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| (* ========================================================================= *) | |
| (* Fermat, weak Euler and Euler-Jacobi pseudoprimes, Carmichael numbers etc. *) | |
| (* ========================================================================= *) | |
| needs "Library/jacobi.ml";; | |
| needs "Examples/miller_rabin.ml";; | |
| (* ------------------------------------------------------------------------- *) | |
| (* A little set cardinality lemma we use repeatedly. In an explicitly group *) | |
| (* theoretic setting, Lagrange's theorem takes the place of this. *) | |
| (* ------------------------------------------------------------------------- *) | |
| let CARD_SUBSET_HALF_LEMMA = prove | |
| (`!f s (t:A->bool) n. | |
| FINITE t /\ CARD t <= n /\ | |
| s SUBSET t /\ IMAGE f s SUBSET t DIFF s /\ | |
| (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) | |
| ==> 2 * CARD s <= n`, | |
| REPEAT STRIP_TAC THEN | |
| MATCH_MP_TAC(ARITH_RULE | |
| `!f t. CARD t <= n /\ | |
| CARD(IMAGE f s) <= CARD(t DIFF s) /\ | |
| CARD(t DIFF s) + CARD(s) = CARD t /\ | |
| CARD(IMAGE (f:A->A) s) = CARD s | |
| ==> 2 * CARD s <= n`) THEN | |
| MAP_EVERY EXISTS_TAC [`f:A->A`; `t:A->bool`] THEN | |
| ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL | |
| [MATCH_MP_TAC CARD_SUBSET THEN ASM_SIMP_TAC[FINITE_DIFF] THEN ASM SET_TAC[]; | |
| MATCH_MP_TAC CARD_UNION_EQ THEN ASM SET_TAC[]; | |
| MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_MESON_TAC[FINITE_SUBSET]]);; | |
| (* ------------------------------------------------------------------------- *) | |
| (* Fermat pseudoprimes and Carmichael numbers. *) | |
| (* ------------------------------------------------------------------------- *) | |
| let fermat_pseudoprime = new_definition | |
| `fermat_pseudoprime a n <=> | |
| (a EXP (n - 1) == 1) (mod n)`;; | |
| let carmichael_number = new_definition | |
| `carmichael_number n <=> | |
| 2 <= n /\ ~prime n /\ !a. coprime(a,n) ==> fermat_pseudoprime a n`;; | |
| let CARMICHAEL_NUMBER,CARMICHAEL_NUMBER_KORSELT_ALT = (CONJ_PAIR o prove) | |
| (`(!n. carmichael_number n <=> | |
| 2 <= n /\ ~prime n /\ !a. (a EXP n == a) (mod n)) /\ | |
| (!n. carmichael_number n <=> | |
| 2 <= n /\ ~prime n /\ ODD n /\ | |
| !p. prime p /\ p divides n | |
| ==> ~(p EXP 2 divides n) /\ (p - 1) divides (n - 1))`, | |
| REWRITE_TAC[carmichael_number; fermat_pseudoprime; AND_FORALL_THM] THEN | |
| X_GEN_TAC `n:num` THEN | |
| ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[ARITH] THEN MATCH_MP_TAC(TAUT | |
| `(q ==> p) /\ (p ==> r) /\ (r ==> q) ==> (p <=> q) /\ (p <=> r)`) THEN | |
| REPEAT CONJ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL | |
| [GEN_TAC THEN MATCH_MP_TAC(NUMBER_RULE | |
| `(a * x == a) (mod n) ==> coprime(a,n) ==> (x == 1) (mod n)`) THEN | |
| ASM_SIMP_TAC[GSYM(CONJUNCT2 EXP); ARITH_RULE `2 <= n ==> SUC(n - 1) = n`]; | |
| MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL | |
| [REWRITE_TAC[GSYM NOT_EVEN] THEN REPEAT STRIP_TAC THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN ANTS_TAC THENL | |
| [MATCH_MP_TAC COPRIME_MINUS1 THEN ASM_ARITH_TAC; | |
| REWRITE_TAC[fermat_pseudoprime]] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (NUMBER_RULE | |
| `(a EXP k == 1) (mod n) ==> (a * a EXP k == a) (mod n)`)) THEN | |
| ASM_SIMP_TAC[GSYM(CONJUNCT2 EXP); ARITH_RULE | |
| `2 <= n ==> SUC(n - 1) = n`] THEN | |
| SUBGOAL_THEN | |
| `?m. (n - 1) EXP n = (n - 1) EXP (2 * m)` | |
| (X_CHOOSE_THEN `m:num` SUBST1_TAC) THENL | |
| [ASM_MESON_TAC[EVEN_EXISTS]; REWRITE_TAC[GSYM EXP_EXP]] THEN | |
| MATCH_MP_TAC(MESON[CONG_TRANS; CONG_SYM] | |
| `(x == 1) (mod n) /\ ~(y == 1) (mod n) ==> ~(x == y) (mod n)`) THEN | |
| ASM_SIMP_TAC[CONG_EXP_1; CONG_MINUS1_SQUARED] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONG_SUB)) THEN | |
| DISCH_THEN(MP_TAC o SPECL [`1`; `1`]) THEN | |
| REWRITE_TAC[CONG_REFL; LE_REFL; CONG_0_DIVIDES; SUB_REFL; NOT_IMP] THEN | |
| CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE_STRONG) THEN | |
| ASM_CASES_TAC `n = 2` THENL [ASM_MESON_TAC[PRIME_2]; ASM_ARITH_TAC]; | |
| DISCH_TAC] THEN | |
| X_GEN_TAC `p:num` THEN STRIP_TAC THEN | |
| ASM_CASES_TAC `ODD p` THENL | |
| [ALL_TAC; ASM_MESON_TAC[DIVIDES_TRANS; DIVIDES_2; NOT_ODD]] THEN | |
| ASM_CASES_TAC `p = 0` THENL [ASM_MESON_TAC[PRIME_0]; ALL_TAC] THEN | |
| ASM_CASES_TAC `p = 1` THENL [ASM_MESON_TAC[PRIME_1]; ALL_TAC] THEN | |
| MP_TAC(SPECL [`n:num`; `p:num`] INDEX_DECOMPOSITION_PRIME) THEN | |
| ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `m:num` THEN | |
| ABBREV_TAC `k = index p n` THEN STRIP_TAC THEN | |
| SUBGOAL_THEN `~(k = 0)` ASSUME_TAC THENL | |
| [EXPAND_TAC "k" THEN REWRITE_TAC[INDEX_EQ_0] THEN ASM_REWRITE_TAC[]; | |
| ALL_TAC] THEN | |
| MP_TAC(snd(EQ_IMP_RULE(SPEC `p EXP k` PRIMITIVE_ROOT_EXISTS))) THEN | |
