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calc_num.ml
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(* ========================================================================= *)
(* Calculation with naturals. *)
(* *)
(* John Harrison, University of Cambridge Computer Laboratory *)
(* *)
(* (c) Copyright, University of Cambridge 1998 *)
(* (c) Copyright, John Harrison 1998-2007 *)
(* (c) Copyright, Mario Carneiro 2020 *)
(* ========================================================================= *)
needs "wf.ml";;
(* ------------------------------------------------------------------------- *)
(* Simple rule to get rid of NUMERAL constant. *)
(* ------------------------------------------------------------------------- *)
let DENUMERAL = GEN_REWRITE_RULE DEPTH_CONV [NUMERAL];;
(* ------------------------------------------------------------------------- *)
(* Big collection of rewrites to do trivial arithmetic. *)
(* *)
(* Note that we have none for DIV and MOD, and that PRE and SUB are a bit *)
(* inefficient; log(n)^2 instead of log(n). *)
(* ------------------------------------------------------------------------- *)
let ARITH_ZERO = prove
(`(NUMERAL 0 = 0) /\
(BIT0 _0 = _0)`,
REWRITE_TAC[NUMERAL; BIT0; DENUMERAL ADD_CLAUSES]);;
let BIT0_0 = prove
(`BIT0 0 = 0`,
REWRITE_TAC [NUMERAL; ARITH_ZERO]);;
let BIT1_0 = prove
(`BIT1 0 = 1`,
REWRITE_TAC [NUMERAL]);;
let ARITH_SUC = prove
(`(!n. SUC(NUMERAL n) = NUMERAL(SUC n)) /\
(SUC _0 = BIT1 _0) /\
(!n. SUC (BIT0 n) = BIT1 n) /\
(!n. SUC (BIT1 n) = BIT0 (SUC n))`,
REWRITE_TAC[NUMERAL; BIT0; BIT1; DENUMERAL ADD_CLAUSES]);;
let ARITH_PRE = prove
(`(!n. PRE(NUMERAL n) = NUMERAL(PRE n)) /\
(PRE _0 = _0) /\
(!n. PRE(BIT0 n) = if n = _0 then _0 else BIT1 (PRE n)) /\
(!n. PRE(BIT1 n) = BIT0 n)`,
REWRITE_TAC[NUMERAL; BIT1; BIT0; DENUMERAL PRE] THEN INDUCT_TAC THEN
REWRITE_TAC[NUMERAL; DENUMERAL PRE; DENUMERAL ADD_CLAUSES; DENUMERAL NOT_SUC;
ARITH_ZERO]);;
let ARITH_ADD = prove
(`(!m n. NUMERAL(m) + NUMERAL(n) = NUMERAL(m + n)) /\
(_0 + _0 = _0) /\
(!n. _0 + BIT0 n = BIT0 n) /\
(!n. _0 + BIT1 n = BIT1 n) /\
(!n. BIT0 n + _0 = BIT0 n) /\
(!n. BIT1 n + _0 = BIT1 n) /\
(!m n. BIT0 m + BIT0 n = BIT0 (m + n)) /\
(!m n. BIT0 m + BIT1 n = BIT1 (m + n)) /\
(!m n. BIT1 m + BIT0 n = BIT1 (m + n)) /\
(!m n. BIT1 m + BIT1 n = BIT0 (SUC(m + n)))`,
PURE_REWRITE_TAC[NUMERAL; BIT0; BIT1; DENUMERAL ADD_CLAUSES; SUC_INJ] THEN
REWRITE_TAC[ADD_AC]);;
let ARITH_MULT = prove
(`(!m n. NUMERAL(m) * NUMERAL(n) = NUMERAL(m * n)) /\
(_0 * _0 = _0) /\
(!n. _0 * BIT0 n = _0) /\
(!n. _0 * BIT1 n = _0) /\
(!n. BIT0 n * _0 = _0) /\
(!n. BIT1 n * _0 = _0) /\
(!m n. BIT0 m * BIT0 n = BIT0 (BIT0 (m * n))) /\
(!m n. BIT0 m * BIT1 n = BIT0 m + BIT0 (BIT0 (m * n))) /\
(!m n. BIT1 m * BIT0 n = BIT0 n + BIT0 (BIT0 (m * n))) /\
(!m n. BIT1 m * BIT1 n = BIT1 m + BIT0 n + BIT0 (BIT0 (m * n)))`,
PURE_REWRITE_TAC[NUMERAL; BIT0; BIT1; DENUMERAL MULT_CLAUSES;
DENUMERAL ADD_CLAUSES; SUC_INJ] THEN
REWRITE_TAC[LEFT_ADD_DISTRIB; RIGHT_ADD_DISTRIB; ADD_AC]);;
let ARITH_EXP = prove
(`(!m n. (NUMERAL m) EXP (NUMERAL n) = NUMERAL(m EXP n)) /\
(_0 EXP _0 = BIT1 _0) /\
(!m. (BIT0 m) EXP _0 = BIT1 _0) /\
(!m. (BIT1 m) EXP _0 = BIT1 _0) /\
(!n. _0 EXP (BIT0 n) = (_0 EXP n) * (_0 EXP n)) /\
(!m n. (BIT0 m) EXP (BIT0 n) = ((BIT0 m) EXP n) * ((BIT0 m) EXP n)) /\
(!m n. (BIT1 m) EXP (BIT0 n) = ((BIT1 m) EXP n) * ((BIT1 m) EXP n)) /\
(!n. _0 EXP (BIT1 n) = _0) /\
(!m n. (BIT0 m) EXP (BIT1 n) =
BIT0 m * ((BIT0 m) EXP n) * ((BIT0 m) EXP n)) /\
(!m n. (BIT1 m) EXP (BIT1 n) =
BIT1 m * ((BIT1 m) EXP n) * ((BIT1 m) EXP n))`,
REWRITE_TAC[NUMERAL] THEN REPEAT STRIP_TAC THEN
TRY(GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [BIT0; BIT1]) THEN
REWRITE_TAC[DENUMERAL EXP; DENUMERAL MULT_CLAUSES; EXP_ADD]);;
let ARITH_EVEN = prove
(`(!n. EVEN(NUMERAL n) <=> EVEN n) /\
(EVEN _0 <=> T) /\
(!n. EVEN(BIT0 n) <=> T) /\
(!n. EVEN(BIT1 n) <=> F)`,
REWRITE_TAC[NUMERAL; BIT1; BIT0; DENUMERAL EVEN; EVEN_ADD]);;
let ARITH_ODD = prove
(`(!n. ODD(NUMERAL n) <=> ODD n) /\
(ODD _0 <=> F) /\
(!n. ODD(BIT0 n) <=> F) /\
(!n. ODD(BIT1 n) <=> T)`,
REWRITE_TAC[NUMERAL; BIT1; BIT0; DENUMERAL ODD; ODD_ADD]);;
let ARITH_LE = prove
(`(!m n. NUMERAL m <= NUMERAL n <=> m <= n) /\
((_0 <= _0) <=> T) /\
(!n. (BIT0 n <= _0) <=> n <= _0) /\
(!n. (BIT1 n <= _0) <=> F) /\
(!n. (_0 <= BIT0 n) <=> T) /\
(!n. (_0 <= BIT1 n) <=> T) /\
(!m n. (BIT0 m <= BIT0 n) <=> m <= n) /\
(!m n. (BIT0 m <= BIT1 n) <=> m <= n) /\
(!m n. (BIT1 m <= BIT0 n) <=> m < n) /\
(!m n. (BIT1 m <= BIT1 n) <=> m <= n)`,
REWRITE_TAC[NUMERAL; BIT1; BIT0; DENUMERAL NOT_SUC;
DENUMERAL(GSYM NOT_SUC); SUC_INJ] THEN
REWRITE_TAC[DENUMERAL LE_0] THEN REWRITE_TAC[DENUMERAL LE; GSYM MULT_2] THEN
REWRITE_TAC[LE_MULT_LCANCEL; SUC_INJ;
DENUMERAL MULT_EQ_0; DENUMERAL NOT_SUC] THEN
REWRITE_TAC[DENUMERAL NOT_SUC] THEN REWRITE_TAC[LE_SUC_LT] THEN
