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proof-pile / formal /hol /100 /bernoulli.ml
Zhangir Azerbayev
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(* ========================================================================= *)
(* Bernoulli numbers and polynomials; sum of kth powers. *)
(* ========================================================================= *)
needs "Library/binomial.ml";;
needs "Library/analysis.ml";;
needs "Library/transc.ml";;
prioritize_real();;
(* ------------------------------------------------------------------------- *)
(* A couple of basic lemmas about new-style sums. *)
(* ------------------------------------------------------------------------- *)
let SUM_DIFFS = prove
(`!a m n. m <= n + 1 ==> sum(m..n) (\i. a(i + 1) - a(i)) = a(n + 1) - a(m)`,
GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN
REWRITE_TAC[SUM_CLAUSES_NUMSEG] THENL
[REWRITE_TAC[ARITH_RULE `m <= 0 + 1 <=> m = 0 \/ m = 1`] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[ARITH; ADD_CLAUSES; REAL_SUB_REFL];
SIMP_TAC[ARITH_RULE `m <= SUC n + 1 <=> m <= n + 1 \/ m = SUC n + 1`] THEN
STRIP_TAC THEN ASM_SIMP_TAC[ADD1] THENL [REAL_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[REAL_SUB_REFL; ARITH_RULE `~((n + 1) + 1 <= n + 1)`] THEN
MATCH_MP_TAC SUM_TRIV_NUMSEG THEN ARITH_TAC]);;
let DIFF_SUM = prove
(`!f f' a b.
(!k. a <= k /\ k <= b ==> ((\x. f x k) diffl f'(k)) x)
==> ((\x. sum(a..b) (f x)) diffl (sum(a..b) f')) x`,
REPLICATE_TAC 3 GEN_TAC THEN INDUCT_TAC THEN
REWRITE_TAC[SUM_CLAUSES_NUMSEG] THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[ARITH; DIFF_CONST; SUM_TRIV_NUMSEG;
ARITH_RULE `~(a <= SUC b) ==> b < a`] THEN
DISCH_TAC THEN MATCH_MP_TAC DIFF_ADD THEN
ASM_SIMP_TAC[LE_REFL; ARITH_RULE `k <= b ==> k <= SUC b`]);;
(* ------------------------------------------------------------------------- *)
(* Bernoulli numbers. *)
(* ------------------------------------------------------------------------- *)
let bernoulli = define
`(bernoulli 0 = &1) /\
(!n. bernoulli(SUC n) =
--sum(0..n) (\j. &(binom(n + 2,j)) * bernoulli j) / (&n + &2))`;;
(* ------------------------------------------------------------------------- *)
(* A slightly tidier-looking form of the recurrence. *)
(* ------------------------------------------------------------------------- *)
let BERNOULLI = prove
(`!n. sum(0..n) (\j. &(binom(n + 1,j)) * bernoulli j) =
if n = 0 then &1 else &0`,
INDUCT_TAC THEN
REWRITE_TAC[bernoulli; SUM_CLAUSES_NUMSEG; GSYM ADD1; ADD_CLAUSES; binom;
REAL_MUL_LID; LE_0; NOT_SUC] THEN
SIMP_TAC[BINOM_LT; ARITH_RULE `n < SUC n`; BINOM_REFL; REAL_ADD_LID] THEN
REWRITE_TAC[ADD_CLAUSES] THEN REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN
REWRITE_TAC[ARITH_RULE `SUC(SUC n) = n + 2`] THEN
MATCH_MP_TAC(REAL_FIELD `x = &n + &2 ==> s + x * --s / (&n + &2) = &0`) THEN
REWRITE_TAC[ADD1; BINOM_TOP_STEP_REAL; ARITH_RULE `~(n = n + 1)`] THEN
REWRITE_TAC[BINOM_REFL] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Bernoulli polynomials. *)
(* ------------------------------------------------------------------------- *)
let bernpoly = new_definition
`bernpoly n x = sum(0..n) (\k. &(binom(n,k)) * bernoulli k * x pow (n - k))`;;
(* ------------------------------------------------------------------------- *)
(* The key derivative recurrence. *)
(* ------------------------------------------------------------------------- *)
let DIFF_BERNPOLY = prove
(`!n x. ((bernpoly (SUC n)) diffl (&(SUC n) * bernpoly n x)) x`,
REPEAT GEN_TAC THEN
GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM ETA_AX] THEN
REWRITE_TAC[bernpoly; SUM_CLAUSES_NUMSEG; LE_0] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM REAL_ADD_RID] THEN
MATCH_MP_TAC DIFF_ADD THEN REWRITE_TAC[SUB_REFL; real_pow; DIFF_CONST] THEN
REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC DIFF_SUM THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[ADD1; BINOM_TOP_STEP_REAL] THEN