| ANTS_TAC THENL [ASM_MESON_TAC[ODD_PRIME]; ALL_TAC] THEN | |
| DISCH_THEN(X_CHOOSE_TAC `g:num`) THEN | |
| FIRST_ASSUM(MP_TAC o MATCH_MP PRIMITIVE_ROOT_IMP_COPRIME) THEN | |
| REWRITE_TAC[EXP_EQ_0; COPRIME_LEXP; ARITH_EQ] THEN | |
| ASM_REWRITE_TAC[] THEN DISCH_TAC THEN | |
| MP_TAC(ISPECL [`p EXP k`; `m:num`; `g:num`; `1`] | |
| CHINESE_REMAINDER_USUAL) THEN | |
| ASM_REWRITE_TAC[COPRIME_LEXP; LEFT_IMP_EXISTS_THM] THEN | |
| X_GEN_TAC `t:num` THEN STRIP_TAC THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `t:num`) THEN | |
| EXPAND_TAC "n" THEN ANTS_TAC THENL | |
| [REWRITE_TAC[COPRIME_RMUL; COPRIME_REXP] THEN CONJ_TAC THENL | |
| [DISJ1_TAC THEN UNDISCH_TAC `(t == g) (mod (p EXP k))` THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP CONG_COPRIME) THEN | |
| ASM_REWRITE_TAC[COPRIME_LEXP] THEN MESON_TAC[COPRIME_SYM]; | |
| ASM_MESON_TAC[NUMBER_RULE `(t == 1) (mod m) ==> coprime(t,m)`]]; | |
| DISCH_THEN(MP_TAC o MATCH_MP (NUMBER_RULE | |
| `(a == 1) (mod (n * m)) ==> (a == 1) (mod n)`))] THEN | |
| ASM_REWRITE_TAC[ORDER_DIVIDES] THEN | |
| SUBGOAL_THEN `order (p EXP k) t = order (p EXP k) g` SUBST1_TAC THENL | |
| [ASM_MESON_TAC[ORDER_CONG]; ASM_REWRITE_TAC[]] THEN | |
| ASM_SIMP_TAC[PHI_PRIMEPOW_ALT] THEN DISCH_THEN(MP_TAC o MATCH_MP | |
| (NUMBER_RULE `(a:num) * b divides c ==> a divides c /\ b divides c`)) THEN | |
| MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN | |
| MP_TAC(SPECL [`p EXP (k - 1)`; `n:num`] CONG_1_DIVIDES_EQ) THEN | |
| ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN | |
| EXPAND_TAC "n" THEN DISCH_THEN(MP_TAC o MATCH_MP (NUMBER_RULE | |
| `(p * m == 1) (mod q) ==> q divides p ==> q = 1`)) THEN | |
| ASM_SIMP_TAC[DIVIDES_EXP_LE; PRIME_GE_2; EXP_EQ_1] THEN | |
| ASM_REWRITE_TAC[PRIMEPOW_DIVIDES_INDEX; ARITH_RULE `k - 1 <= k`] THEN | |
| ARITH_TAC; | |
| X_GEN_TAC `a:num` THEN | |
| ABBREV_TAC `b = a EXP n` THEN | |
| SUBGOAL_THEN | |
| `!m:num. m divides n ==> (b == a) (mod m)` | |
| (fun th -> MESON_TAC[th; DIVIDES_REFL]) THEN | |
| MATCH_MP_TAC INDUCT_COPRIME_ALT THEN | |
| ASM_REWRITE_TAC[DIVIDES_ZERO] THEN CONJ_TAC THENL | |
| [ASM_MESON_TAC[DIVIDES_LMUL2; DIVIDES_RMUL2; CONG_CHINESE]; ALL_TAC] THEN | |
| MAP_EVERY X_GEN_TAC [`p:num`; `k:num`] THEN DISCH_TAC THEN | |
| ASM_CASES_TAC `k = 0` THEN ASM_REWRITE_TAC[EXP; CONG_MOD_1] THEN | |
| ASM_CASES_TAC `k = 1` THENL | |
| [ALL_TAC; | |
| FIRST_X_ASSUM(MP_TAC o SPEC `p:num`) THEN ASM_REWRITE_TAC[] THEN | |
| MATCH_MP_TAC(TAUT | |
| `(dk ==> d2) /\ (dk ==> d) | |
| ==> (d ==> ~d2 /\ p) ==> dk ==> q`) THEN | |
| GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o LAND_CONV) [GSYM EXP_1] THEN | |
| CONJ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] DIVIDES_TRANS) THEN | |
| MATCH_MP_TAC DIVIDES_EXP_LE_IMP THEN ASM_ARITH_TAC] THEN | |
| ASM_REWRITE_TAC[EXP_1] THEN DISCH_TAC THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `p:num`) THEN | |
| ASM_REWRITE_TAC[] THEN STRIP_TAC THEN | |
| ASM_CASES_TAC `coprime(a:num,p)` THENL | |
| [SUBGOAL_THEN `b = a * a EXP (n - 1)` SUBST1_TAC THENL | |
| [REWRITE_TAC[GSYM(CONJUNCT2 EXP)] THEN | |
| ASM_SIMP_TAC[ARITH_RULE `~(n = 0) ==> SUC(n - 1) = n`]; | |
| ALL_TAC] THEN | |
| MATCH_MP_TAC(NUMBER_RULE | |
| `(b == 1) (mod n) ==> (a * b == a) (mod n)`) THEN | |
| UNDISCH_TAC `p - 1 divides n - 1` THEN | |
| SIMP_TAC[divides; LEFT_IMP_EXISTS_THM; GSYM EXP_EXP] THEN | |
| REPEAT STRIP_TAC THEN MATCH_MP_TAC CONG_EXP_1 THEN | |
| ASM_SIMP_TAC[FERMAT_LITTLE_PRIME]; | |
| MATCH_MP_TAC(NUMBER_RULE | |
| `p divides a /\ p divides b ==> (b:num == a) (mod p)`) THEN | |
| EXPAND_TAC "b" THEN ASM_SIMP_TAC[PRIME_DIVEXP_EQ] THEN | |
| ASM_MESON_TAC[PRIME_COPRIME_EQ; COPRIME_SYM]]]);; | |
| let CARMICHAEL_NUMBER_KORSELT = prove | |
| (`!n. carmichael_number n <=> | |
| 2 <= n /\ ~prime n /\ ODD n /\ squarefree n /\ | |
| !p. prime p /\ p divides n ==> (p - 1) divides (n - 1)`, | |
| REWRITE_TAC[CARMICHAEL_NUMBER_KORSELT_ALT; SQUAREFREE_PRIME_DIVISOR] THEN | |
| MESON_TAC[]);; | |
| let CARMICHAEL_NUMBER_IMP_ODD = prove | |
| (`!n. carmichael_number n ==> ODD n`, | |
| SIMP_TAC[CARMICHAEL_NUMBER_KORSELT]);; | |
| let CARMICHAEL_NUMBER_IMP_SQUAREFREE = prove | |
| (`!n. carmichael_number n ==> squarefree n`, | |
| SIMP_TAC[CARMICHAEL_NUMBER_KORSELT]);; | |
| let CARMICHAEL_NUMBER_IMP_NZ = prove | |
| (`!n. carmichael_number n ==> ~(n = 0)`, | |
| MESON_TAC[CARMICHAEL_NUMBER_IMP_ODD; ODD]);; | |
| let CARMICHAEL_NUMBER_IMP_TRIPLET = prove | |
| (`!n. carmichael_number n ==> CARD {p | prime p /\ p divides n} >= 3`, | |
| REPEAT STRIP_TAC THEN | |
| REWRITE_TAC[ARITH_RULE `n >= 3 <=> ~(n = 0) /\ ~(n = 1) /\ ~(n = 2)`] THEN | |
| ASM_SIMP_TAC[MESON[HAS_SIZE] `FINITE s ==> (CARD s = n <=> s HAS_SIZE n)`; | |
| FINITE_SPECIAL_DIVISORS; CARMICHAEL_NUMBER_IMP_NZ] THEN | |
| CONV_TAC(ONCE_DEPTH_CONV HAS_SIZE_CONV) THEN | |