REWRITE_TAC[LT_MULT_LCANCEL] THEN
SUBGOAL_THEN `2 = SUC 1` (fun th -> REWRITE_TAC[th]) THENL
[REWRITE_TAC[NUMERAL; BIT0; BIT1; DENUMERAL ADD_CLAUSES];
REWRITE_TAC[DENUMERAL NOT_SUC; NOT_SUC; EQ_MULT_LCANCEL] THEN
REWRITE_TAC[ONCE_REWRITE_RULE[DISJ_SYM] LE_LT] THEN
MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN
SUBGOAL_THEN `~(SUC 1 * m = SUC (SUC 1 * n))`
(fun th -> REWRITE_TAC[th]) THEN
DISCH_THEN(MP_TAC o AP_TERM `EVEN`) THEN
REWRITE_TAC[EVEN_MULT; EVEN_ADD; NUMERAL; BIT1; EVEN]]);;
let ARITH_LT = prove
(`(!m n. NUMERAL m < NUMERAL n <=> m < n) /\
((_0 < _0) <=> F) /\
(!n. (BIT0 n < _0) <=> F) /\
(!n. (BIT1 n < _0) <=> F) /\
(!n. (_0 < BIT0 n) <=> _0 < n) /\
(!n. (_0 < BIT1 n) <=> T) /\
(!m n. (BIT0 m < BIT0 n) <=> m < n) /\
(!m n. (BIT0 m < BIT1 n) <=> m <= n) /\
(!m n. (BIT1 m < BIT0 n) <=> m < n) /\
(!m n. (BIT1 m < BIT1 n) <=> m < n)`,
REWRITE_TAC[NUMERAL; GSYM NOT_LE; ARITH_LE] THEN
REWRITE_TAC[DENUMERAL LE]);;
let ARITH_GE = REWRITE_RULE[GSYM GE; GSYM GT] ARITH_LE;;
let ARITH_GT = REWRITE_RULE[GSYM GE; GSYM GT] ARITH_LT;;
let ARITH_EQ = prove
(`(!m n. (NUMERAL m = NUMERAL n) <=> (m = n)) /\
((_0 = _0) <=> T) /\
(!n. (BIT0 n = _0) <=> (n = _0)) /\
(!n. (BIT1 n = _0) <=> F) /\
(!n. (_0 = BIT0 n) <=> (_0 = n)) /\
(!n. (_0 = BIT1 n) <=> F) /\
(!m n. (BIT0 m = BIT0 n) <=> (m = n)) /\
(!m n. (BIT0 m = BIT1 n) <=> F) /\
(!m n. (BIT1 m = BIT0 n) <=> F) /\
(!m n. (BIT1 m = BIT1 n) <=> (m = n))`,
REWRITE_TAC[NUMERAL; GSYM LE_ANTISYM; ARITH_LE] THEN
REWRITE_TAC[LET_ANTISYM; LTE_ANTISYM; DENUMERAL LE_0]);;
let ARITH_SUB = prove
(`(!m n. NUMERAL m - NUMERAL n = NUMERAL(m - n)) /\
(_0 - _0 = _0) /\
(!n. _0 - BIT0 n = _0) /\
(!n. _0 - BIT1 n = _0) /\
(!n. BIT0 n - _0 = BIT0 n) /\
(!n. BIT1 n - _0 = BIT1 n) /\
(!m n. BIT0 m - BIT0 n = BIT0 (m - n)) /\
(!m n. BIT0 m - BIT1 n = PRE(BIT0 (m - n))) /\
(!m n. BIT1 m - BIT0 n = if n <= m then BIT1 (m - n) else _0) /\
(!m n. BIT1 m - BIT1 n = BIT0 (m - n))`,
REWRITE_TAC[NUMERAL; DENUMERAL SUB_0] THEN PURE_REWRITE_TAC[BIT0; BIT1] THEN
REWRITE_TAC[GSYM MULT_2; SUB_SUC; LEFT_SUB_DISTRIB] THEN
REWRITE_TAC[SUB] THEN REPEAT GEN_TAC THEN COND_CASES_TAC THEN
REWRITE_TAC[DENUMERAL SUB_EQ_0] THEN
RULE_ASSUM_TAC(REWRITE_RULE[NOT_LE]) THEN
ASM_REWRITE_TAC[LE_SUC_LT; LT_MULT_LCANCEL; ARITH_EQ] THEN
POP_ASSUM(CHOOSE_THEN SUBST1_TAC o REWRITE_RULE[LE_EXISTS]) THEN
REWRITE_TAC[ADD1; LEFT_ADD_DISTRIB] THEN
REWRITE_TAC[ADD_SUB2; GSYM ADD_ASSOC]);;
let ARITH = end_itlist CONJ
[ARITH_ZERO; ARITH_SUC; ARITH_PRE;
ARITH_ADD; ARITH_MULT; ARITH_EXP;
ARITH_EVEN; ARITH_ODD;
ARITH_EQ; ARITH_LE; ARITH_LT; ARITH_GE; ARITH_GT;
ARITH_SUB];;
let EXP_2_NE_0 = prove
(`!n. ~(2 EXP n = 0)`,
REWRITE_TAC [EXP_EQ_0; ARITH_EQ]);;
(* ------------------------------------------------------------------------- *)
(* Now more delicate conversions for situations where efficiency matters. *)
(* ------------------------------------------------------------------------- *)
let NUM_EVEN_CONV =
let tth,rths = CONJ_PAIR ARITH_EVEN in
GEN_REWRITE_CONV I [tth] THENC GEN_REWRITE_CONV I [rths];;
let NUM_ODD_CONV =
let tth,rths = CONJ_PAIR ARITH_ODD in
GEN_REWRITE_CONV I [tth] THENC GEN_REWRITE_CONV I [rths];;
let NUM_SUC_CONV,NUM_ADD_CONV,NUM_MULT_CONV,NUM_EXP_CONV,
NUM_LT_CONV,NUM_LE_CONV,NUM_EQ_CONV =
let num_ty = type_of(lhand(concl ZERO_DEF)) in
let Comb(NUMERAL_tm,Comb(BIT0_tm,Comb(BIT1_tm,zero_tm))) =
mk_small_numeral 2
and suc_tm = rator(rand(concl TWO))
and one_tm = rand(mk_small_numeral 1)
and add_tm = rator(rator(lhand(snd(strip_forall(concl ADD_0)))))
and mul_tm = rator(rator(rand(snd(strip_forall(concl EXP_2)))))
and exp_tm = rator(rator(lhand(snd(strip_forall(concl EXP_2)))))
and eq_tm = rator(rator(concl TWO)) in
let num_0 = Int 0 and num_1 = Int 1 and num_2 = Int 2 in
let a_tm = mk_var("a",num_ty)
and b_tm = mk_var("b",num_ty)
and c_tm = mk_var("c",num_ty)
and d_tm = mk_var("d",num_ty)
and e_tm = mk_var("e",num_ty)
and h_tm = mk_var("h",num_ty)
and l_tm = mk_var("l",num_ty)
and m_tm = mk_var("m",num_ty)
and n_tm = mk_var("n",num_ty)
and p_tm = mk_var("p",num_ty) in
let STANDARDIZE =
let ilist = [BIT0_tm,BIT0_tm; BIT1_tm,BIT1_tm; zero_tm,zero_tm;
suc_tm,suc_tm; add_tm,add_tm; mul_tm,mul_tm;
exp_tm,exp_tm; eq_tm,eq_tm; NUMERAL_tm,NUMERAL_tm;
a_tm,a_tm; b_tm,b_tm; c_tm,c_tm; d_tm,d_tm; e_tm,e_tm;
h_tm,h_tm; l_tm,l_tm; m_tm,m_tm; n_tm,n_tm; p_tm,p_tm] in
let rec replace tm =
match tm with
Var(_,_) | Const(_,_) -> rev_assocd tm ilist tm
| Comb(s,t) -> mk_comb(replace s,replace t)
| Abs(_,_) -> failwith "replace" in
fun th -> let tm' = replace (concl th) in EQ_MP (REFL tm') th in
let REFL_bit0 = STANDARDIZE(REFL BIT0_tm)
and REFL_bit1 = STANDARDIZE(REFL BIT1_tm) in
let AP_BIT0 th = MK_COMB(REFL_bit0,th)
and AP_BIT1 th = MK_COMB(REFL_bit1,th)
and QUICK_PROVE_HYP ath bth = EQ_MP (DEDUCT_ANTISYM_RULE ath bth) ath in
let rec dest_raw_numeral tm =
match tm with
