PATTERN]) in
    let A = `A:bool` and B = `B:bool` in
    fun A' B' -> INST [A',A; B',B] th in

  fun f -> function
  | Comb(Comb(Const("_BITMATCH",ty),e'),
      Comb(Comb(Const("_SEQPATTERN",_),c),rs')) ->
    let Tyapp(_, [N']),Tyapp(_, [_; B']) = dest_fun_ty ty in
    let th = PINST [N',N;B',B] [e',e; c,r; rs',rs] th in
    let ex,h = dest_comb (lhs (hd (hyp th))) in
    let th2 = BETA_CONV h in
    let th3 = AP_TERM ex th2 in
    let rec skip_exs = function
    | Comb(Const("?",_),Abs(y,c)) -> strip_ex y (skip_exs c)
    | Comb(Comb(Const("_UNGUARDED_PATTERN",_),p),res) ->
      PROVE_HYP (f p) (skip_guard p res)
    | _ -> failwith "skip_exs" in
    PROVE_HYP (TRANS th3 (skip_exs (rhs (concl th3)))) th
  | _ -> failwith "bm_skip_clause";;

type 'a discrim_tree =
    Leaf_dt of 'a
  | Split_dt of int * term * 'a discrim_tree * 'a discrim_tree;;

let rec map_dt f = function
| Leaf_dt a -> Leaf_dt (f a)
| Split_dt(i, bit, tr1, tr2) ->
  Split_dt(i, bit, map_dt f tr1, map_dt f tr2);;

let bm_build_tree' =
  let bit_tm =
    let N = `:N` in
    fun i e ->
      let Tyapp("word",[N']) = type_of e in
      let bit = mk_const("bit",[N',N]) in
      mk_comb(mk_comb(bit,i),e) in

  (* (seqp_rand `r` `A |- _BITMATCH e rs1 = _BITMATCH e rs2`) proves
     `A |- _BITMATCH e (_SEQPATTERN r rs1) =
           _BITMATCH e (_SEQPATTERN r rs2)` *)
  let seqp_rand =
    let th = (UNDISCH o prove)
      (`_BITMATCH (e:N word) (rs1:num->B->bool) = _BITMATCH e rs2 ==>
        _BITMATCH e (_SEQPATTERN r rs1) = _BITMATCH e (_SEQPATTERN r rs2)`,
      DISCH_TAC THEN ASM_REWRITE_TAC [BITMATCH_SEQPATTERN]) in
    let [rs1; e; r; rs2] = frees (concl th) and N,B = `:N`,`:B` in
    fun r' th' ->
      match dest_eq (concl th') with
      | Comb(Comb(Const(_,ty),e'),rs1'),Comb(_,rs2') ->
      let Tyapp(_, [N']),Tyapp(_, [_; B']) = dest_fun_ty ty in
      PROVE_HYP th' (PINST [N',N; B',B] [e',e; r',r; rs1',rs1; rs2',rs2] th)
      | _ -> failwith "seqp_rand" in

  let aMerge a b = match a,b with
  | None,_ -> a
  | _,None -> None
  | Some(a0,a1),Some false -> Some(a0+1, a1)
  | Some(a0,a1),Some true -> Some(a0, a1+1) in

  fun sz cls -> function
  | Comb(Comb(Const("_BITMATCH",_),e) as m, cs) as tm ->
    let rec build eqth cls cs =
      let analysis = Array.make sz (Some(0,0)) in
      let _ = List.iter (Array.iteri
        (fun i a -> analysis.(i) <- aMerge analysis.(i) a)) cls in
      let r =
        let r = ref None in
        let f i a = match a with
        | Some(n1,n2) when n1 != 0 && n2 != 0 ->
          let v = abs (n1 - n2) in
          (match !r with
          | Some(v',_) when v' <= v -> ()
          | _ -> r := Some(v, i))
        | _ -> () in
        (Array.iteri f analysis; !r) in
      match r with
      | None -> Leaf_dt eqth
      | Some(_,i) ->
        let ii = mk_numeral (Int i) in
        let bit = bit_tm ii e in
        let skip_th sc th =
          let sm, rs' = dest_comb (lhs (concl th)) in
          let tm = mk_comb (sm, mk_comb (sc, rs')) in
          TRANS (bm_skip_clause (pat_to_bit false ii) tm) th in
        let rec split_ths = function
        | [], cs -> let th = REFL (mk_comb(m,cs)) in [],[],th,th
        | cl::cls, Comb(Comb(Const("_SEQPATTERN",_),c) as sc,cs) ->
          let cls1,cls2,th1,th2 = split_ths(cls, cs) in
          if let Some(b) = cl.(i) in b then
            cls1, cl::cls2, skip_th sc th1, seqp_rand c th2
          else
            cl::cls1, cls2, seqp_rand c th1, skip_th sc th2
        | cl::cls, cs ->
          let cls1,cls2,th1,th2 = split_ths(cls, cs) in
          if let Some(b) = cl.(i) in b then
            cls1, cl::cls2, th1, th2
          else
            cl::cls1, cls2, th1, th2
        | _ -> failwith "split_ths" in
        let cls1,cls2,th1,th2 = split_ths(cls, cs) in
        let tr1 = build (TRANS eqth th1) cls1 (rand (rhs (concl th1)))
        and tr2 = build (TRANS eqth th2) cls2 (rand (rhs (concl th2))) in
        Split_dt(i, bit, tr1, tr2) in
    build (REFL tm) cls cs
  | _ -> failwith "bm_build_tree'";;