| REWRITE_TAC[NOT_EXISTS_THM] THEN REPEAT CONJ_TAC THENL | |
| [ALL_TAC; X_GEN_TAC `p:num`; MAP_EVERY X_GEN_TAC [`p:num`; `q:num`]] THEN | |
| STRIP_TAC THEN | |
| FIRST_ASSUM(MP_TAC o MATCH_MP CARMICHAEL_NUMBER_IMP_SQUAREFREE) THEN | |
| ASM_SIMP_TAC[SQUAREFREE_EXPAND_EQ; NPRODUCT_CLAUSES; FINITE_INSERT; | |
| FINITE_EMPTY; IN_INSERT; NOT_IN_EMPTY; MULT_CLAUSES] | |
| THENL | |
| [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [carmichael_number]) THEN | |
| ARITH_TAC; | |
| FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [carmichael_number]) THEN | |
| ASM SET_TAC[]; | |
| POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev)] THEN | |
| MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`q:num`; `p:num`] THEN | |
| MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[] THEN | |
| CONJ_TAC THENL [REWRITE_TAC[MULT_SYM] THEN SET_TAC[]; ALL_TAC] THEN | |
| MAP_EVERY X_GEN_TAC [`p:num`; `q:num`] THEN STRIP_TAC THEN | |
| REWRITE_TAC[CARMICHAEL_NUMBER_KORSELT_ALT] THEN REPEAT STRIP_TAC THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `q:num`) THEN | |
| ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN | |
| SUBGOAL_THEN `n - 1 = p * (q - 1) + p - 1` SUBST1_TAC THENL | |
| [EXPAND_TAC "n" THEN REWRITE_TAC[LEFT_SUB_DISTRIB] THEN | |
| MATCH_MP_TAC(ARITH_RULE | |
| `~(p = 0) /\ p * 1 <= pq ==> pq - 1 = (pq - p * 1) + p - 1`) THEN | |
| REWRITE_TAC[LE_MULT_LCANCEL; ARITH_RULE `1 <= q <=> ~(q = 0)`] THEN | |
| MP_TAC PRIME_IMP_NZ THEN ASM SET_TAC[]; | |
| DISCH_THEN(MP_TAC o MATCH_MP (NUMBER_RULE | |
| `(q:num) divides p * q + r ==> q divides r`)) THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE_STRONG) THEN | |
| SUBGOAL_THEN `2 <= p /\ 2 <= q` MP_TAC THENL [ALL_TAC; ASM_ARITH_TAC] THEN | |
| CONJ_TAC THEN MATCH_MP_TAC PRIME_GE_2 THEN ASM SET_TAC[]]);; | |
| let CARMICHAEL_NUMBER_PRIME_FACTOR_CONG_1 = prove | |
| (`!n p q. | |
| carmichael_number n /\ | |
| prime p /\ p divides n /\ | |
| prime q /\ q divides n | |
| ==> ~((q == 1) (mod p))`, | |
| REPEAT STRIP_TAC THEN | |
| FIRST_X_ASSUM(STRIP_ASSUME_TAC o | |
| GEN_REWRITE_RULE I [CARMICHAEL_NUMBER_KORSELT]) THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `q:num`) THEN ASM_REWRITE_TAC[] THEN | |
| FIRST_X_ASSUM(MP_TAC o MATCH_MP CONG_1_DIVIDES) THEN | |
| UNDISCH_TAC `(p:num) divides n` THEN | |
| REWRITE_TAC[TAUT `p ==> ~q <=> ~(p /\ q)`] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (NUMBER_RULE | |
| `q divides n /\ q divides q' /\ q' divides n' | |
| ==> (n' + 1 == n + 1) (mod q)`)) THEN | |
| ASM_SIMP_TAC[ARITH_RULE `2 <= n ==> n - 1 + 1 = n`] THEN | |
| REWRITE_TAC[NUMBER_RULE `(n == n + 1) (mod q) <=> q = 1`] THEN | |
| ASM_MESON_TAC[PRIME_1]);; | |
| let FERMAT_PSEUDOPRIME_IMP_COPRIME = prove | |
| (`!a n. fermat_pseudoprime a n ==> n = 0 \/ coprime(a,n)`, | |
| REPEAT GEN_TAC THEN ASM_CASES_TAC `n = 0` THEN REWRITE_TAC[] THEN | |
| ASM_CASES_TAC `n = 1` THEN ASM_REWRITE_TAC[COPRIME_1] THEN | |
| REWRITE_TAC[fermat_pseudoprime] THEN DISCH_THEN(MP_TAC o MATCH_MP | |
| (NUMBER_RULE `(a == 1) (mod n) ==> coprime(a,n)`)) THEN | |
| ASM_REWRITE_TAC[COPRIME_LEXP] THEN | |
| STRIP_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC);; | |
| let PRIME_IMP_FERMAT_PSEUDOPRIME = prove | |
| (`!p a. prime p /\ ~(p divides a) ==> fermat_pseudoprime a p`, | |
| REPEAT STRIP_TAC THEN REWRITE_TAC[fermat_pseudoprime] THEN | |
| MATCH_MP_TAC FERMAT_LITTLE_PRIME THEN | |
| ASM_MESON_TAC[PRIME_COPRIME_EQ; COPRIME_SYM]);; | |
| let PRIME_EQ_FERMAT_PSEUDOPRIME = prove | |
| (`!p. prime p <=> 2 <= p /\ (!a. 0 < a /\ a < p ==> fermat_pseudoprime a p)`, | |
| GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[PRIME_GE_2] THENL | |
| [MATCH_MP_TAC PRIME_IMP_FERMAT_PSEUDOPRIME THEN ASM_REWRITE_TAC[] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE_STRONG) THEN ASM_ARITH_TAC; | |
| REWRITE_TAC[PRIME] THEN | |
| REPEAT(CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN | |
| X_GEN_TAC `a:num` THEN STRIP_TAC THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `a:num`) THEN | |
| ASM_REWRITE_TAC[] THEN | |
| DISCH_THEN(STRIP_ASSUME_TAC o MATCH_MP FERMAT_PSEUDOPRIME_IMP_COPRIME) THEN | |
| ONCE_REWRITE_TAC[COPRIME_SYM] THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC]);; | |
| let ABSOLUTE_FERMAT_PSEUDOPRIME = prove | |
| (`!n. (!a. coprime(a,n) ==> fermat_pseudoprime a n) <=> | |
| n = 0 \/ n = 1 \/ prime n \/ carmichael_number n`, | |
| GEN_TAC THEN ASM_CASES_TAC `n = 0` THENL | |
| [ASM_SIMP_TAC[fermat_pseudoprime; COPRIME_0; CONG_MOD_0] THEN | |
| CONV_TAC NUM_REDUCE_CONV; | |
| ALL_TAC] THEN | |
| ASM_CASES_TAC `n = 1` THENL | |
| [ASM_SIMP_TAC[fermat_pseudoprime; CONG_MOD_1]; | |
| ALL_TAC] THEN | |
| ASM_REWRITE_TAC[carmichael_number; ARITH_RULE | |
| `2 <= n <=> ~(n = 0 \/ n = 1)`] THEN | |
| ASM_CASES_TAC `prime n` THEN ASM_REWRITE_TAC[] THEN | |