Comb(Const("BIT1",_),t) -> num_2 */ dest_raw_numeral t +/ num_1
| Comb(Const("BIT0",_),t) -> num_2 */ dest_raw_numeral t
| Const("_0",_) -> num_0 in
let bitcounts =
let rec bctr w z tm =
match tm with
Const("_0",_) -> (w,z)
| Comb(Const("BIT0",_),t) -> bctr w (z + 1) t
| Comb(Const("BIT1",_),t) -> bctr (w + 1) z t
| _ -> failwith "malformed numeral" in
bctr 0 0 in
let rec wellformed tm =
match tm with
Const("_0",_) -> true
| Comb(Const("BIT0",_),t)|Comb(Const("BIT1",_),t) -> wellformed t
| _ -> false in
let rec orderrelation mtm ntm =
if mtm == ntm then
if wellformed mtm then 0 else failwith "orderrelation"
else
match (mtm,ntm) with
Const("_0",_),Const("_0",_) -> 0
| Const("_0",_),_ ->
if wellformed ntm then -1 else failwith "orderrelation"
| _, Const("_0",_) ->
if wellformed ntm then 1 else failwith "orderrelation"
| Comb(Const("BIT0",_),mt),Comb(Const("BIT0",_),nt)
| Comb(Const("BIT1",_),mt),Comb(Const("BIT1",_),nt) ->
orderrelation mt nt
| Comb(Const("BIT0",_),mt),Comb(Const("BIT1",_),nt) ->
if orderrelation mt nt > 0 then 1 else -1
| Comb(Const("BIT1",_),mt),Comb(Const("BIT0",_),nt) ->
if orderrelation mt nt < 0 then -1 else 1 in
let doublebn tm = if tm = zero_tm then tm else mk_comb(BIT0_tm,tm) in
let rec subbn mtm ntm =
match (mtm,ntm) with
(_,Const("_0",_)) -> mtm
| (Comb(Const("BIT0",_),mt),Comb(Const("BIT0",_),nt)) ->
doublebn (subbn mt nt)
| (Comb(Const("BIT1",_),mt),Comb(Const("BIT1",_),nt)) ->
doublebn (subbn mt nt)
| (Comb(Const("BIT1",_),mt),Comb(Const("BIT0",_),nt)) ->
mk_comb(BIT1_tm,subbn mt nt)
| (Comb(Const("BIT0",_),mt),Comb(Const("BIT1",_),nt)) ->
mk_comb(BIT1_tm,sbcbn mt nt)
| _ -> failwith "malformed numeral or wrong relation"
and sbcbn mtm ntm =
match (mtm,ntm) with
| (Comb(Const("BIT0",_),mt),Const("_0",_)) ->
mk_comb(BIT1_tm,sbcbn mt ntm)
| (Comb(Const("BIT1",_),mt),Const("_0",_)) ->
doublebn mt
| (Comb(Const("BIT0",_),mt),Comb(Const("BIT0",_),nt)) ->
mk_comb(BIT1_tm,sbcbn mt nt)
| (Comb(Const("BIT1",_),mt),Comb(Const("BIT1",_),nt)) ->
mk_comb(BIT1_tm,sbcbn mt nt)
| (Comb(Const("BIT1",_),mt),Comb(Const("BIT0",_),nt)) ->
doublebn (subbn mt nt)
| (Comb(Const("BIT0",_),mt),Comb(Const("BIT1",_),nt)) ->
doublebn (sbcbn mt nt)
| _ -> failwith "malformed numeral or wrong relation" in
let topsplit tm =
match tm with
Const("_0",_) -> 0,zero_tm
| Comb(Const("BIT1",_),Const("_0",_)) -> 1,zero_tm
| Comb(Const("BIT0",_),Comb(Const("BIT1",_),Const("_0",_))) -> 2,zero_tm
| Comb(Const("BIT1",_),Comb(Const("BIT1",_),Const("_0",_))) -> 3,zero_tm
| Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),Const("_0",_)))) -> 4,zero_tm
| Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),Const("_0",_)))) -> 5,zero_tm
| Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),Const("_0",_)))) -> 6,zero_tm
| Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),Const("_0",_)))) -> 7,zero_tm
| Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT0",_),n)))) -> 0,n
| Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT0",_),n)))) -> 1,n
| Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT0",_),n)))) -> 2,n
| Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT0",_),n)))) -> 3,n
| Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT0",_),n)))) -> 4,n
| Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT0",_),n)))) -> 5,n
| Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT0",_),n)))) -> 6,n
| Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT0",_),n)))) -> 7,n
| Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),n)))) -> 8,n
| Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),n)))) -> 9,n
| Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),n)))) -> 10,n
| Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),n)))) -> 11,n
| Comb(Const("BIT0",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),n)))) -> 12,n
| Comb(Const("BIT1",_),Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),n)))) -> 13,n
| Comb(Const("BIT0",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),n)))) -> 14,n
| Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),Comb(Const("BIT1",_),n)))) -> 15,n
| _ -> failwith "malformed numeral" in
let NUM_ADD_RULE,NUM_ADC_RULE =
let rec mk_compnumeral k base =
if k = 0 then base else
let t = mk_compnumeral (k / 2) base in
if k mod 2 = 1 then mk_comb(BIT1_tm,t) else mk_comb(BIT0_tm,t) in
let bases v =
let part2 = map (fun k -> mk_compnumeral k v) (8--15) in
let part1 = map (subst[mk_comb(BIT0_tm,v),mk_comb(BIT1_tm,v)])
part2
and part0 = map (fun k -> mk_compnumeral k zero_tm) (0--15) in
part0 @ part1 @ part2 in
let starts =
allpairs (fun mtm ntm ->
mk_comb(mk_comb(add_tm,mtm),ntm)) (bases m_tm) (bases n_tm) in
let BITS_INJ = (STANDARDIZE o prove)
(`(BIT0 m = BIT0 n <=> m = n) /\
(BIT1 m = BIT1 n <=> m = n)`,
REWRITE_TAC[BIT0; BIT1] THEN
REWRITE_TAC[GSYM MULT_2] THEN
REWRITE_TAC[SUC_INJ; EQ_MULT_LCANCEL; ARITH_EQ]) in