DIFF_TAC THEN ASM_SIMP_TAC[ARITH_RULE `k <= n ==> ~(k = n + 1)`] THEN
REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_LID] THEN
ASM_SIMP_TAC[ARITH_RULE `k <= n ==> (n + 1) - k - 1 = n - k`] THEN
ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; ARITH_RULE `k <= n ==> k <= n + 1`] THEN
UNDISCH_TAC `k <= n:num` THEN
REWRITE_TAC[GSYM REAL_OF_NUM_ADD; GSYM REAL_OF_NUM_LE] THEN
ABBREV_TAC `z = x pow (n - k)` THEN CONV_TAC REAL_FIELD);;
(* ------------------------------------------------------------------------- *)
(* Hence the key stepping recurrence. *)
(* ------------------------------------------------------------------------- *)
let INTEGRALS_EQ = prove
(`!f g. (!x. ((\x. f(x) - g(x)) diffl &0) x) /\ f(&0) = g(&0)
==> !x. f(x) = g(x)`,
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`\x:real. f(x) - g(x)`; `x:real`; `&0`] DIFF_ISCONST_ALL) THEN
ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let RECURRENCE_BERNPOLY = prove
(`!n x. bernpoly n (x + &1) - bernpoly n x = &n * x pow (n - 1)`,
INDUCT_TAC THENL
[REWRITE_TAC[bernpoly; SUM_SING_NUMSEG; REAL_SUB_REFL; SUB_REFL;
real_pow; REAL_MUL_LZERO];
ALL_TAC] THEN
MATCH_MP_TAC INTEGRALS_EQ THEN CONJ_TAC THENL
[X_GEN_TAC `x:real` THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real`) THEN
ONCE_REWRITE_TAC[GSYM REAL_SUB_0] THEN
DISCH_THEN(MP_TAC o AP_TERM `(*) (&(SUC n))`) THEN
REWRITE_TAC[REAL_MUL_RZERO] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[REAL_SUB_LDISTRIB] THEN
REPEAT(MATCH_MP_TAC DIFF_SUB THEN CONJ_TAC) THEN
SIMP_TAC[SUC_SUB1; DIFF_CMUL; DIFF_POW; DIFF_BERNPOLY; ETA_AX] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM REAL_MUL_RID] THEN
MATCH_MP_TAC DIFF_CHAIN THEN REWRITE_TAC[DIFF_BERNPOLY] THEN
DIFF_TAC THEN REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[bernpoly; GSYM SUM_SUB_NUMSEG] THEN
REWRITE_TAC[REAL_ADD_LID; REAL_POW_ONE; GSYM REAL_SUB_LDISTRIB] THEN
REWRITE_TAC[SUM_CLAUSES_NUMSEG; LE_0; SUB_REFL; real_pow] THEN
REWRITE_TAC[REAL_SUB_REFL; REAL_MUL_RZERO; REAL_ADD_RID] THEN
SIMP_TAC[ARITH_RULE `i <= n ==> SUC n - i = SUC(n - i)`] THEN
REWRITE_TAC[real_pow; REAL_MUL_LZERO; REAL_SUB_RZERO; REAL_MUL_RID] THEN
REWRITE_TAC[BERNOULLI; ADD1] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[ARITH; real_pow; REAL_MUL_LID] THEN
CONV_TAC SYM_CONV THEN REWRITE_TAC[REAL_ENTIRE; REAL_POW_EQ_0] THEN
ASM_REWRITE_TAC[ADD_SUB]);;
(* ------------------------------------------------------------------------- *)
(* Hence we get the main result. *)
(* ------------------------------------------------------------------------- *)
let SUM_OF_POWERS = prove
(`!n. sum(0..n) (\k. &k pow m) =
(bernpoly(SUC m) (&n + &1) - bernpoly(SUC m) (&0)) / (&m + &1)`,
GEN_TAC THEN ASM_SIMP_TAC[REAL_EQ_RDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
ONCE_REWRITE_TAC[GSYM REAL_MUL_SYM] THEN
REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
`sum(0..n) (\i. bernpoly (SUC m) (&(i + 1)) - bernpoly (SUC m) (&i))` THEN
CONJ_TAC THENL
[REWRITE_TAC[RECURRENCE_BERNPOLY; GSYM REAL_OF_NUM_ADD] THEN
REWRITE_TAC[GSYM REAL_OF_NUM_SUC; SUC_SUB1];
SIMP_TAC[SUM_DIFFS; LE_0] THEN REWRITE_TAC[REAL_OF_NUM_ADD]]);;
(* ------------------------------------------------------------------------- *)
(* Now explicit computations of the various terms on specific instances. *)
(* ------------------------------------------------------------------------- *)
let SUM_CONV =
let pth = prove
(`sum(0..0) f = f 0 /\ sum(0..SUC n) f = sum(0..n) f + f(SUC n)`,
SIMP_TAC[SUM_CLAUSES_NUMSEG; LE_0]) in
let econv_0 = GEN_REWRITE_CONV I [CONJUNCT1 pth]
and econv_1 = GEN_REWRITE_CONV I [CONJUNCT2 pth] in
let rec sconv tm =
(econv_0 ORELSEC
(LAND_CONV(RAND_CONV num_CONV) THENC econv_1 THENC
COMB2_CONV (RAND_CONV sconv) (RAND_CONV NUM_SUC_CONV))) tm in
sconv;;
let BINOM_CONV =
let pth = prove
(`a * b * x = FACT c ==> x = (FACT c) DIV (a * b)`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC DIV_UNIQ THEN EXISTS_TAC `0` THEN CONJ_TAC THENL