let bm_build_tree = function
| Comb(Comb(Const("_BITMATCH",_),e),cs) as tm ->
  let sz = Num.int_of_num (dest_finty (dest_word_ty (type_of e))) in
  let cls = bm_analyze_clauses sz cs in
  cls, bm_build_tree' sz cls tm
| _ -> failwith "bm_build_tree";;

let BM_IF_CONV =
  let rec of_tree = function
  | Leaf_dt th -> th
  | Split_dt(_, bit, tr1, tr2) -> SPLIT_IF bit (of_tree tr2) (of_tree tr1) in
  of_tree o snd o bm_build_tree;;

let MATCH_EQ = prove
 (`(r:A->B->bool) e = (\y. x = y) ==> _MATCH e r = x`,
  REWRITE_TAC [_MATCH] THEN DISCH_THEN SUBST1_TAC THEN METIS_TAC[]);;

let bitpat_inverts = new_definition
 `bitpat_inverts p f (x:A) <=> (!y. pat_set p y ==> f y = x)`;;

let bitpat_down = new_definition
 `bitpat_down(:N) (f:num->A) (n:num) = f (n DIV 2 EXP dimindex(:N))`;;

let CONSPAT_down_inverts = prove
 (`bitpat_inverts p f (x:A) ==>
   bitpat_inverts (CONSPAT p (a:N word)) (bitpat_down(:N) f) x`,
  REWRITE_TAC [bitpat_inverts; bitpat_down; CONSPAT_pat_set] THEN
  REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
  FIRST_X_ASSUM SUBST1_TAC THEN
  IMP_REWRITE_TAC [DIV_MULT_ADD; DIV_LT; VAL_BOUND; EXP_2_NE_0] THEN
  ASM_REWRITE_TAC [ADD]);;

let CONSPAT_word_inverts = prove
 (`bitpat_inverts (CONSPAT p (a:N word)) word a`,
  REWRITE_TAC [bitpat_inverts; CONSPAT_pat_set] THEN
  REPEAT STRIP_TAC THEN FIRST_X_ASSUM SUBST1_TAC THEN
  REWRITE_TAC [WORD_VAL_GALOIS; MOD_MULT_ADD] THEN
  REWRITE_TAC [GSYM WORD_VAL_GALOIS; WORD_VAL]);;

let bitpat_inverts_comp = MESON [bitpat_inverts; o_DEF]
 `bitpat_inverts p f (x:A) ==> bitpat_inverts p (g o f) (g x:B)`;;

let CONSPAT_word1_inverts =
  (* `bitpat_inverts (CONSPAT p (word1 b)) (bit 0 o word) b` *)
  REWRITE_RULE [BIT0_WORD1]
    (PART_MATCH rand (MATCH_MP bitpat_inverts_comp CONSPAT_word_inverts)
      (lhs (concl BIT0_WORD1)));;

(* Given a pattern `p` and a target variable `z`,
  (build_inverts `p`) produces an association list mapping `z`
  to `|- bitpat_inverts p f z` for some term `f` not containing `z`. *)
let build_inverts =
  let N,A,p,a,f,x,b =
    `:N`,`:A`,`p:bitpat`,`a:N word`,`f:num->A`,`x:A`,`b:bool`
  and th0 = ASSUME `bitpat_inverts p f (x:A)`
  and th1 = UNDISCH CONSPAT_down_inverts in
  let rec go = function
  | Const("NILPAT",_),_ -> []
  | (Comb(Comb(Const("CONSPAT",_),p'),a') as cp),thunk ->
    let Tyapp(_, [N']) = type_of a' in
    let next th =
      let th' = PINST [N',N] [a',a] th1 in
      let Comb(Comb(_,p'),f') = rator (concl th') in
      PROVE_HYP th' (INST [p',p; f',f] th) in
    let var v thv =
      let th = thunk() in
      let f' = rand (rator (concl thv)) in
      let th' = PROVE_HYP thv (PINST [type_of v,A] [cp,p; f',f; v,x] th) in
      (v,th') :: go (p', fun () -> next th) in
    (match a' with
    | Var(_,_) -> var a' (PINST [N',N] [p',p; a',a] CONSPAT_word_inverts)
    | Comb(Const("word1",_), (Var(_,_) as b')) ->
      var b' (INST [p',p; b',b] CONSPAT_word1_inverts)
    | _ -> go (p', next o thunk))
  | _ -> failwith "build_inverts" in
  fun p' -> go (p', fun () -> th0);;