| ASM_MESON_TAC[PRIME_IMP_FERMAT_PSEUDOPRIME; PRIME_COPRIME_EQ; COPRIME_SYM]);; | |
| let FERMAT_PSEUDOPRIME_BOUND_PHI_ALT = prove | |
| (`!n. ~(n = 1) /\ ~prime n /\ ~carmichael_number n | |
| ==> 2 * CARD {a | a < n /\ fermat_pseudoprime a n} <= phi n`, | |
| GEN_TAC THEN ASM_CASES_TAC `n = 0` THEN | |
| ASM_REWRITE_TAC[LT; EMPTY_GSPEC; CARD_CLAUSES; MULT_CLAUSES; LE_0] THEN | |
| REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN | |
| ASM_REWRITE_TAC[carmichael_number; NOT_FORALL_THM; NOT_IMP; | |
| LEFT_IMP_EXISTS_THM; ARITH_RULE `2 <= n <=> ~(n = 0 \/ n = 1)`] THEN | |
| X_GEN_TAC `b:num` THEN STRIP_TAC THEN | |
| MATCH_MP_TAC CARD_SUBSET_HALF_LEMMA THEN | |
| EXISTS_TAC `\a. (a * b) MOD n` THEN | |
| EXISTS_TAC `{a:num | coprime(a,n) /\ a < n}` THEN | |
| SIMP_TAC[PHI_ALT; LE_REFL; SUBSET; FORALL_IN_IMAGE] THEN | |
| REWRITE_TAC[IN_ELIM_THM; IN_DIFF] THEN REPEAT CONJ_TAC THENL | |
| [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{i:num | i < n}` THEN | |
| REWRITE_TAC[FINITE_NUMSEG_LT] THEN SET_TAC[]; | |
| ASM_MESON_TAC[FERMAT_PSEUDOPRIME_IMP_COPRIME; LT]; | |
| ALL_TAC; | |
| REWRITE_TAC[GSYM CONG] THEN REPEAT STRIP_TAC THEN | |
| MATCH_MP_TAC CONG_IMP_EQ THEN EXISTS_TAC `n:num` THEN | |
| ASM_MESON_TAC[CONG_MULT_RCANCEL]] THEN | |
| X_GEN_TAC `a:num` THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[LT] THEN | |
| STRIP_TAC THEN | |
| FIRST_ASSUM(MP_TAC o MATCH_MP FERMAT_PSEUDOPRIME_IMP_COPRIME) THEN | |
| ASM_REWRITE_TAC[COPRIME_LMOD; COPRIME_LMUL] THEN | |
| STRIP_TAC THEN ASM_REWRITE_TAC[MOD_LT_EQ; fermat_pseudoprime] THEN | |
| RULE_ASSUM_TAC(REWRITE_RULE[fermat_pseudoprime]) THEN | |
| REWRITE_TAC[MOD_EXP_MOD; CONG] THEN REWRITE_TAC[GSYM CONG; MULT_EXP] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (NUMBER_RULE | |
| `(a * b == 1) (mod n) ==> (a == 1) (mod n) ==> (b == 1) (mod n)`)) THEN | |
| ASM_REWRITE_TAC[]);; | |
| let FERMAT_PSEUDOPRIME_BOUND_PHI = prove | |
| (`!n. ~(n = 1) /\ ~prime n /\ ~carmichael_number n | |
| ==> CARD {a | a < n /\ fermat_pseudoprime a n} <= phi n DIV 2`, | |
| REWRITE_TAC[ARITH_RULE `a <= b DIV 2 <=> 2 * a <= b`] THEN | |
| REWRITE_TAC[FERMAT_PSEUDOPRIME_BOUND_PHI_ALT]);; | |
| let FERMAT_PSEUDOPRIME_BOUND_LT = prove | |
| (`!n. ~(n = 0) /\ ~(n = 1) /\ ~prime n /\ ~carmichael_number n | |
| ==> CARD {a | a < n /\ fermat_pseudoprime a n} < n DIV 2`, | |
| REPEAT STRIP_TAC THEN | |
| MP_TAC(SPEC `n:num` FERMAT_PSEUDOPRIME_BOUND_PHI_ALT) THEN | |
| ASM_REWRITE_TAC[] THEN | |
| MATCH_MP_TAC(ARITH_RULE `p < n - 1 ==> 2 * a <= p ==> a < n DIV 2`) THEN | |
| MATCH_MP_TAC PHI_LIMIT_COMPOSITE THEN ASM_REWRITE_TAC[]);; | |
| let MILLER_RABIN_IMP_FERMAT_PSEUDOPRIME = prove | |
| (`!a q. miller_rabin_pseudoprime a q /\ ~(q = 2) ==> fermat_pseudoprime a q`, | |
| REWRITE_TAC[fermat_pseudoprime] THEN | |
| REWRITE_TAC[MILLER_RABIN_IMP_FERMAT_PSEUDOPRIME_EXPLICIT]);; | |
| let MILLER_RABIN_EQ_FERMAT_PSEUDOPRIME = prove | |
| (`!a q. (?p k. prime p /\ ODD p /\ p EXP k = q) | |
| ==> (miller_rabin_pseudoprime a q <=> fermat_pseudoprime a q)`, | |
| REWRITE_TAC[fermat_pseudoprime] THEN | |
| REWRITE_TAC[MILLER_RABIN_EQ_FERMAT_PSEUDOPRIME_EXPLICIT]);; | |
| (* ------------------------------------------------------------------------- *) | |
| (* Weak Euler pseudoprimes. *) | |
| (* ------------------------------------------------------------------------- *) | |
| let weak_euler_pseudoprime = new_definition | |
| `weak_euler_pseudoprime a n <=> | |
| ODD n /\ | |
| ((&a pow ((n - 1) DIV 2) == (&1:int)) (mod &n) \/ | |
| (&a pow ((n - 1) DIV 2) == (-- &1:int)) (mod &n))`;; | |
| let WEAK_EULER_IMP_FERMAT_PSEUDOPRIME = prove | |
| (`!a n. weak_euler_pseudoprime a n ==> fermat_pseudoprime a n`, | |
| REWRITE_TAC[fermat_pseudoprime; weak_euler_pseudoprime] THEN | |
| REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (INTEGER_RULE | |
| `(x:int == y) (mod m) ==> (x pow 2 == y pow 2) (mod m)`)) THEN | |
| CONV_TAC INT_REDUCE_CONV THEN | |
| REWRITE_TAC[num_congruent; GSYM INT_OF_NUM_POW; INT_POW_POW] THEN | |
| MATCH_MP_TAC(INTEGER_RULE | |
| `x:int = y ==> (x == &1) (mod n) ==> (y == &1) (mod n)`) THEN | |
| AP_TERM_TAC THEN | |
| FIRST_X_ASSUM(CHOOSE_THEN SUBST1_TAC o REWRITE_RULE[ODD_EXISTS]) THEN | |
| REWRITE_TAC[SUC_SUB1; ARITH_RULE `(2 * m) DIV 2 = m`] THEN ARITH_TAC);; | |
| let WEAK_EULER_PSEUDOPRIME_IMP_ODD = prove | |
| (`!a n. weak_euler_pseudoprime a n ==> ODD n`, | |
| SIMP_TAC[weak_euler_pseudoprime]);; | |
| let WEAK_EULER_PSEUDOPRIME_IMP_COPRIME = prove | |
| (`!a n. weak_euler_pseudoprime a n ==> coprime(a,n)`, | |
| MESON_TAC[WEAK_EULER_IMP_FERMAT_PSEUDOPRIME; FERMAT_PSEUDOPRIME_IMP_COPRIME; | |
| ODD; weak_euler_pseudoprime]);; | |
| let PRIME_IMP_WEAK_EULER_PSEUDOPRIME = prove | |
| (`!p a. prime p /\ ~(p = 2) /\ ~(p divides a) ==> weak_euler_pseudoprime a p`, | |
| REPEAT STRIP_TAC THEN REWRITE_TAC[weak_euler_pseudoprime] THEN | |