[POP_ASSUM MP_TAC THEN ARITH_TAC;
POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
SIMP_TAC[LT_NZ; MULT_ASSOC; MULT_CLAUSES] THEN
MESON_TAC[LT_NZ; FACT_LT]]) in
let match_pth = MATCH_MP pth
and binom_tm = `binom` in
fun tm ->
let bop,lr = dest_comb tm in
if bop <> binom_tm then failwith "BINOM_CONV" else
let l,r = dest_pair lr in
let n = dest_numeral l and k = dest_numeral r in
if n </ k then
let th = SPECL [l;r] BINOM_LT in
MP th (EQT_ELIM(NUM_LT_CONV(lhand(concl th))))
else
let d = n -/ k in
let th1 = match_pth(SPECL [mk_numeral d; r] BINOM_FACT) in
CONV_RULE NUM_REDUCE_CONV th1;;
let BERNOULLIS =
let th_0,th_1 = CONJ_PAIR bernoulli
and b_tm = `bernoulli` in
let conv_1 = GEN_REWRITE_CONV I [th_1] in
let rec bconv n =
if n <= 0 then [th_0] else
let bths = bconv (n - 1)
and tm = mk_comb(b_tm,mk_small_numeral n) in
(RAND_CONV num_CONV THENC conv_1 THENC
LAND_CONV(RAND_CONV SUM_CONV) THENC
ONCE_DEPTH_CONV BETA_CONV THENC
DEPTH_CONV(NUM_RED_CONV ORELSEC BINOM_CONV) THENC
GEN_REWRITE_CONV ONCE_DEPTH_CONV bths THENC
REAL_RAT_REDUCE_CONV) tm :: bths in
bconv;;
let BERNOULLI_CONV =
let b_tm = `bernoulli` in
fun tm -> let op,n = dest_comb tm in
if op <> b_tm || not(is_numeral n) then failwith "BERNOULLI_CONV"
else hd(BERNOULLIS(dest_small_numeral n));;
let BERNPOLY_CONV =
let conv_1 =
REWR_CONV bernpoly THENC SUM_CONV THENC
TOP_DEPTH_CONV BETA_CONV THENC NUM_REDUCE_CONV
and conv_3 =
ONCE_DEPTH_CONV BINOM_CONV THENC REAL_POLY_CONV in
fun tm ->
let n = dest_small_numeral(lhand tm) in
let conv_2 = GEN_REWRITE_CONV ONCE_DEPTH_CONV (BERNOULLIS n) in
(conv_1 THENC conv_2 THENC conv_3) tm;;
let SOP_CONV =
let pth = prove
(`sum(0..n) (\k. &k pow m) =
(\p. (p(&n + &1) - p(&0)) / (&m + &1))
(\x. bernpoly (SUC m) x)`,
REWRITE_TAC[SUM_OF_POWERS]) in
let conv_0 = REWR_CONV pth in
REWR_CONV pth THENC
RAND_CONV(ABS_CONV(LAND_CONV NUM_SUC_CONV THENC BERNPOLY_CONV)) THENC
TOP_DEPTH_CONV BETA_CONV THENC
REAL_POLY_CONV;;
let SOP_NUM_CONV =
let pth = prove
(`sum(0..n) (\k. &k pow p) = &m ==> nsum(0..n) (\k. k EXP p) = m`,
REWRITE_TAC[REAL_OF_NUM_POW; GSYM REAL_OF_NUM_SUM_NUMSEG;
REAL_OF_NUM_EQ]) in
let rule_1 = PART_MATCH (lhs o rand) pth in
fun tm ->
let th1 = rule_1 tm in
let th2 = SOP_CONV(lhs(lhand(concl th1))) in
MATCH_MP th1 th2;;
(* ------------------------------------------------------------------------- *)
(* The example Bernoulli bragged about. *)
(* ------------------------------------------------------------------------- *)
time SOP_NUM_CONV `nsum(0..1000) (\k. k EXP 10)`;;
(* ------------------------------------------------------------------------- *)
(* The general formulas for moderate powers. *)
(* ------------------------------------------------------------------------- *)
time SOP_CONV `sum(0..n) (\k. &k pow 0)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 1)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 2)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 3)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 4)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 5)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 6)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 7)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 8)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 9)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 10)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 11)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 12)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 13)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 14)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 15)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 16)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 17)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 18)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 19)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 20)`;;
time SOP_CONV `sum(0..n) (\k. &k pow 21)`;;