(* Given `bitmatch e with p -> res | ...` proves
   `bit_set p (val e) |- (bitmatch e with p -> res | ...) = res`,
  and given `bitmatch e with _ -> res` proves
  `(bitmatch e with _ -> res) = res`. *)
let BM_FIRST_CASE_CONV =
  let th1 = (UNDISCH o prove)
   (`r (val (e:N word)) = (\y:A. x = y) ==> _BITMATCH e (_SEQPATTERN r s) = x`,
    DISCH_TAC THEN
    ASM_REWRITE_TAC [BITMATCH_SEQPATTERN; _BITMATCH; MESON[] `?y:A. x=y`] THEN
    POP_ASSUM (ACCEPT_TAC o MATCH_MP MATCH_EQ)) in
  let th2 = (UNDISCH o METIS[_UNGUARDED_PATTERN])
    `A ==> (_UNGUARDED_PATTERN A B <=> B)` in
  let th3 = EQ_MP (SYM th2) (ASSUME `B:bool`) in
  let th4 = (UNDISCH_ALL o prove)
   (`bitpat_inverts p f (x:A) ==> pat_set p e ==> f e = x`,
    DISCH_THEN (MATCH_ACCEPT_TAC o REWRITE_RULE [bitpat_inverts])) in
  let thg1,thg2 = (CONJ_PAIR o UNDISCH o METIS[_UNGUARDED_PATTERN])
    `_UNGUARDED_PATTERN A B ==> A /\ B` in
  let nty,eA,eB,ep,ef,ex = `:N`,`A:bool`,`B:bool`,`p:bitpat`,`f:num->A`,`x:A`
  and ee,ea,er,es = `e:N word`,`a:A`,`r:num->A->bool`,`s:num->A->bool` in
  function
  | Comb(Comb(Const("_BITMATCH",_),e),
      Comb(Comb(Const("_SEQPATTERN",_), (Abs(x,Abs(y,c')) as c)),cs)) as tm ->
    let A' = type_of tm in
    let N = dest_word_ty (type_of e) in
    let val_e = mk_comb (mk_const("val", [N,nty]), e) in
    let zs, Comb(Comb(Const("_UNGUARDED_PATTERN",_),
      (Comb((Comb(_,p) as mp),_) as ps)),restm) = strip_exists c' in
    let res = lhand restm in
    let instAB = INST [ps,eA; restm,eB] in
    let ps' = mk_comb (mp, val_e) in
    let th' = if zs = [] then instAB th2 else
      let inverts = build_inverts p in
      let rec prove_ex c1 pr eqth = match c1 with
      | Comb(Const("?",_),Abs(z,c')) ->
        let inv = assoc z inverts in
        let f = lhand (concl inv) in
        let inv = PROVE_HYP inv (PINST [type_of z,aty]
          [p,ep; f,ef; z,ex; x,`e:num`] th4) in
        let abr = mk_abs (z, lhs (concl eqth)) in
        let eqth1 = TRANS (AP_TERM abr (ASSUME (concl inv)))
          (BETA (mk_comb (abr, z))) in
        let pr',th = prove_ex c' (PROVE_HYP inv o pr) (TRANS eqth1 eqth) in
        pr', TRANS (SYM eqth1) (TRIV_CHOOSE (ASSUME c1) th)
      | _ -> pr, TRANS (PROVE_HYP (instAB thg1) (pr eqth)) (instAB thg2) in
      let pr,thR = prove_ex c' I (REFL res) in
      DEDUCT_ANTISYM_RULE (itlist SIMPLE_EXISTS zs (instAB th3)) (pr thR) in
    let th' = INST [val_e,x] (TRANS (BETA (mk_comb (c, x))) (ABS y th')) in
    PROVE_HYP th' (PINST [N,nty; A',aty]
      [e,ee; res,ex; c,er; cs,es] th1)
  | Comb(Comb(Const("_BITMATCH",_),e), Comb(Const("_ELSEPATTERN",_), a)) ->
    let A' = type_of a in
    let N = dest_word_ty (type_of e) in
    PINST [N,nty; A',aty] [e,ee; a,ea] BITMATCH_ELSEPATTERN
  | _ -> failwith "BM_FIRST_CASE_CONV";;