let ARITH_0 = (STANDARDIZE o MESON[NUMERAL; ADD_CLAUSES])
`m + _0 = m /\ _0 + n = n` in
let patadj = subst[`SUC(m + _0)`,`SUC m`; `SUC(_0 + n)`,`SUC n`] in
let mkclauses sucflag t =
let tm = if sucflag then mk_comb(suc_tm,t) else t in
let th1 = PURE_REWRITE_CONV[ARITH_ADD; ARITH_SUC; ARITH_0] tm in
let tm1 = patadj(rand(concl th1)) in
if not(free_in add_tm tm1) then th1,
(if free_in m_tm tm1 then 0 else 1) else
let ptm = rand(rand(rand(rand tm1))) in
let tmc = mk_eq(mk_eq(ptm,p_tm),mk_eq(tm,subst[p_tm,ptm] tm1)) in
EQT_ELIM(REWRITE_CONV[ARITH_ADD; ARITH_SUC; ARITH_0; BITS_INJ] tmc),
(if free_in suc_tm tm1 then 3 else 2) in
let add_clauses,add_flags =
let l1,l2 = unzip(map (mkclauses false) starts) in
Array.fromList(map STANDARDIZE l1),Array.fromList l2 in
let adc_clauses,adc_flags =
let l1,l2 = unzip(map (mkclauses true) starts) in
Array.fromList(map STANDARDIZE l1),Array.fromList l2 in
let rec NUM_ADD_RULE mtm ntm =
let m_lo,m_hi = topsplit mtm
and n_lo,n_hi = topsplit ntm in
let m_ind = if m_hi = zero_tm then m_lo else m_lo + 16
and n_ind = if n_hi = zero_tm then n_lo else n_lo + 16 in
let ind = 32 * m_ind + n_ind in
let th1 = Array.sub add_clauses ind
and fl = Array.sub add_flags ind in
match fl with
0 -> INST [m_hi,m_tm] th1
| 1 -> INST [n_hi,n_tm] th1
| 2 -> let th2 = NUM_ADD_RULE m_hi n_hi in
(match concl th2 with Comb(_,ptm) ->
let th3 = INST [m_hi,m_tm; n_hi,n_tm;ptm,p_tm] th1 in
EQ_MP th3 th2)
| 3 -> let th2 = NUM_ADC_RULE m_hi n_hi in
(match concl th2 with Comb(_,ptm) ->
let th3 = INST [m_hi,m_tm; n_hi,n_tm;ptm,p_tm] th1 in
EQ_MP th3 th2)
and NUM_ADC_RULE mtm ntm =
let m_lo,m_hi = topsplit mtm
and n_lo,n_hi = topsplit ntm in
let m_ind = if m_hi = zero_tm then m_lo else m_lo + 16
and n_ind = if n_hi = zero_tm then n_lo else n_lo + 16 in
let ind = 32 * m_ind + n_ind in
let th1 = Array.sub adc_clauses ind
and fl = Array.sub adc_flags ind in
match fl with
0 -> INST [m_hi,m_tm] th1
| 1 -> INST [n_hi,n_tm] th1
| 2 -> let th2 = NUM_ADD_RULE m_hi n_hi in
(match concl th2 with Comb(_,ptm) ->
let th3 = INST [m_hi,m_tm; n_hi,n_tm;ptm,p_tm] th1 in
EQ_MP th3 th2)
| 3 -> let th2 = NUM_ADC_RULE m_hi n_hi in
(match concl th2 with Comb(_,ptm) ->
let th3 = INST [m_hi,m_tm; n_hi,n_tm;ptm,p_tm] th1 in
EQ_MP th3 th2) in
NUM_ADD_RULE,NUM_ADC_RULE in
let NUM_SHIFT_CONV =
let pth_0 = (STANDARDIZE o prove)
(`(n = a + p * b <=> BIT0 n = BIT0 a + BIT0 p * b)`,
REWRITE_TAC[BIT0; BIT1] THEN
REWRITE_TAC[GSYM MULT_2; GSYM MULT_ASSOC; GSYM LEFT_ADD_DISTRIB] THEN
REWRITE_TAC[EQ_MULT_LCANCEL; ARITH_EQ])
and pth_z = (STANDARDIZE o prove)
(`n = _0 + p * b <=> BIT0 n = _0 + BIT0 p * b`,
SUBST1_TAC(SYM(SPEC `_0` NUMERAL)) THEN
REWRITE_TAC[BIT1; BIT0] THEN
REWRITE_TAC[ADD_CLAUSES; GSYM MULT_2] THEN
REWRITE_TAC[GSYM MULT_ASSOC; EQ_MULT_LCANCEL; ARITH_EQ])
and pth_1 = (STANDARDIZE o prove)
(`(n = a + p * b <=> BIT1 n = BIT1 a + BIT0 p * b)`,
REWRITE_TAC[BIT0; BIT1] THEN
REWRITE_TAC[GSYM MULT_2; GSYM MULT_ASSOC; GSYM LEFT_ADD_DISTRIB;
ADD_CLAUSES; SUC_INJ] THEN
REWRITE_TAC[EQ_MULT_LCANCEL; ARITH_EQ])
and pth_base = (STANDARDIZE o prove)
(`n = _0 + BIT1 _0 * n`,
MESON_TAC[ADD_CLAUSES; MULT_CLAUSES; NUMERAL])
and pth_triv = (STANDARDIZE o prove)
(`_0 = a + p * b <=> _0 = a + BIT0 p * b`,
CONV_TAC(BINOP_CONV SYM_CONV) THEN
SUBST1_TAC(SYM(SPEC `_0` NUMERAL)) THEN
REWRITE_TAC[ADD_EQ_0; MULT_EQ_0; BIT0])
and pths_1 = (Array.fromList o CONJUNCTS o STANDARDIZE o prove)
(`(n = a + p * b <=>
BIT0(BIT0(BIT0(BIT0 n))) =
BIT0(BIT0(BIT0(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT1(BIT0(BIT0(BIT0 n))) =
BIT1(BIT0(BIT0(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT0(BIT1(BIT0(BIT0 n))) =
BIT0(BIT1(BIT0(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT1(BIT1(BIT0(BIT0 n))) =
BIT1(BIT1(BIT0(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT0(BIT0(BIT1(BIT0 n))) =
BIT0(BIT0(BIT1(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT1(BIT0(BIT1(BIT0 n))) =
BIT1(BIT0(BIT1(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT0(BIT1(BIT1(BIT0 n))) =
BIT0(BIT1(BIT1(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT1(BIT1(BIT1(BIT0 n))) =
BIT1(BIT1(BIT1(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT0(BIT0(BIT0(BIT1 n))) =
BIT0(BIT0(BIT0(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT1(BIT0(BIT0(BIT1 n))) =
BIT1(BIT0(BIT0(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT0(BIT1(BIT0(BIT1 n))) =
BIT0(BIT1(BIT0(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT1(BIT1(BIT0(BIT1 n))) =
BIT1(BIT1(BIT0(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT0(BIT0(BIT1(BIT1 n))) =
BIT0(BIT0(BIT1(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT1(BIT0(BIT1(BIT1 n))) =
BIT1(BIT0(BIT1(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT0(BIT1(BIT1(BIT1 n))) =
BIT0(BIT1(BIT1(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = a + p * b <=>
BIT1(BIT1(BIT1(BIT1 n))) =