| CONJ_TAC THENL [ASM_MESON_TAC[PRIME_ODD]; ONCE_REWRITE_TAC[DISJ_SYM]] THEN | |
| MP_TAC(SPECL [`a:num`; `p:num`] JACOBI_CASES) THEN | |
| MP_TAC(SPECL [`a:num`; `p:num`] JACOBI_EULER) THEN | |
| ASM_REWRITE_TAC[JACOBI_EQ_0] THEN ONCE_REWRITE_TAC[COPRIME_SYM] THEN | |
| ASM_SIMP_TAC[PRIME_COPRIME_EQ] THEN ASM_MESON_TAC | |
| [INTEGER_RULE `(j == a) (mod p) ==> j:int = z ==> (a == z) (mod p)`]);; | |
| let PRIME_EQ_WEAK_EULER_PSEUDOPRIME = prove | |
| (`!p. prime p <=> | |
| p = 2 \/ 2 <= p /\ | |
| (!a. 0 < a /\ a < p ==> weak_euler_pseudoprime a p)`, | |
| GEN_TAC THEN ASM_CASES_TAC `p = 2` THEN ASM_REWRITE_TAC[PRIME_2] THEN | |
| EQ_TAC THENL | |
| [SIMP_TAC[PRIME_GE_2] THEN REPEAT STRIP_TAC THEN | |
| MATCH_MP_TAC PRIME_IMP_WEAK_EULER_PSEUDOPRIME THEN ASM_REWRITE_TAC[] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE_STRONG) THEN ASM_ARITH_TAC; | |
| MESON_TAC[WEAK_EULER_IMP_FERMAT_PSEUDOPRIME; | |
| PRIME_EQ_FERMAT_PSEUDOPRIME]]);; | |
| let MILLER_RABIN_IMP_WEAK_EULER_PSEUDOPRIME = prove | |
| (`!a q. miller_rabin_pseudoprime a q /\ ~(q = 2) | |
| ==> weak_euler_pseudoprime a q`, | |
| MAP_EVERY X_GEN_TAC [`a:num`; `n:num`] THEN | |
| REWRITE_TAC[miller_rabin_pseudoprime; weak_euler_pseudoprime] THEN | |
| ASM_CASES_TAC `n = 2` THEN ASM_REWRITE_TAC[] THEN | |
| ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[ODD] THEN | |
| ASM_CASES_TAC `n = 1` THEN ASM_REWRITE_TAC[INT_CONG_MOD_1; ARITH] THEN | |
| REWRITE_TAC[num_congruent; GSYM INT_OF_NUM_POW] THEN | |
| ASM_SIMP_TAC[GSYM INT_OF_NUM_SUB; LE_1] THEN | |
| REWRITE_TAC[INTEGER_RULE | |
| `(x:int == n - z) (mod n) <=> (x == --z) (mod n)`] THEN | |
| MP_TAC(SPECL [`n - 1`; `2`] INDEX_DECOMPOSITION) THEN | |
| ASM_REWRITE_TAC[ARITH_EQ; ARITH_RULE `n - 1 = 0 <=> n = 0 \/ n = 1`] THEN | |
| ABBREV_TAC `e = index 2 (n - 1)` THEN | |
| REWRITE_TAC[DIVIDES_2; NOT_EVEN; LEFT_IMP_EXISTS_THM] THEN | |
| X_GEN_TAC `m:num` THEN STRIP_TAC THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN | |
| SUBST1_TAC(SYM(ASSUME `2 EXP e * m = n - 1`)) THEN | |
| SIMP_TAC[DIV_MULT; EXP_EQ_0; ARITH_EQ] THEN | |
| DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN | |
| SUBGOAL_THEN `(2 EXP e * m) DIV 2 = 2 EXP (e - 1) * m` SUBST1_TAC THENL | |
| [MATCH_MP_TAC DIV_UNIQ THEN EXISTS_TAC `0` THEN | |
| REWRITE_TAC[ARITH_RULE `(ee * m) * 2 + 0 = (2 * ee) * m`] THEN | |
| REWRITE_TAC[ARITH] THEN AP_THM_TAC THEN AP_TERM_TAC THEN | |
| REWRITE_TAC[GSYM(CONJUNCT2 EXP)] THEN AP_TERM_TAC THEN | |
| REWRITE_TAC[ARITH_RULE `e = SUC(e - 1) <=> ~(e = 0)`] THEN | |
| EXPAND_TAC "e" THEN REWRITE_TAC[INDEX_EQ_0] THEN | |
| ASM_REWRITE_TAC[NOT_EVEN; DIVIDES_2; ODD_SUB] THEN ASM_ARITH_TAC; | |
| ALL_TAC] THEN | |
| DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC MP_TAC) THENL | |
| [DISJ1_TAC THEN | |
| REWRITE_TAC[ONCE_REWRITE_RULE[MULT_SYM] (GSYM INT_POW_POW)] THEN | |
| ASM_SIMP_TAC[INT_CONG_POW_1]; | |
| DISCH_THEN(X_CHOOSE_THEN `i:num` STRIP_ASSUME_TAC)] THEN | |
| FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE | |
| `i < e ==> e - 1 = SUC(i + (e - i - 2)) \/ e - 1 = i`)) THEN | |
| MATCH_MP_TAC MONO_OR THEN ASM_SIMP_TAC[] THEN DISCH_THEN SUBST1_TAC THEN | |
| REWRITE_TAC[EXP; EXP_ADD] THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `2 * 2 EXP (e - i - 2)` o | |
| MATCH_MP INT_CONG_POW) THEN | |
| REWRITE_TAC[GSYM INT_POW_POW; INT_POW_NEG; INT_POW_ONE] THEN | |
| REWRITE_TAC[EVEN_MULT; ARITH; INT_POW_POW] THEN REWRITE_TAC[MULT_AC]);; | |
| let MILLER_RABIN_EQ_WEAK_EULER_PSEUDOPRIME = prove | |
| (`!a n. ~(n = 2) /\ (?p k. prime p /\ ODD p /\ p EXP k = n) \/ | |
| (n == 3) (mod 4) | |
| ==> (miller_rabin_pseudoprime a n <=> weak_euler_pseudoprime a n)`, | |
| REPEAT GEN_TAC THEN ASM_CASES_TAC `n = 2` THEN ASM_REWRITE_TAC[] THENL | |
| [REWRITE_TAC[CONG] THEN CONV_TAC NUM_REDUCE_CONV; ALL_TAC] THEN | |
| DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC MP_TAC) THENL | |
| [EQ_TAC THEN ASM_SIMP_TAC[MILLER_RABIN_IMP_WEAK_EULER_PSEUDOPRIME] THEN | |
| ASM_SIMP_TAC[MILLER_RABIN_EQ_FERMAT_PSEUDOPRIME] THEN | |
| REWRITE_TAC[WEAK_EULER_IMP_FERMAT_PSEUDOPRIME]; | |
| REWRITE_TAC[CONG] THEN | |
| REWRITE_TAC[weak_euler_pseudoprime; miller_rabin_pseudoprime] THEN | |
| ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[] THEN | |
| CONV_TAC NUM_REDUCE_CONV THEN | |
| ASM_CASES_TAC `ODD n` THEN ASM_REWRITE_TAC[CONG_MINUS1] THEN | |
| REWRITE_TAC[num_divides; GSYM INT_OF_NUM_ADD; num_congruent] THEN | |
| REWRITE_TAC[GSYM INT_OF_NUM_POW; INTEGER_RULE | |
| `n divides (x + &1:int) <=> (x == -- &1) (mod n)`] THEN | |
| DISCH_THEN(SUBST1_TAC o MATCH_MP (ARITH_RULE | |
| `n MOD 4 = 3 ==> n - 1 = 2 * (2 * (n DIV 4) + 1)`)) THEN | |
| ABBREV_TAC `m = n DIV 4` THEN | |
| SIMP_TAC[INDEX_MUL; PRIME_2; INDEX_REFL; ARITH; | |
| ARITH_RULE `~(2 * n + 1 = 0)`] THEN | |
| SIMP_TAC[INDEX_ZERO; DIVIDES_2; EVEN_ADD; EVEN_MULT; ARITH] THEN | |
| CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN CONV_TAC NUM_REDUCE_CONV THEN | |
| REWRITE_TAC[ARITH_RULE `(2 * n) DIV 2 = n`] THEN | |
| REWRITE_TAC[ARITH_RULE `i < 1 <=> i = 0`; UNWIND_THM2] THEN | |
| REWRITE_TAC[EXP; MULT_CLAUSES]]);; | |
| let ABSOLUTE_WEAK_EULER_PSEUDOPRIME,ABSOLUTE_WEAK_EULER_PSEUDOPRIME_ALT = | |
| (CONJ_PAIR o prove) | |
| (`(!n. (!a. coprime(a,n) ==> weak_euler_pseudoprime a n) <=> | |
| ODD n /\ | |
| (prime n \/ | |
| squarefree n /\ | |
| !p. prime p /\ p divides n ==> (p - 1) divides ((n - 1) DIV 2))) /\ | |
| (!n. (!a. coprime(a,n) ==> weak_euler_pseudoprime a n) <=> | |
| ODD n /\ | |
| (prime n \/ | |
| !a. coprime(a,n) ==> (a EXP ((n - 1) DIV 2) == 1) (mod n)))`, | |
| REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `n:num` THEN | |
| ASM_CASES_TAC `ODD n` THENL | |
| [ASM_REWRITE_TAC[]; | |
| ASM_MESON_TAC[WEAK_EULER_PSEUDOPRIME_IMP_ODD; COPRIME_1]] THEN | |
| ASM_CASES_TAC `prime n` THEN ASM_REWRITE_TAC[] THENL | |
| [ASM_MESON_TAC[COPRIME_SYM; PRIME_COPRIME_EQ; | |
| PRIME_IMP_WEAK_EULER_PSEUDOPRIME; | |
| NUM_REDUCE_CONV `ODD 2`]; | |
| ALL_TAC] THEN | |
| ASM_CASES_TAC `n = 1` THENL | |
| [ASM_REWRITE_TAC[weak_euler_pseudoprime; CONG_MOD_1; INT_CONG_MOD_1] THEN | |
| CONV_TAC NUM_REDUCE_CONV THEN | |
| REWRITE_TAC[DIVIDES_0; SQUAREFREE_1]; | |
| ALL_TAC] THEN | |
| MATCH_MP_TAC(TAUT | |
| `(r ==> p) /\ (q ==> r) /\ (p ==> q) | |
| ==> (p <=> q) /\ (p <=> r)`) THEN | |
| CONJ_TAC THENL | |
| [ASM_SIMP_TAC[weak_euler_pseudoprime; INT_OF_NUM_POW; GSYM num_congruent]; | |
| ALL_TAC] THEN | |
| CONJ_TAC THENL | |
| [STRIP_TAC THEN X_GEN_TAC `a:num` THEN STRIP_TAC THEN | |
| MATCH_MP_TAC CONG_MOD_SQUAREFREE THEN ASM_REWRITE_TAC[] THEN | |
| X_GEN_TAC `p:num` THEN STRIP_TAC THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `p:num`) THEN | |
| ASM_REWRITE_TAC[divides] THEN | |
| DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST1_TAC) THEN | |
| REWRITE_TAC[GSYM EXP_EXP] THEN MATCH_MP_TAC CONG_EXP_1 THEN | |
| MATCH_MP_TAC FERMAT_LITTLE_PRIME THEN | |
| MAP_EVERY UNDISCH_TAC [`coprime(a:num,n)`; `(p:num) divides n`] THEN | |
| ASM_REWRITE_TAC[] THEN CONV_TAC NUMBER_RULE; | |
| DISCH_TAC] THEN | |
| MP_TAC(SPEC `n:num` ABSOLUTE_FERMAT_PSEUDOPRIME) THEN | |
| ASM_SIMP_TAC[WEAK_EULER_IMP_FERMAT_PSEUDOPRIME] THEN | |
| STRIP_TAC THENL [ASM_MESON_TAC[ODD]; ALL_TAC] THEN | |
| FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CARMICHAEL_NUMBER_KORSELT]) THEN | |
| ASM_REWRITE_TAC[] THEN | |
| REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN | |
| ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `p:num` THEN | |
| DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN | |
| GEN_REWRITE_TAC LAND_CONV [divides] THEN | |
| DISCH_THEN(X_CHOOSE_THEN `m:num` MP_TAC) THEN | |
| DISJ_CASES_TAC(SPEC `m:num` EVEN_OR_ODD) THENL | |
| [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EVEN_EXISTS]) THEN | |
| DISCH_THEN(X_CHOOSE_THEN `q:num` SUBST1_TAC) THEN | |
| DISCH_THEN SUBST1_TAC THEN | |
| REWRITE_TAC[ARITH_RULE `(a * 2 * b) DIV 2 = a * b`] THEN | |
| CONV_TAC NUMBER_RULE; | |
| DISCH_THEN(ASSUME_TAC o SYM)] THEN | |
| MATCH_MP_TAC(TAUT `F ==> p`) THEN | |
| SUBGOAL_THEN `?q:num. p * q = n` STRIP_ASSUME_TAC THENL | |
| [ASM_MESON_TAC[divides]; ALL_TAC] THEN | |
| SUBGOAL_THEN `coprime(p:num,q)` ASSUME_TAC THENL | |
| [ASM_MESON_TAC[SQUAREFREE_MUL]; ALL_TAC] THEN | |
| MP_TAC(snd(EQ_IMP_RULE(SPEC `p:num` QUADRATIC_NONRESIDUE_EXISTS))) THEN | |
| ANTS_TAC THENL | |
| [ASM_MESON_TAC[ODD_PRIME; PRIME_ODD; CARMICHAEL_NUMBER_IMP_ODD; | |
| ODD_MULT; NUM_ODD_CONV `ODD 2`]; | |
| DISCH_THEN(X_CHOOSE_THEN `b:num` STRIP_ASSUME_TAC)] THEN | |
| MP_TAC(ISPECL [`p:num`; `q:num`; `b:num`; `1`] | |
| CHINESE_REMAINDER_USUAL) THEN | |
| ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `a:num` THEN STRIP_TAC THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `a:num`) THEN ANTS_TAC THENL | |
| [EXPAND_TAC "n" THEN REWRITE_TAC[COPRIME_RMUL] THEN | |
| ASM_SIMP_TAC[NUMBER_RULE `(a == 1) (mod q) ==> coprime(a,q)`] THEN | |
| MAP_EVERY UNDISCH_TAC [`coprime(p:num,b)`; `(a:num == b) (mod p)`] THEN | |
| CONV_TAC NUMBER_RULE; | |
| ASM_REWRITE_TAC[weak_euler_pseudoprime; DE_MORGAN_THM]] THEN | |
| EXPAND_TAC "n" THEN REWRITE_TAC[GSYM INT_OF_NUM_MUL] THEN | |
| ASM_REWRITE_TAC[] THEN CONJ_TAC THENL | |
| [ALL_TAC; | |
| DISCH_THEN(MP_TAC o MATCH_MP (INTEGER_RULE | |
| `(a:int == --z) (mod (p * q)) | |
| ==> (a == z) (mod q) ==> q divides (&2 * z)`)) THEN | |
| ASM_SIMP_TAC[INT_OF_NUM_POW; GSYM num_congruent; CONG_EXP_1] THEN | |
| REWRITE_TAC[INT_OF_NUM_MUL; MULT_CLAUSES; GSYM num_divides] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE) THEN | |
| REWRITE_TAC[ARITH_RULE `q <= 2 \/ 2 = 0 <=> q = 0 \/ q = 1 \/ q = 2`] THEN | |
| ASM_MESON_TAC[NUM_REDUCE_CONV `~ODD 0 /\ ~ODD 2`; | |