let bm_add_pos tr = function
| Comb(Comb(Const("_BITMATCH",_),e),cs) ->
  let N = dest_word_ty (type_of e) in
  let val_e = mk_comb (mk_const("val", [N,`:N`]), e) in
  let rec build_cases stk mth = function
  | Comb(Comb(Const("_SEQPATTERN",_), Abs(x,Abs(y,c'))),cs) ->
    let ps' = mk_comb (rator (lhand (snd (strip_exists c'))), val_e) in
    let th = itlist (fun n ->
      try PROVE_HYP (pat_to_bit true n ps') with Failure _ -> I) stk mth in
    TRANS th (BM_FIRST_CASE_CONV (rhs (concl th))) :: build_cases stk mth cs
  | _ -> [] in
  let rec build stk = function
  | Leaf_dt mth ->
    Leaf_dt (mth, build_cases stk mth (rand (rhs (concl mth))))
  | Split_dt (i, bit, tr1, tr2) ->
    let stk' = lhand bit :: stk in
    Split_dt (i, bit, build stk' tr1, build stk' tr2) in
  build [] tr
| _ -> failwith "bm_build_pos_tree";;

let bm_build_pos_tree tm =
  let A, tr = bm_build_tree tm in A, bm_add_pos tr tm;;

let rec get_dt A = function
| Leaf_dt a -> [], a
| Split_dt(i, bit, tr1, tr2) ->
  match A.(i) with
  | None -> failwith ("get_dt splitting on " ^ string_of_int i)
  | Some b ->
    let stk,r = get_dt A (if b then tr2 else tr1) in
    (b,bit)::stk, r;;

let BM_CASES tm =
  let A, tr = bm_build_pos_tree tm in
  map (fun cl -> hd (snd (snd (get_dt cl tr)))) A;;

(* (bitpat_matches `p` n) returns None if the pattern `p` would match
   `word n`, and Some(i) where i is the smallest differing bit otherwise.
   It throws if n >= 2^pat_size p. *)
let rec bitpat_matches p i = match p with
| Comb(Comb(Const("CONSPAT",_),p),a) ->
  let N = dest_word_ty (type_of a) in
  let n = Num.int_of_num (dest_finty N) in
  let m = power_num (Int 2) (Int n) in
  let i' = quo_num i m and a' = mod_num i m in
  let r = match a with
  | Comb(Const("word1",_),Const("T",_)) -> if a' = Int 1 then None else Some 0
  | Comb(Const("word1",_),Const("F",_)) -> if a' = Int 0 then None else Some 0
  | Comb(Const("word1",_),Var(_,_)) -> None
  | Comb(Const("word",_),n) ->
    let n' = dest_numeral n in
    if a' = n' then None else
    let rec f i r = if i land 1 != 0 then r else f (i lsr 1) (r+1) in
    Some (f ((Num.int_of_num a') lxor (Num.int_of_num n')) 0)
  | Var(_,_) -> None
  | _ -> failwith "bitpat_matches" in
  (match r with
  | Some j -> Some j
  | None ->
    match bitpat_matches p i' with
    | Some j -> Some (j + n)
    | None -> None)
| Const("NILPAT",_) -> if i = Int 0 then None else
  failwith "bitpat_matches: out of range"
| Abs(_,c) -> bitpat_matches c i
| Comb(Const("?",_),c) -> bitpat_matches c i
| Comb(Comb(Const("_UNGUARDED_PATTERN",_),c),_) -> bitpat_matches c i
| Comb(Comb(Const("pat_set",_),c),_) -> bitpat_matches c i
| _ -> failwith "bitpat_matches";;

(* (inst_bitpat_numeral `pat_set p (val e)` n) will produce an instantiation
   theta for p and e such that e[theta] = word n, and a proof of
    `|- (pat_set p (val e))[theta]`.

   (inst_bitpat_numeral `pat_set p e` n) will produce an instantiation
   theta for p and e such that e[theta] = n, and a proof of
    `|- (pat_set p e)[theta]`. *)
let inst_bitpat_numeral =
  let en,ep,ex,ei,ea = `n:num`,`p:bitpat`,`x:num`,`i:num`,`a:num`
  and eN,T,F = `:N`,`T`,`F` in
  let dim =
    let dN = `dimindex(:N)` in
    fun N -> DIMINDEX_CONV (inst [N,eN] dN) in