BIT1(BIT1(BIT1(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b)`,
MP_TAC(REWRITE_RULE[GSYM MULT_2] BIT0) THEN
MP_TAC(REWRITE_RULE[GSYM MULT_2] BIT1) THEN
ABBREV_TAC `two = 2` THEN
DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN
DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
REWRITE_TAC[ADD_CLAUSES; SUC_INJ; EQ_MULT_LCANCEL; ARITH_EQ;
GSYM LEFT_ADD_DISTRIB; GSYM MULT_ASSOC])
and pths_0 = (Array.fromList o CONJUNCTS o STANDARDIZE o prove)
(`(n = _0 + p * b <=>
BIT0(BIT0(BIT0(BIT0 n))) =
_0 + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT1(BIT0(BIT0(BIT0 n))) =
BIT1 _0 + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT0(BIT1(BIT0(BIT0 n))) =
BIT0(BIT1 _0) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT1(BIT1(BIT0(BIT0 n))) =
BIT1(BIT1 _0) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT0(BIT0(BIT1(BIT0 n))) =
BIT0(BIT0(BIT1 _0)) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT1(BIT0(BIT1(BIT0 n))) =
BIT1(BIT0(BIT1 _0)) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT0(BIT1(BIT1(BIT0 n))) =
BIT0(BIT1(BIT1 _0)) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT1(BIT1(BIT1(BIT0 n))) =
BIT1(BIT1(BIT1 _0)) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT0(BIT0(BIT0(BIT1 n))) =
BIT0(BIT0(BIT0(BIT1 _0))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT1(BIT0(BIT0(BIT1 n))) =
BIT1(BIT0(BIT0(BIT1 _0))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT0(BIT1(BIT0(BIT1 n))) =
BIT0(BIT1(BIT0(BIT1 _0))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT1(BIT1(BIT0(BIT1 n))) =
BIT1(BIT1(BIT0(BIT1 _0))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT0(BIT0(BIT1(BIT1 n))) =
BIT0(BIT0(BIT1(BIT1 _0))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT1(BIT0(BIT1(BIT1 n))) =
BIT1(BIT0(BIT1(BIT1 _0))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT0(BIT1(BIT1(BIT1 n))) =
BIT0(BIT1(BIT1(BIT1 _0))) + BIT0(BIT0(BIT0(BIT0 p))) * b) /\
(n = _0 + p * b <=>
BIT1(BIT1(BIT1(BIT1 n))) =
BIT1(BIT1(BIT1(BIT1 _0))) + BIT0(BIT0(BIT0(BIT0 p))) * b)`,
SUBST1_TAC(MESON[NUMERAL] `_0 = 0`) THEN
MP_TAC(REWRITE_RULE[GSYM MULT_2] BIT0) THEN
MP_TAC(REWRITE_RULE[GSYM MULT_2] BIT1) THEN
ABBREV_TAC `two = 2` THEN
DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN
DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
REWRITE_TAC[ADD_CLAUSES; SUC_INJ; EQ_MULT_LCANCEL; ARITH_EQ;
GSYM LEFT_ADD_DISTRIB; GSYM MULT_ASSOC]) in
let rec NUM_SHIFT_CONV k tm =
if k <= 0 then INST [tm,n_tm] pth_base else
match tm with
Comb(_,Comb(_,Comb(_,Comb(_,_)))) when k >= 4 ->
let i,ntm = topsplit tm in
let th1 = NUM_SHIFT_CONV (k - 4) ntm in
(match concl th1 with
Comb(_,Comb(Comb(_,Const("_0",_)),Comb(Comb(_,ptm),btm))) ->
let th2 = Array.sub pths_0 i in
let th3 = INST [ntm,n_tm; btm,b_tm; ptm,p_tm] th2 in
EQ_MP th3 th1
| Comb(_,Comb(Comb(_,atm),Comb(Comb(_,ptm),btm))) ->
let th2 = Array.sub pths_1 i in
let th3 = INST[ntm,n_tm; atm,a_tm; btm,b_tm; ptm,p_tm] th2 in
EQ_MP th3 th1)
| Comb(Const("BIT0",_),ntm) ->
let th1 = NUM_SHIFT_CONV (k - 1) ntm in
(match concl th1 with
Comb(_,Comb(Comb(_,Const("_0",_)),Comb(Comb(_,ptm),btm))) ->
EQ_MP (INST [ntm,n_tm; btm,b_tm; ptm,p_tm] pth_z) th1
| Comb(_,Comb(Comb(_,atm),Comb(Comb(_,ptm),btm))) ->
EQ_MP
(INST[ntm,n_tm; atm,a_tm; btm,b_tm; ptm,p_tm] pth_0) th1)
| Comb(Const("BIT1",_),ntm) ->
let th1 = NUM_SHIFT_CONV (k - 1) ntm in
(match concl th1 with
Comb(_,Comb(Comb(_,atm),Comb(Comb(_,ptm),btm))) ->
EQ_MP
(INST [ntm,n_tm; atm,a_tm; btm,b_tm; ptm,p_tm] pth_1) th1)
| Const("_0",_) ->
let th1 = NUM_SHIFT_CONV (k - 1) tm in
(match concl th1 with
Comb(_,Comb(Comb(_,atm),Comb(Comb(_,ptm),btm))) ->
EQ_MP (INST [atm,a_tm; btm,b_tm; ptm,p_tm] pth_triv)
th1)
| _ -> failwith "malformed numeral" in
NUM_SHIFT_CONV in
let NUM_UNSHIFT_CONV =
let pth_triv = (STANDARDIZE o prove)
(`a + p * _0 = a`,
SUBST1_TAC(SYM(SPEC `_0` NUMERAL)) THEN
REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES])
and pth_base = (STANDARDIZE o prove)
(`a + BIT1 _0 * b = a + b`,
SUBST1_TAC(SYM(SPEC `BIT1 _0` NUMERAL)) THEN
REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES])
and pth_0 = (STANDARDIZE o prove)
(`BIT0 a + BIT0 p * b = BIT0(a + p * b)`,
REWRITE_TAC[BIT0] THEN REWRITE_TAC[GSYM MULT_2] THEN
REWRITE_TAC[GSYM MULT_ASSOC; GSYM LEFT_ADD_DISTRIB])
and pth_1 = (STANDARDIZE o prove)
(`BIT1 a + BIT0 p * b = BIT1(a + p * b)`,
REWRITE_TAC[BIT0; BIT1] THEN REWRITE_TAC[GSYM MULT_2] THEN
REWRITE_TAC[ADD_CLAUSES; SUC_INJ] THEN
REWRITE_TAC[GSYM MULT_ASSOC; GSYM LEFT_ADD_DISTRIB] THEN
REWRITE_TAC[EQ_MULT_LCANCEL; ARITH_EQ])
and pth_z = (STANDARDIZE o prove)
(`_0 + BIT0 p * b = BIT0(_0 + p * b)`,
SUBST1_TAC(SYM(SPEC `_0` NUMERAL)) THEN
REWRITE_TAC[BIT1; BIT0] THEN REWRITE_TAC[ADD_CLAUSES] THEN
REWRITE_TAC[RIGHT_ADD_DISTRIB])
and puths_1 = (Array.fromList o CONJUNCTS o STANDARDIZE o prove)
(`(a + p * b = n <=>
BIT0(BIT0(BIT0(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT0(BIT0(BIT0(BIT0 n)))) /\