| ODD_MULT; MULT_CLAUSES]] THEN | |
| MP_TAC(SPECL [`a:num`; `p:num`] JACOBI_EULER) THEN ANTS_TAC THENL | |
| [ASM_MESON_TAC[ODD_MULT; NUM_REDUCE_CONV `ODD 2`]; ALL_TAC] THEN | |
| SUBGOAL_THEN `jacobi(a,p) = jacobi(b,p)` SUBST1_TAC THENL | |
| [ASM_MESON_TAC[JACOBI_CONG]; ASM_SIMP_TAC[JACOBI_PRIME]] THEN | |
| COND_CASES_TAC THENL | |
| [MP_TAC(SPECL [`p:num`; `b:num`] DIVIDES_LE) THEN | |
| ASM_REWRITE_TAC[GSYM NOT_LT] THEN ASM_MESON_TAC[COPRIME_0; PRIME_1]; | |
| DISCH_THEN(MP_TAC o SPEC `m:num` o MATCH_MP INT_CONG_POW)] THEN | |
| REWRITE_TAC[INT_POW_POW] THEN | |
| SUBGOAL_THEN `(p - 1) DIV 2 * m = (n - 1) DIV 2` SUBST1_TAC THENL | |
| [SUBST1_TAC(SYM(ASSUME `(p - 1) * m = n - 1`)) THEN | |
| SUBGOAL_THEN `ODD p` MP_TAC THENL [ASM_MESON_TAC[ODD_MULT]; ALL_TAC] THEN | |
| REWRITE_TAC[ODD_EXISTS; LEFT_IMP_EXISTS_THM] THEN | |
| X_GEN_TAC `r:num` THEN DISCH_THEN SUBST1_TAC THEN | |
| REWRITE_TAC[SUC_SUB1; GSYM MULT_ASSOC] THEN | |
| REWRITE_TAC[ARITH_RULE `(2 * r) DIV 2 = r`]; | |
| REWRITE_TAC[TAUT `p ==> ~q <=> ~(p /\ q)`]] THEN | |
| ASM_REWRITE_TAC[INT_POW_NEG; INT_POW_ONE; GSYM NOT_ODD] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (INTEGER_RULE | |
| `(--z:int == a) (mod p) /\ (a == z) (mod (p * q)) | |
| ==> p divides &2 * z`)) THEN | |
| REWRITE_TAC[INT_OF_NUM_MUL; MULT_CLAUSES; GSYM num_divides] THEN | |
| ASM_MESON_TAC[DIVIDES_PRIME_PRIME; ODD_MULT; | |
| NUM_REDUCE_CONV `ODD 2`; PRIME_2]);; | |
| let WEAK_EULER_PSEUDOPRIME_BOUND_PHI_ALT = prove | |
| (`!n. ~prime n /\ | |
| ~(squarefree n /\ | |
| !p. prime p /\ p divides n ==> p - 1 divides (n - 1) DIV 2) | |
| ==> 2 * CARD {a | a < n /\ weak_euler_pseudoprime a n} <= phi n`, | |
| GEN_TAC THEN ASM_CASES_TAC `n = 0` THEN | |
| ASM_REWRITE_TAC[LT; EMPTY_GSPEC; CARD_CLAUSES; MULT_CLAUSES; LE_0] THEN | |
| STRIP_TAC THEN | |
| MP_TAC(SPEC `n:num` ABSOLUTE_WEAK_EULER_PSEUDOPRIME) THEN | |
| ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN | |
| X_GEN_TAC `b:num` THEN STRIP_TAC THEN | |
| MATCH_MP_TAC CARD_SUBSET_HALF_LEMMA THEN | |
| EXISTS_TAC `\a. (a * b) MOD n` THEN | |
| EXISTS_TAC `{a:num | coprime(a,n) /\ a < n}` THEN | |
| SIMP_TAC[PHI_ALT; LE_REFL; SUBSET; FORALL_IN_IMAGE] THEN | |
| REWRITE_TAC[IN_ELIM_THM; IN_DIFF] THEN REPEAT CONJ_TAC THENL | |
| [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{i:num | i < n}` THEN | |
| REWRITE_TAC[FINITE_NUMSEG_LT] THEN SET_TAC[]; | |
| ASM_MESON_TAC[WEAK_EULER_PSEUDOPRIME_IMP_COPRIME; LT]; | |
| ALL_TAC; | |
| REWRITE_TAC[GSYM CONG] THEN REPEAT STRIP_TAC THEN | |
| MATCH_MP_TAC CONG_IMP_EQ THEN EXISTS_TAC `n:num` THEN | |
| ASM_MESON_TAC[CONG_MULT_RCANCEL]] THEN | |
| X_GEN_TAC `a:num` THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[LT] THEN | |
| STRIP_TAC THEN | |
| FIRST_ASSUM(MP_TAC o MATCH_MP WEAK_EULER_PSEUDOPRIME_IMP_COPRIME) THEN | |
| ASM_REWRITE_TAC[COPRIME_LMOD; COPRIME_LMUL] THEN | |
| STRIP_TAC THEN ASM_REWRITE_TAC[MOD_LT_EQ; weak_euler_pseudoprime] THEN | |
| REWRITE_TAC[GSYM INT_OF_NUM_REM; INT_POW_REM; GSYM INT_REM_EQ] THEN | |
| REWRITE_TAC[INT_REM_EQ; GSYM INT_OF_NUM_MUL; INT_POW_MUL] THEN | |
| FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [weak_euler_pseudoprime]) THEN | |
| FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV | |
| [weak_euler_pseudoprime]) THEN | |
| ASM_CASES_TAC `ODD n` THEN ASM_REWRITE_TAC[] THEN | |
| MESON_TAC (map INTEGER_RULE | |
| [`(a * b:int == &1) (mod n) | |
| ==> (a == &1) (mod n) ==> (b == &1) (mod n)`; | |
| `(a * b:int == &1) (mod n) | |
| ==> (a == -- &1) (mod n) ==> (b == -- &1) (mod n)`; | |
| `(a * b:int == -- &1) (mod n) | |
| ==> (a == &1) (mod n) ==> (b == -- &1) (mod n)`; | |
| `(a * b:int == -- &1) (mod n) | |
| ==> (a == -- &1) (mod n) ==> (b == &1) (mod n)`]));; | |
| let WEAK_EULER_PSEUDOPRIME_BOUND_PHI = prove | |
| (`!n. ~prime n /\ | |
| ~(squarefree n /\ | |
| !p. prime p /\ p divides n ==> p - 1 divides (n - 1) DIV 2) | |
| ==> CARD {a | a < n /\ weak_euler_pseudoprime a n} <= phi n DIV 2`, | |
| REWRITE_TAC[ARITH_RULE `a <= b DIV 2 <=> 2 * a <= b`] THEN | |
| REWRITE_TAC[WEAK_EULER_PSEUDOPRIME_BOUND_PHI_ALT]);; | |
| let WEAK_EULER_PSEUDOPRIME_BOUND_LT = prove | |
| (`!n. ~(n = 0) /\ ~(n = 1) /\ ~prime n /\ | |
| ~(squarefree n /\ | |
| !p. prime p /\ p divides n ==> p - 1 divides (n - 1) DIV 2) | |
| ==> CARD {a | a < n /\ weak_euler_pseudoprime a n} < n DIV 2`, | |
| REPEAT STRIP_TAC THEN | |
| MP_TAC(SPEC `n:num` WEAK_EULER_PSEUDOPRIME_BOUND_PHI_ALT) THEN | |
| ASM_REWRITE_TAC[] THEN | |
| MATCH_MP_TAC(ARITH_RULE `p < n - 1 ==> 2 * a <= p ==> a < n DIV 2`) THEN | |
| MATCH_MP_TAC PHI_LIMIT_COMPOSITE THEN ASM_REWRITE_TAC[]);; | |
| (* ------------------------------------------------------------------------- *) | |
| (* Euler (-Jacobi) pseudoprimes, as used in the Solovay-Strassen test. *) | |
| (* ------------------------------------------------------------------------- *) | |