  let w0 = prove (`pat_set NILPAT _0`,
    REWRITE_TAC [NILPAT_pat_set; NUMERAL])
  and wS,(w1T,w1F) =
    let pth = prove
     (`pat_set p x ==>
       pat_set (CONSPAT p (word a:N word)) (num_shift_add a x (dimindex(:N)))`,
      REWRITE_TAC [CONSPAT_pat_set; num_shift_add] THEN
      DISCH_THEN (fun th ->
        EXISTS_TAC ex THEN REWRITE_TAC [th; VAL_WORD; MULT_SYM])) in
    (UNDISCH_ALL o prove)
     (`dimindex(:N) = NUMERAL n ==> num_shift_add a x n = i ==> pat_set p x ==>
       pat_set (CONSPAT p (word (NUMERAL a):N word)) i`,
      REWRITE_TAC [NUMERAL] THEN
      REPEAT (DISCH_THEN (SUBST1_TAC o SYM)) THEN ACCEPT_TAC pth),
    (CONJ_PAIR o UNDISCH o prove)
     (`pat_set p x ==>
       pat_set (CONSPAT p (word1 T)) (BIT1 x) /\
       pat_set (CONSPAT p (word1 F)) (BIT0 x)`,
      REWRITE_TAC [word1; bitval] THEN
      DISCH_THEN (fun th -> CONJ_TAC THEN ASSUME_TAC th) THENL [
        POP_ASSUM (MP_TAC o MP (PINST [`:1`,eN] [`1`,ea] pth)) THEN
        SUBGOAL_THEN
          `num_shift_add 1 x (dimindex(:1)) = num_shift_add (BIT1 0) x (SUC 0)`
          (fun th -> REWRITE_TAC [th; num_shift_add_SUC; num_shift_add_0]) THEN
        CONV_TAC (ONCE_DEPTH_CONV DIMINDEX_CONV) THEN
        REWRITE_TAC [BIT1_0; ONE];
        POP_ASSUM (MP_TAC o MP (PINST [`:1`,eN] [`0`,ea] pth)) THEN
        SUBGOAL_THEN
          `num_shift_add 0 x (dimindex(:1)) = num_shift_add (BIT0 0) x (SUC 0)`
          (fun th -> REWRITE_TAC [th; num_shift_add_SUC; num_shift_add_0]) THEN
        CONV_TAC (ONCE_DEPTH_CONV DIMINDEX_CONV) THEN
        REWRITE_TAC [BIT0_0; ONE]]) in
  let w1F0 = REWRITE_RULE [ARITH_ZERO] (INST [`_0`,ex] w1F) in

  let rec go i = function
  | Comb(Comb(Const("CONSPAT",_),p),a) ->
    let N = dest_word_ty (type_of a) in
    let n = Num.int_of_num (dest_finty N) in
    let m = power_num (Int 2) (Int n) in
    let i' = quo_num i m and a' = mod_num i m in
    let ls, th' = go i' p in
    let p',x = dest_comb (concl th') in let p' = rand p' in
    (match a with
    | Comb(Const("word1",_),a) ->
      let ls, b = match a with
      | Const("T",_) -> ls,true
      | Const("F",_) -> ls,false
      | Var(_,_) -> let b = a' = Int 1 in ((if b then T else F),a)::ls, b
      | _ -> failwith "inst_bitpat_numeral" in
      ls, PROVE_HYP th' (
        if b then INST [x,ex; p',ep] w1T
        else if i = Int 0 then INST [p',ep] w1F0
        else INST [x,ex; p',ep] w1F)
    | _ ->
      let thd = dim N in
      let n' = rand (rhs (concl thd)) in
      let ls, a = match a with
      | Comb(Const("word",_),Comb(Const("NUMERAL",_),a)) -> ls, a
      | Var(_,_) ->
        let n = mk_numeral a' in
        (mk_comb (mk_const ("word", [N,eN]), n), a) :: ls, rand n
      | _ -> failwith "inst_bitpat_numeral" in
      let thn = NUM_SHIFT_ADD_CORE a x n' in
      let e = rhs (concl thn) in
      ls, PROVE_HYP th' (PROVE_HYP thn (PROVE_HYP thd
        (INST [n',en; a,ea; x,ex; e,ei; p',ep] (INST_TYPE [N,eN] wS)))))
  | Const("NILPAT",_) -> [], w0
  | _ -> failwith "inst_bitpat_numeral" in

  let pth1 = SYM (SPEC en NUMERAL)
  and pth2 = (UNDISCH_ALL o prove)
   (`pat_size p = NUMERAL n ==> dimindex(:N) = NUMERAL n ==>
     pat_set p x ==> pat_set p (val (word (NUMERAL x):N word))`,
    DISCH_THEN (SUBST1_TAC o SYM) THEN REPEAT STRIP_TAC THEN
    REWRITE_TAC [NUMERAL] THEN IMP_REWRITE_TAC [VAL_WORD_EQ] THEN
    POP_ASSUM (ACCEPT_TAC o MATCH_MP pat_set_lt))
  and conv =
    let ps = `pat_size` in
    REWRITE_CONV [CONSPAT_pat_size; NILPAT_pat_size] THENC
    ONCE_DEPTH_CONV DIMINDEX_CONV THENC REDEPTH_CONV NUM_ADD_CONV o
    mk_comb o (fun tm -> (ps, tm)) in

  let check p i = match bitpat_matches p i with
  | None -> go i p
  | _ -> failwith "inst_bitpat_numeral: number does not match pattern" in