(a + p * b = n <=>
BIT1(BIT0(BIT0(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT1(BIT0(BIT0(BIT0 n)))) /\
(a + p * b = n <=>
BIT0(BIT1(BIT0(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT0(BIT1(BIT0(BIT0 n)))) /\
(a + p * b = n <=>
BIT1(BIT1(BIT0(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT1(BIT1(BIT0(BIT0 n)))) /\
(a + p * b = n <=>
BIT0(BIT0(BIT1(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT0(BIT0(BIT1(BIT0 n)))) /\
(a + p * b = n <=>
BIT1(BIT0(BIT1(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT1(BIT0(BIT1(BIT0 n)))) /\
(a + p * b = n <=>
BIT0(BIT1(BIT1(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT0(BIT1(BIT1(BIT0 n)))) /\
(a + p * b = n <=>
BIT1(BIT1(BIT1(BIT0 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT1(BIT1(BIT1(BIT0 n)))) /\
(a + p * b = n <=>
BIT0(BIT0(BIT0(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT0(BIT0(BIT0(BIT1 n)))) /\
(a + p * b = n <=>
BIT1(BIT0(BIT0(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT1(BIT0(BIT0(BIT1 n)))) /\
(a + p * b = n <=>
BIT0(BIT1(BIT0(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT0(BIT1(BIT0(BIT1 n)))) /\
(a + p * b = n <=>
BIT1(BIT1(BIT0(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT1(BIT1(BIT0(BIT1 n)))) /\
(a + p * b = n <=>
BIT0(BIT0(BIT1(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT0(BIT0(BIT1(BIT1 n)))) /\
(a + p * b = n <=>
BIT1(BIT0(BIT1(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT1(BIT0(BIT1(BIT1 n)))) /\
(a + p * b = n <=>
BIT0(BIT1(BIT1(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT0(BIT1(BIT1(BIT1 n)))) /\
(a + p * b = n <=>
BIT1(BIT1(BIT1(BIT1 a))) + BIT0(BIT0(BIT0(BIT0 p))) * b =
BIT1(BIT1(BIT1(BIT1 n))))`,
SUBST1_TAC(MESON[NUMERAL] `_0 = 0`) THEN
MP_TAC(REWRITE_RULE[GSYM MULT_2] BIT0) THEN
MP_TAC(REWRITE_RULE[GSYM MULT_2] BIT1) THEN
ABBREV_TAC `two = 2` THEN
DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN
DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
REWRITE_TAC[ADD_CLAUSES; SUC_INJ; EQ_MULT_LCANCEL; ARITH_EQ;
GSYM LEFT_ADD_DISTRIB; GSYM MULT_ASSOC]) in
let puths_2 = Array.fromList
(map (fun i -> let th1 = Array.sub puths_1 (i mod 16)
and th2 = Array.sub puths_1 (i / 16) in
let th3 = GEN_REWRITE_RULE RAND_CONV [th1] th2 in
STANDARDIZE th3) (0--255)) in
let rec NUM_UNSHIFT_CONV tm =
match tm with
Comb(Comb(Const("+",_),atm),Comb(Comb(Const("*",_),ptm),btm)) ->
(match (atm,ptm,btm) with
(_,_,Const("_0",_)) ->
INST [atm,a_tm; ptm,p_tm] pth_triv
| (_,Comb(Const("BIT1",_),Const("_0",_)),_) ->
let th1 = INST [atm,a_tm; btm,b_tm] pth_base in
let Comb(_,Comb(Comb(_,mtm),ntm)) = concl th1 in
TRANS th1 (NUM_ADD_RULE mtm ntm)
| (Comb(_,Comb(_,Comb(_,Comb(_,atm')))),
Comb(_,Comb(_,Comb(_,Comb(_,(Comb(_,_) as ptm'))))),_) ->
let i,_ = topsplit atm in
(match (atm',ptm') with
(Comb(_,Comb(_,Comb(_,Comb(_,atm'')))),
Comb(_,Comb(_,Comb(_,Comb(_,(Comb(_,_) as ptm'')))))) ->
let j,_ = topsplit atm' in
let tm' = mk_comb(mk_comb(add_tm,atm''),
mk_comb(mk_comb(mul_tm,ptm''),btm)) in
let th1 = NUM_UNSHIFT_CONV tm' in
let th2 = INST [atm'',a_tm; ptm'',p_tm; btm,b_tm;
rand(concl th1),n_tm]
(Array.sub puths_2 (16 * j + i)) in
EQ_MP th2 th1
| _ ->
let tm' = mk_comb(mk_comb(add_tm,atm'),
mk_comb(mk_comb(mul_tm,ptm'),btm)) in
let th1 = NUM_UNSHIFT_CONV tm' in
let th2 = INST [atm',a_tm; ptm',p_tm; btm,b_tm;
rand(concl th1),n_tm]
(Array.sub puths_1 i) in
EQ_MP th2 th1)
| (Const("_0",_),Comb(Const("BIT0",_),qtm),_) ->
let th1 = INST [btm,b_tm; qtm,p_tm] pth_z in
CONV_RULE(RAND_CONV(RAND_CONV NUM_UNSHIFT_CONV)) th1
| (Comb(Const("BIT0",_),ctm),Comb(Const("BIT0",_),qtm),_) ->
let th1 = INST [ctm,a_tm; btm,b_tm; qtm,p_tm] pth_0 in
CONV_RULE(RAND_CONV(RAND_CONV NUM_UNSHIFT_CONV)) th1
| (Comb(Const("BIT1",_),ctm),Comb(Const("BIT0",_),qtm),_) ->
let th1 = INST [ctm,a_tm; btm,b_tm; qtm,p_tm] pth_1 in
CONV_RULE(RAND_CONV(RAND_CONV NUM_UNSHIFT_CONV)) th1
| _ -> failwith "malformed numeral")
| _ -> failwith "malformed numeral" in
NUM_UNSHIFT_CONV in
let NUM_SQUARE_RULE =
let pth_0 = (STANDARDIZE o prove)
(`_0 EXP 2 = _0`,
MESON_TAC[NUMERAL; REWRITE_CONV[ARITH] `0 EXP 2`])
and pth_1 = (STANDARDIZE o prove)
(`(BIT1 _0) EXP 2 = BIT1 _0`,
MESON_TAC[NUMERAL; REWRITE_CONV[ARITH] `1 EXP 2`])
and pth_even = (STANDARDIZE o prove)
(`m EXP 2 = n <=> (BIT0 m) EXP 2 = BIT0(BIT0 n)`,
ABBREV_TAC `two = 2` THEN
REWRITE_TAC[BIT0] THEN EXPAND_TAC "two" THEN
REWRITE_TAC[GSYM MULT_2] THEN REWRITE_TAC[EXP_2] THEN
REWRITE_TAC[AC MULT_AC `(2 * m) * (2 * n) = 2 * 2 * m * n`] THEN
REWRITE_TAC[EQ_MULT_LCANCEL; ARITH_EQ])
and pth_odd = (STANDARDIZE o prove)
(`m EXP 2 = n <=> (BIT1 m) EXP 2 = BIT1(BIT0(m + n))`,
ABBREV_TAC `two = 2` THEN
REWRITE_TAC[NUMERAL; BIT0; BIT1] THEN
EXPAND_TAC "two" THEN REWRITE_TAC[GSYM MULT_2] THEN
REWRITE_TAC[EXP_2; MULT_CLAUSES; ADD_CLAUSES] THEN