  function
  | Comb(Comb(Const("pat_set",_),p), Comb(Const("val",_), e)) ->
    let N' = dest_word_ty (type_of e) in
    let thd = dim N' in
    let pth2 = (PROVE_HYP thd o PROVE_HYP (conv p) o
       INST [p,ep; rand (rhs (concl thd)),en] o INST_TYPE [N',eN]) pth2 in
    fun i ->
      let ls, th = check p i in
      let e' = rand (concl th) in
      let th1 = PROVE_HYP th (INST ((e',ex)::ls) pth2) in
      (match e with
      | Var(_,_) -> (rand (rand (concl th1)),e)::ls, th1
      | _ when aconv e (rand (rand (concl th1))) -> ls, th1
      | _ -> failwith "inst_bitpat_numeral: pattern failed")
  | Comb(Comb(Const("pat_set",_),p), e) ->
    fun i ->
      let ls, th = check p i in
      let f, e' = dest_comb (concl th) in
      let th1 = INST [e',en] pth1 in
      let th2 = EQ_MP (AP_TERM f th1) th in
      (match e with
      | Var(_,_) -> (rhs (concl th1), e)::ls, th2
      | _ when aconv e (rhs (concl th1)) -> ls, th2
      | _ -> failwith "inst_bitpat_numeral: pattern failed")
  | _ -> failwith "inst_bitpat_numeral";;

let BITMATCH_CONV =
  fun tm -> match tm with
  | Comb(Comb(Const("_BITMATCH",_),
      Comb(Const("word",Tyapp(_,[_;Tyapp(_,[N])])),n)),_) when is_numeral n ->
    let A, tr = bm_build_pos_tree tm in
    let n = Num.int_of_num (dest_numeral n)
    and sz = Num.int_of_num (dest_finty N) in
    let a = Array.init sz (fun i -> Some (n land (1 lsl i) != 0)) in
    (match snd (snd (get_dt a tr)) with
    | th::_ ->
      let ps = hd (hyp th) in
      let ls, th' = inst_bitpat_numeral ps (Int n) in
      PROVE_HYP th' (INST ls th)
    | _ -> failwith "BITMATCH_CONV")
  | _ -> failwith "BITMATCH_CONV";;

let BITMATCH_SIMP_CONV asl =
  let pos,neg =
    let rec go = function
    | [] -> [],[]
    | th::ths ->
      let pos,neg = go ths in
      match concl th with
      | Comb(Const("~",_),c) when
        (match snd (strip_exists c) with
        | Comb(Comb(Const("pat_set",_),_),_) -> true
        | _ -> false) -> pos,th::neg
      | Comb(Comb(Const("pat_set",_),_),_) -> th::pos,neg
      | _ -> pos,neg in
    go asl in
  let rec conv = function
  | Comb(Comb(Const("_BITMATCH",_),_),Comb(Const("_ELSEPATTERN",_),_)) as tm ->
    PART_MATCH lhs BITMATCH_ELSEPATTERN tm
  | Comb(Comb(Const("_BITMATCH",_),
      Comb(Const("word",_),n)),_) as tm when is_numeral n -> BITMATCH_CONV tm
  | Comb(Comb(Const("_BITMATCH",_),e),
      Comb(Comb(Const("_SEQPATTERN",_),c),cs)) as tm ->
    let pat = mk_comb (rator (lhand (snd (strip_exists (body (body c))))),
        mk_comb (mk_const ("val", [dest_word_ty (type_of e),`:N`]), e)) in
    let vars = frees (lhand pat) in
    let rec check_pos = function
    | th::ths -> (try
        let _,ls,_ = term_unify vars pat (concl th) in
        PROVE_HYP th (INST ls (BM_FIRST_CASE_CONV tm))
      with Failure _ -> check_pos ths)
    | [] ->
      let rec check_neg = function
      | th::ths -> (try
          let pat' = snd (strip_exists (rand (concl th))) in
          let _,ls,_ = term_unify (frees (lhand pat')) pat' pat in
          let ath = INST ls (SPEC_ALL
            (PURE_REWRITE_RULE [NOT_EXISTS_THM] (ASSUME (concl th)))) in
          let th' = PROVE_HYP th (bm_skip_clause (K ath) tm) in
          TRANS th' (TRY_CONV conv (rhs (concl th')))
        with Failure _ -> check_neg ths)
      | [] ->
        let sz = Num.int_of_num (dest_finty (dest_word_ty (type_of e))) in
        let a = bm_analyze_pat sz pat in
        let rec check_disj = function
        | th::ths -> (try
            let h = concl th in
            let a' = bm_analyze_pat sz h in
            let r = ref None in
            Array.iteri (fun i x -> match x,a'.(i),!r with
              | Some b, Some c, None when b != c -> r := Some i
              | _ -> ()) a;
            let i = match !r with
            | Some i -> mk_numeral (Int i)
            | _ -> fail () in
            let th' = PROVE_HYP th (PROVE_HYP (pat_to_bit true i h)
              (bm_skip_clause (pat_to_bit false i) tm)) in
            TRANS th' (TRY_CONV conv (rhs (concl th')))
          with Failure _ -> check_disj ths)
        | [] -> failwith "BITMATCH_SIMP_CONV" in
        check_disj pos in
      check_neg neg in
    check_pos pos
  | _ -> failwith "BITMATCH_SIMP_CONV" in
  conv;;

let rec bitpat_irrefutable = function
| Comb(Comb(Const("CONSPAT",_),p),a) ->
  (match a with
  | Comb(Const("word1",_),Var(_,_)) -> bitpat_irrefutable p
  | Var(_,_) -> bitpat_irrefutable p
  | _ -> false)
| Const("NILPAT",_) -> true
| Abs(_,c) -> bitpat_irrefutable c
| Comb(Const("?",_),c) -> bitpat_irrefutable c
| Comb(Comb(Const("_UNGUARDED_PATTERN",_),c),_) -> bitpat_irrefutable c
| Comb(Comb(Const("pat_set",_),c),_) -> bitpat_irrefutable c
| _ -> failwith "bitpat_irrefutable";;

let bitpat_irrefutable_thm =
  let eN,ee,ee',en,em,ek = `:N`,`e:num`,`e:N word`,`n:num`,`m:num`,`k:num`
  and ep,dN,pl,_1 = `p:bitpat`,`dimindex(:N)`,`(+)`,`1`
  and e2n = `e DIV 2 EXP n`
  and pth,pth1 =
    let pth = prove
     (`dimindex(:N) = n ==> pat_set p (e DIV 2 EXP n) ==>
       pat_set (CONSPAT p (word e:N word)) e`,
      REWRITE_TAC [CONSPAT_pat_set] THEN DISCH_THEN (SUBST1_TAC o SYM) THEN
      DISCH_THEN (fun th -> EXISTS_TAC `e DIV 2 EXP dimindex(:N)` THEN
        REWRITE_TAC [th; VAL_WORD; ADD_SYM; MULT_SYM;
          GSYM (MATCH_MP DIVISION (SPEC `n:num` EXP_2_NE_0))])) in
    UNDISCH_ALL pth,
    (UNDISCH o prove)
     (`pat_set p (e DIV 2 EXP 1) ==> pat_set (CONSPAT p (word1 (ODD e))) e`,
      REWRITE_TAC [WORD1_ODD] THEN
      ACCEPT_TAC (MATCH_MP pth (DIMINDEX_CONV `dimindex(:1)`)))
  and pth0 = (UNDISCH o prove) (`e < 2 EXP 0 ==> pat_set NILPAT e`,
    REWRITE_TAC [EXP; ARITH_RULE `n < 1 <=> n = 0`; NILPAT_pat_set])
  and pthS = (UNDISCH_ALL o prove) (`n + m = k ==>
    e < 2 EXP k ==> e DIV 2 EXP n < 2 EXP m`,
    DISCH_THEN (SUBST1_TAC o SYM) THEN
    IMP_REWRITE_TAC [EXP_ADD; RDIV_LT_EQ; EXP_2_NE_0])
  and pthW = (UNDISCH_ALL o prove)
   (`dimindex(:N) = n ==> val (e:N word) < 2 EXP n`,
    DISCH_THEN (SUBST1_TAC o SYM) THEN REWRITE_TAC [VAL_BOUND]) in

  let rec build = function
  | Const("NILPAT",_),e -> [], INST [e,ee] pth0
  | Comb(Comb(Const("CONSPAT",_),p),v),e ->
    let pthS n m =
      let th = NUM_ADD_CONV (mk_comb (mk_comb (pl, n), m)) in
      PROVE_HYP th (INST [n,en; m,em; rhs (concl th),ek; e,ee] pthS) in
    (match v with
    | Comb(Const("word1",_),(Var(_,_) as v)) ->
      let ls, th = build (p, vsubst [e,ee; _1,en] e2n) in
      let th' = PROVE_HYP th (INST [lhand (concl th),ep; e,ee] pth1) in
      (rand (rand (lhand (concl th'))),v)::ls,
      PROVE_HYP (pthS _1 (rand (rand (hd (hyp th))))) th'
    | Var(_,ty) ->
      let N = dest_word_ty ty in
      let th1 = DIMINDEX_CONV (inst [N,eN] dN) in
      let n = rhs (concl th1) in
      let ls, th = build (p, vsubst [e,ee; n,en] e2n) in
      let th' = PROVE_HYP th1
        (INST [n,en; lhand (concl th),ep; e,ee] (INST_TYPE [N,eN] pth)) in
      let th' = PROVE_HYP th th' in
      (rand (lhand (concl th')),v)::ls,
      PROVE_HYP (pthS n (rand (rand (hd (hyp th))))) th'
    | _ -> failwith "bitpat_irrefutable_thm: not irrefutable")
  | _ -> failwith "bitpat_irrefutable_thm" in