REWRITE_TAC[SUC_INJ; GSYM MULT_ASSOC; GSYM LEFT_ADD_DISTRIB] THEN
REWRITE_TAC[AC ADD_AC `(m + m * 2 * m) + m = m * 2 * m + m + m`] THEN
REWRITE_TAC[GSYM MULT_2; AC MULT_AC `m * 2 * m = 2 * m * m`] THEN
REWRITE_TAC[GSYM MULT_ASSOC; GSYM LEFT_ADD_DISTRIB] THEN
REWRITE_TAC[EQ_MULT_LCANCEL; ARITH_EQ] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [ADD_SYM] THEN
REWRITE_TAC[EQ_ADD_RCANCEL])
and pth_qstep = (UNDISCH o STANDARDIZE o prove)
(`n + BIT1 _0 = m /\
m EXP 2 = p /\
m + a = BIT0(BIT0 p)
==> (BIT1(BIT1(BIT1 n))) EXP 2 = BIT1(BIT0(BIT0(BIT0 a)))`,
ABBREV_TAC `two = 2` THEN
SUBST1_TAC(MESON[NUMERAL] `_0 = 0`) THEN
REWRITE_TAC[BIT1; BIT0] THEN EXPAND_TAC "two" THEN
REWRITE_TAC[GSYM MULT_2] THEN
REWRITE_TAC[ADD1; LEFT_ADD_DISTRIB; GSYM ADD_ASSOC] THEN
REWRITE_TAC[MULT_ASSOC] THEN REWRITE_TAC[ARITH] THEN
REWRITE_TAC[IMP_CONJ] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN DISCH_TAC THEN
MATCH_MP_TAC(MESON[EQ_ADD_LCANCEL]
`!m:num. m + n = m + p ==> n = p`) THEN
EXISTS_TAC `16 * (n + 1)` THEN
ASM_REWRITE_TAC[ADD_ASSOC; GSYM LEFT_ADD_DISTRIB] THEN
EXPAND_TAC "two" THEN REWRITE_TAC[EXP_2] THEN
REWRITE_TAC[LEFT_ADD_DISTRIB; RIGHT_ADD_DISTRIB] THEN
REWRITE_TAC[MULT_CLAUSES; MULT_ASSOC] THEN
REWRITE_TAC[AC MULT_AC `(8 * n) * NUMERAL p = (8 * NUMERAL p) * n`] THEN
REWRITE_TAC[ARITH] THEN
REWRITE_TAC[AC ADD_AC
`(n + 16) + p + q + 49 = (n + p + q) + (16 + 49)`] THEN
REWRITE_TAC[GSYM ADD_ASSOC] THEN REWRITE_TAC[ARITH] THEN
REWRITE_TAC[ADD_ASSOC; EQ_ADD_RCANCEL] THEN
REWRITE_TAC[GSYM ADD_ASSOC; GSYM MULT_2; MULT_ASSOC] THEN
ONCE_REWRITE_TAC[AC ADD_AC `a + b + c:num = b + a + c`] THEN
REWRITE_TAC[GSYM RIGHT_ADD_DISTRIB] THEN
REWRITE_TAC[ARITH])
and pth_rec = (UNDISCH o STANDARDIZE o prove)
(`n = l + p * h /\
h + l = m /\
h EXP 2 = a /\
l EXP 2 = c /\
m EXP 2 = d /\
a + c = e /\
e + b = d
==> n EXP 2 = c + p * (b + p * a)`,
REWRITE_TAC[IMP_CONJ] THEN
DISCH_THEN SUBST1_TAC THEN
REPLICATE_TAC 5 (DISCH_THEN(SUBST1_TAC o SYM)) THEN
REWRITE_TAC[EXP_2; LEFT_ADD_DISTRIB; RIGHT_ADD_DISTRIB] THEN
REWRITE_TAC[MULT_AC] THEN CONV_TAC(BINOP_CONV NUM_CANCEL_CONV) THEN
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[RIGHT_ADD_DISTRIB] THEN
REWRITE_TAC[MULT_AC] THEN REWRITE_TAC[ADD_AC])
and pth_toom3 = (STANDARDIZE o prove)
(`h EXP 2 = e /\
l EXP 2 = a /\
(l + BIT1 _0 * (m + BIT1 _0 * h)) EXP 2 =
a + BIT1 _0 * (b + BIT1 _0 * (c + BIT1 _0 * (d + BIT1 _0 * e))) /\
(l + BIT0(BIT1 _0) * (m + BIT0(BIT1 _0) * h)) EXP 2 =
a + BIT0(BIT1 _0) * (b + BIT0(BIT1 _0) *
(c + BIT0(BIT1 _0) * (d + BIT0(BIT1 _0) * e))) /\
(h + BIT0(BIT1 _0) * (m + BIT0(BIT1 _0) * l)) EXP 2 =
e + BIT0(BIT1 _0) * (d + BIT0(BIT1 _0) *
(c + BIT0(BIT1 _0) * (b + BIT0(BIT1 _0) * a)))
==> (l + p * (m + p * h)) EXP 2 =
a + p * (b + p * (c + p * (d + p * e)))`,
ABBREV_TAC `two = 2` THEN
SUBST1_TAC(MESON[NUMERAL] `_0 = 0`) THEN
REWRITE_TAC[BIT1; BIT0] THEN
EXPAND_TAC "two" THEN REWRITE_TAC[GSYM MULT_2] THEN
REWRITE_TAC[ARITH] THEN
SUBGOAL_THEN
`!p x y z. (x + p * (y + p * z)) EXP 2 =
x * x + p * (2 * x * y + p * ((2 * x * z + y * y) +
p * (2 * y * z + p * z * z)))`
(fun th -> REWRITE_TAC[th])
THENL
[REWRITE_TAC[EXP_2; MULT_2; LEFT_ADD_DISTRIB; RIGHT_ADD_DISTRIB] THEN
REWRITE_TAC[MULT_AC] THEN REWRITE_TAC[ADD_AC];
REWRITE_TAC[EXP_2]] THEN
MAP_EVERY ABBREV_TAC
[`a':num = l * l`; `b' = 2 * l * m`; `c' = 2 * l * h + m * m`;
`d' = 2 * m * h`; `e':num = h * h`] THEN
SUBST1_TAC(AC MULT_AC `2 * m * l = 2 * l * m`) THEN
SUBST1_TAC(AC MULT_AC `2 * h * l = 2 * l * h`) THEN
SUBST1_TAC(AC MULT_AC `2 * h * m = 2 * m * h`) THEN
ASM_REWRITE_TAC[] THEN EXPAND_TAC "two" THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
ASM_CASES_TAC `a':num = a` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `e':num = e` THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
REWRITE_TAC[EQ_ADD_LCANCEL; EQ_MULT_LCANCEL] THEN
REWRITE_TAC[LEFT_ADD_DISTRIB; MULT_ASSOC] THEN
REWRITE_TAC[ARITH] THEN
REWRITE_TAC[MULT_CLAUSES; EQ_ADD_LCANCEL] THEN
REWRITE_TAC[ADD_ASSOC; EQ_ADD_RCANCEL] THEN
REWRITE_TAC[GSYM ADD_ASSOC] THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (MESON[]
`b = b' /\ c = c' /\ d = d'
==> 5 * b + c' + d' = 5 * b' + c + d`)) THEN
REWRITE_TAC[LEFT_ADD_DISTRIB; MULT_ASSOC] THEN
REWRITE_TAC(map (fun k ->
SYM(REWRITE_CONV[ARITH_SUC]
(mk_comb(suc_tm,mk_small_numeral(k - 1)))))
(1--5)) THEN
REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES] THEN
CONV_TAC(LAND_CONV NUM_CANCEL_CONV) THEN DISCH_THEN SUBST_ALL_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (MESON[]
`b = b' /\ c = c' /\ d = d'
==> b + d':num = b' + d /\ 4 * b + d' = 4 * b' + d`)) THEN
REWRITE_TAC[LEFT_ADD_DISTRIB; MULT_ASSOC] THEN
REWRITE_TAC(map (fun k ->
SYM(REWRITE_CONV[ARITH_SUC]
(mk_comb(suc_tm,mk_small_numeral(k - 1)))))
(1--4)) THEN
REWRITE_TAC[MULT_CLAUSES; ADD_CLAUSES] THEN
CONV_TAC(LAND_CONV(BINOP_CONV NUM_CANCEL_CONV)) THEN
REWRITE_TAC[GSYM MULT_2] THEN ONCE_REWRITE_TAC[ADD_SYM] THEN