  fun tm -> match snd (strip_exists tm) with
  | Comb(Comb(Const("pat_set",_),p),(Comb(Const("val",_),e') as e)) ->
    let ls,th = build (p, e) in
    let rec build_ex = function
    | Comb(Const("?",_),Abs(v,c)) as tm ->
      let e = rev_assoc v ls in
      EXISTS (tm, e) (build_ex (vsubst [e,v] c))
    | tm when aconv (concl th) tm -> th
    | _ -> failwith "bitpat_irrefutable_thm: nonlinear pattern" in
    let th = build_ex tm in
    let N = dest_word_ty (type_of e') in
    let th1 = DIMINDEX_CONV (inst [N,eN] dN) in
    let n = rhs (concl th1) in
    if aconv n (rand (rand (hd (hyp th)))) then
      PROVE_HYP (PROVE_HYP th1 (PINST [N,eN] [n,en; e',ee'] pthW)) th
    else failwith "bitpat_irrefutable_thm: incorrect bit length"
  | _ -> failwith "bitpat_irrefutable_thm";;

let ONLY_BITMATCH_CASES_THEN thltac = WITH_GOAL (fun w ->
  let e,cs =
    let f = function
    | Comb(Comb(Const("_BITMATCH",_),_),_) -> true
    | _ -> false in
    (rand F_F I) (dest_comb (find_term f w)) in
  let rec tac thl = function
  | Comb(Comb(Const("_SEQPATTERN",_),Abs(_,Abs(_,c))),cs) ->
    let rec go = function
    | Comb((Const("?",_) as f),Abs(z,c)) ->
      let tm, tac1 = go c in
      mk_comb (f, mk_abs (z, tm)), POP_ASSUM CHOOSE_TAC THEN tac1
    | tm ->
      mk_comb (rator (lhand tm),
        mk_comb (mk_const ("val", [dest_word_ty (type_of e),`:N`]), e)),
      ALL_TAC in
    let tm, tac1 = go c in
    if bitpat_irrefutable c then
      ASSUME_TAC (bitpat_irrefutable_thm tm) THEN
      tac1 THEN POP_ASSUM (fun th -> thltac (th::thl))
    else
      ASM_CASES_TAC tm THENL [
        tac1 THEN POP_ASSUM (fun th -> thltac (th::thl));
        POP_ASSUM (fun th -> tac (th::thl) cs)]
  | _ -> thltac thl in
  tac [] cs);;

let BITMATCH_ASM_CASES_TAC =
  ONLY_BITMATCH_CASES_THEN (fun thl ->
    CONV_TAC (TOP_SWEEP_CONV (BITMATCH_SIMP_CONV thl)) THEN
    MAP_EVERY ASSUME_TAC thl);;

let BITMATCH_CASES_TAC =
  ONLY_BITMATCH_CASES_THEN (CONV_TAC o
    TOP_SWEEP_CONV o BITMATCH_SIMP_CONV);;

(* (bm_seq_numeral `bitmatch e with ...` n) will
   return `word n` and `(bitmatch word n with ...) = res` where `res` is the
   appropriate match branch. Unlike BITMATCH_CONV this also works with matches
   with non-disjoint cases. *)
let bm_seq_numeral = function
| Comb((Comb(Const("_BITMATCH",_),e) as me),cs) ->
  let N = dest_word_ty (type_of e) in
  let sz = Num.int_of_num (dest_finty N) in
  let word = mk_const ("word", [N,`:N`]) in
  let rec mk_fun cs =
    let tm = mk_comb (me, cs) in
    let th = BM_FIRST_CASE_CONV tm in
    let inst e' th = if is_var e then INST [e',e] th else th in
    match cs with
    | Comb(Comb(Const("_SEQPATTERN",_),c),cs') ->
      let ps = hd (hyp th) in
      let pats = Array.init sz (fun i -> try
        Some (bm_skip_clause (pat_to_bit false (mk_numeral (Int i))) tm)
      with Failure _ -> None) in
      let f = mk_fun cs' in
      fun n e' ->
        (match bitpat_matches c n with
        | None ->
          let ls, th' = inst_bitpat_numeral ps n in
          PROVE_HYP th' (INST ls th)
        | Some i ->
          let Some th' = pats.(i) in
          let th1 = inst e' th' in
          let th2 = match hd (hyp th1) with
          | Comb(Const("~",_),p) -> EQF_ELIM (WORD_RED_CONV p)
          | p -> EQT_ELIM (WORD_RED_CONV p) in
          TRANS (PROVE_HYP th2 th1) (f n e'))
    | Comb(Const("_ELSEPATTERN",_),_) -> fun _ e' -> inst e' th
    | _ -> failwith "bm_seq_numeral" in
  let f = mk_fun cs in
  fun n -> let e = mk_comb (word, mk_numeral n) in e, f n e
| _ -> failwith "bm_seq_numeral";;

let BITMATCH_SEQ_CONV = function
| Comb(Comb(Const("_BITMATCH",_), Comb(Const("word",_),n)),_) as tm ->
  snd (bm_seq_numeral tm (dest_numeral n))
| _ -> failwith "BITMATCH_CONV";;