REWRITE_TAC[GSYM(el 4 (CONJUNCTS MULT_CLAUSES))] THEN
SIMP_TAC[EQ_MULT_LCANCEL; NOT_SUC])
and pth_even3 = (STANDARDIZE o prove)
(`m EXP 2 = n <=>
(BIT0(BIT0(BIT0 m))) EXP 2 = BIT0(BIT0(BIT0(BIT0(BIT0(BIT0 n)))))`,
ABBREV_TAC `two = 2` THEN
REWRITE_TAC[BIT0] THEN REWRITE_TAC[GSYM MULT_2] THEN
EXPAND_TAC "two" THEN REWRITE_TAC[EXP_2] THEN
REWRITE_TAC[AC MULT_AC
`(2 * 2 * 2 * m) * 2 * 2 * 2 * m = 2 * 2 * 2 * 2 * 2 * 2 * m * m`] THEN
REWRITE_TAC[EQ_MULT_LCANCEL; ARITH_EQ]) in
let NUM_UNSHIFT2_CONV =
RAND_CONV(RAND_CONV NUM_UNSHIFT_CONV) THENC NUM_UNSHIFT_CONV in
let NUM_UNSHIFT3_CONV =
RAND_CONV(RAND_CONV NUM_UNSHIFT2_CONV) THENC NUM_UNSHIFT_CONV in
let NUM_UNSHIFT4_CONV =
RAND_CONV(RAND_CONV NUM_UNSHIFT3_CONV) THENC NUM_UNSHIFT_CONV in
let BINOP2_CONV conv1 conv2 = COMB2_CONV (RAND_CONV conv1) conv2 in
let TOOM3_CONV = BINOP2_CONV
(LAND_CONV NUM_UNSHIFT2_CONV) NUM_UNSHIFT4_CONV in
let rec GEN_NUM_SQUARE_RULE w z tm =
match tm with
Const("_0",_) -> pth_0
| Comb(Const("BIT0",_),mtm) ->
(match mtm with
Comb(Const("BIT0",_),Comb(Const("BIT0",_),ptm)) ->
let th1 = GEN_NUM_SQUARE_RULE w (z - 3) ptm in
let ntm = rand(concl th1) in
EQ_MP (INST [ptm,m_tm; ntm,n_tm] pth_even3) th1
| _ ->
let th1 = GEN_NUM_SQUARE_RULE w (z - 1) mtm in
let ntm = rand(concl th1) in
EQ_MP (INST [mtm,m_tm; ntm,n_tm] pth_even) th1)
| Comb(Const("BIT1",_),mtm) ->
if mtm = zero_tm then pth_1 else
if (w < 100 || z < 20) && w + z < 150 then
match mtm with
Comb(Const("BIT1",_),Comb(Const("BIT1",_),ntm)) ->
let th1 = NUM_ADD_RULE ntm one_tm in
let mtm = rand(concl th1) in
let th2 = NUM_SQUARE_RULE mtm in
let ptm = rand(concl th2) in
let atm = subbn
(mk_comb(BIT0_tm,mk_comb(BIT0_tm,ptm))) mtm in
let th3 = NUM_ADD_RULE mtm atm in
let th4 = INST
[atm,a_tm; mtm,m_tm; ntm,n_tm; ptm,p_tm] pth_qstep in
QUICK_PROVE_HYP (CONJ th1 (CONJ th2 th3)) th4
| _ ->
let th1 = GEN_NUM_SQUARE_RULE (w - 1) z mtm in
let ntm = rand(concl th1) in
let th2 = EQ_MP (INST [mtm,m_tm; ntm,n_tm] pth_odd) th1 in
(match concl th2 with
Comb(_,Comb(_,Comb(_,Comb(Comb(_,ptm),qtm)))) ->
let th3 = NUM_ADD_RULE ptm qtm in
TRANS th2 (AP_BIT1 (AP_BIT0 th3)))
else if w + z < 800 then
let k2 = (w + z) / 2 in
let th1 = NUM_SHIFT_CONV k2 tm in
let Comb(Comb(_,ltm),Comb(Comb(_,ptm),htm)) = rand(concl th1) in
let th2 = NUM_ADD_RULE htm ltm in
let mtm = rand(concl th2) in
let th3 = NUM_SQUARE_RULE htm
and th4 = NUM_SQUARE_RULE ltm
and th5 = NUM_SQUARE_RULE mtm in
let atm = rand(concl th3)
and ctm = rand(concl th4)
and dtm = rand(concl th5) in
let th6 = NUM_ADD_RULE atm ctm in
let etm = rand(concl th6) in
let btm = subbn dtm etm in
let th7 = NUM_ADD_RULE etm btm in
let dtm = rand(concl th7) in
let th8 = INST [atm,a_tm; btm,b_tm; ctm,c_tm; dtm,d_tm; etm,e_tm;
htm,h_tm; ltm,l_tm; mtm,m_tm; tm,n_tm; ptm,p_tm]
pth_rec in
let th9 = QUICK_PROVE_HYP (end_itlist CONJ
[th1;th2;th3;th4;th5;th6;th7]) th8 in
CONV_RULE(RAND_CONV(RAND_CONV(RAND_CONV NUM_UNSHIFT_CONV) THENC
NUM_UNSHIFT_CONV)) th9
else
let k3 = (w + z) / 3 in
let th0 = (NUM_SHIFT_CONV k3 THENC
RAND_CONV(RAND_CONV(NUM_SHIFT_CONV k3))) tm in
let Comb(Comb(_,ltm),Comb(Comb(_,ptm),
Comb(Comb(_,mtm),Comb(Comb(_,_),htm)))) = rand(concl th0) in
let th1 = NUM_SQUARE_RULE htm
and th2 = NUM_SQUARE_RULE ltm in
let atm = rand(concl th2) and etm = rand(concl th1) in
let lnum = dest_raw_numeral ltm
and mnum = dest_raw_numeral mtm
and hnum = dest_raw_numeral htm in
let btm = rand(mk_numeral(num_2 */ lnum */ mnum))
and ctm = rand(mk_numeral(mnum */ mnum +/ num_2 */ lnum */ hnum))
and dtm = rand(mk_numeral(num_2 */ hnum */ mnum)) in
let th = INST
[atm,a_tm; btm,b_tm; ctm,c_tm; dtm,d_tm; etm,e_tm;
htm,h_tm; mtm,m_tm; ltm,l_tm; ptm,p_tm] pth_toom3 in
let th' = CONV_RULE
(BINOP2_CONV
(RAND_CONV(RAND_CONV
(BINOP2_CONV TOOM3_CONV (BINOP2_CONV TOOM3_CONV TOOM3_CONV))))
TOOM3_CONV) th in
let [tm3;tm4;tm5] = conjuncts(rand(rand(lhand(concl th')))) in
let th3 = NUM_SQUARE_RULE (lhand(lhand tm3))
and th4 = NUM_SQUARE_RULE (lhand(lhand tm4))
and th5 = NUM_SQUARE_RULE (lhand(lhand tm5)) in
MP th' (end_itlist CONJ [th1;th2;th3;th4;th5])
and NUM_SQUARE_RULE tm =
let w,z = bitcounts tm in GEN_NUM_SQUARE_RULE w z tm in
NUM_SQUARE_RULE in
let NUM_MUL_RULE =
let QUICK_PROVE_HYP ath bth =
EQ_MP (DEDUCT_ANTISYM_RULE ath bth) ath
and pth_0l,pth_0r = (CONJ_PAIR o STANDARDIZE o prove)
(`_0 * n = _0 /\ m * _0 = _0`,
MESON_TAC[NUMERAL; MULT_CLAUSES])
and pth_1l,pth_1r = (CONJ_PAIR o STANDARDIZE o prove)
(`(BIT1 _0) * n = n /\ m * (BIT1 _0) = m`,
MESON_TAC[NUMERAL; MULT_CLAUSES])
and pth_evenl,pth_evenr = (CONJ_PAIR o STANDARDIZE o prove)
(`(m * n = p <=> (BIT0 m) * n = BIT0 p) /\
(m * n = p <=> m * BIT0 n = BIT0 p)`,
REWRITE_TAC[BIT0] THEN REWRITE_TAC[GSYM MULT_2] THEN
REWRITE_TAC[AC MULT_AC `m * 2 * n = 2 * m * n`] THEN
REWRITE_TAC[GSYM MULT_ASSOC; EQ_MULT_LCANCEL; ARITH_EQ])
and pth_oddl,pth_oddr = (CONJ_PAIR o STANDARDIZE o prove)