| let euler_jacobi_pseudoprime = new_definition | |
| `euler_jacobi_pseudoprime a n <=> | |
| ODD n /\ coprime(a,n) /\ | |
| (jacobi(a,n) == &a pow ((n - 1) DIV 2)) (mod &n)`;; | |
| let EULER_JACOBI_PSEUDOPRIME_IMP_COPRIME = prove | |
| (`!a n. euler_jacobi_pseudoprime a n ==> coprime(a,n)`, | |
| SIMP_TAC[euler_jacobi_pseudoprime]);; | |
| let EULER_JACOBI_PSEUDOPRIME_IMP_ODD = prove | |
| (`!a n. euler_jacobi_pseudoprime a n ==> ODD n`, | |
| SIMP_TAC[euler_jacobi_pseudoprime]);; | |
| let EULER_JACOBI_IMP_WEAK_EULER_PSEUDOPRIME = prove | |
| (`!a n. euler_jacobi_pseudoprime a n ==> weak_euler_pseudoprime a n`, | |
| REPEAT GEN_TAC THEN | |
| SIMP_TAC[weak_euler_pseudoprime; euler_jacobi_pseudoprime] THEN | |
| MP_TAC(SPECL [`a:num`; `n:num`] JACOBI_CASES) THEN | |
| REWRITE_TAC[JACOBI_EQ_0; COPRIME_SYM] THEN | |
| STRIP_TAC THEN ASM_SIMP_TAC[INT_CONG_SYM]);; | |
| let EULER_JACOBI_IMP_FERMAT_PSEUDOPRIME = prove | |
| (`!a n. euler_jacobi_pseudoprime a n ==> fermat_pseudoprime a n`, | |
| MESON_TAC[EULER_JACOBI_IMP_WEAK_EULER_PSEUDOPRIME; | |
| WEAK_EULER_IMP_FERMAT_PSEUDOPRIME]);; | |
| let PRIME_IMP_EULER_JACOBI_PSEUDOPRIME = prove | |
| (`!p a. prime p /\ ~(p = 2) /\ ~(p divides a) | |
| ==> euler_jacobi_pseudoprime a p`, | |
| SIMP_TAC[euler_jacobi_pseudoprime; JACOBI_EULER] THEN | |
| ASM_MESON_TAC[PRIME_COPRIME_EQ; COPRIME_SYM; PRIME_ODD]);; | |
| let PRIME_EQ_EULER_JACOBI_PSEUDOPRIME = prove | |
| (`!p. prime p <=> | |
| p = 2 \/ 2 <= p /\ | |
| (!a. 0 < a /\ a < p ==> euler_jacobi_pseudoprime a p)`, | |
| GEN_TAC THEN ASM_CASES_TAC `p = 2` THEN ASM_REWRITE_TAC[PRIME_2] THEN | |
| EQ_TAC THENL | |
| [SIMP_TAC[PRIME_GE_2] THEN REPEAT STRIP_TAC THEN | |
| MATCH_MP_TAC PRIME_IMP_EULER_JACOBI_PSEUDOPRIME THEN ASM_REWRITE_TAC[] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE_STRONG) THEN ASM_ARITH_TAC; | |
| MESON_TAC[EULER_JACOBI_IMP_FERMAT_PSEUDOPRIME; | |
| PRIME_EQ_FERMAT_PSEUDOPRIME]]);; | |
| let ABSOLUTE_EULER_JACOBI_PSEUDOPRIME = prove | |
| (`!n. (!a. coprime(a,n) ==> euler_jacobi_pseudoprime a n) <=> | |
| n = 1 \/ ODD n /\ prime n`, | |
| GEN_TAC THEN ASM_CASES_TAC `n = 1` THENL | |
| [ASM_REWRITE_TAC[euler_jacobi_pseudoprime; JACOBI_1] THEN | |
| REWRITE_TAC[INT_CONG_MOD_1; COPRIME_1; ARITH]; | |
| ALL_TAC] THEN | |
| ASM_CASES_TAC `prime n` THEN ASM_REWRITE_TAC[] THENL | |
| [EQ_TAC THENL | |
| [MESON_TAC[EULER_JACOBI_PSEUDOPRIME_IMP_ODD; COPRIME_1]; | |
| DISCH_TAC THEN REPEAT STRIP_TAC] THEN | |
| MATCH_MP_TAC PRIME_IMP_EULER_JACOBI_PSEUDOPRIME THEN | |
| ASM_MESON_TAC[NUM_REDUCE_CONV `ODD 2`; PRIME_COPRIME_EQ; COPRIME_SYM]; | |
| REWRITE_TAC[NOT_FORALL_THM; NOT_IMP]] THEN | |
| ASM_CASES_TAC `n = 0` THENL | |
| [ASM_REWRITE_TAC[euler_jacobi_pseudoprime; ARITH] THEN | |
| MESON_TAC[COPRIME_1]; | |
| ALL_TAC] THEN | |
| ASM_CASES_TAC `carmichael_number n` THENL | |
| [ALL_TAC; | |
| FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [carmichael_number]) THEN | |
| ASM_REWRITE_TAC[ARITH_RULE `2 <= n <=> ~(n = 0) /\ ~(n = 1)`] THEN | |
| MESON_TAC[EULER_JACOBI_IMP_FERMAT_PSEUDOPRIME]] THEN | |
| MP_TAC(SPEC `n:num` PRIME_FACTOR) THEN | |
| ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN | |
| X_GEN_TAC `p:num` THEN STRIP_TAC THEN | |
| FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [divides]) THEN | |
| DISCH_THEN(X_CHOOSE_THEN `q:num` (MP_TAC o SYM)) THEN | |
| ASM_CASES_TAC `q = 0` THEN ASM_REWRITE_TAC[MULT_CLAUSES] THEN | |
| ASM_CASES_TAC `q = 1` THENL [ASM_MESON_TAC[MULT_CLAUSES]; DISCH_TAC] THEN | |
| FIRST_ASSUM(MP_TAC o MATCH_MP CARMICHAEL_NUMBER_IMP_SQUAREFREE) THEN | |
| REWRITE_TAC[SQUAREFREE_COPRIME] THEN | |
| DISCH_THEN(MP_TAC o SPECL [`p:num`; `q:num`]) THEN | |
| ASM_REWRITE_TAC[] THEN DISCH_TAC THEN | |
| MP_TAC(snd(EQ_IMP_RULE(SPEC `p:num` QUADRATIC_NONRESIDUE_EXISTS))) THEN | |
| ANTS_TAC THENL | |
| [ASM_MESON_TAC[ODD_PRIME; PRIME_ODD; CARMICHAEL_NUMBER_IMP_ODD; | |
| ODD_MULT; NUM_ODD_CONV `ODD 2`]; | |
| DISCH_THEN(X_CHOOSE_THEN `a:num` STRIP_ASSUME_TAC)] THEN | |
| MP_TAC(ISPECL [`p:num`; `q:num`; `a:num`; `1`] | |
| CHINESE_REMAINDER_USUAL) THEN | |
| ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:num` THEN | |
| STRIP_TAC THEN EXPAND_TAC "n" THEN REWRITE_TAC[COPRIME_RMUL] THEN | |
| ASM_SIMP_TAC[NUMBER_RULE `(t == 1) (mod q) ==> coprime(t,q)`] THEN | |
| CONJ_TAC THENL [ASM_MESON_TAC[CONG_COPRIME; COPRIME_SYM]; ALL_TAC] THEN | |
| REWRITE_TAC[euler_jacobi_pseudoprime] THEN | |
| DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN | |
| EXPAND_TAC "n" THEN REWRITE_TAC[JACOBI_RMUL] THEN | |
| SUBGOAL_THEN `jacobi(t,q) = &1` SUBST1_TAC THENL | |
| [ASM_MESON_TAC[JACOBI_CONG; JACOBI_1]; ALL_TAC] THEN | |
| REWRITE_TAC[GSYM INT_OF_NUM_MUL; INT_MUL_RID] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (INTEGER_RULE | |
| `(x:int == y) (mod (p * q)) ==> (x == y) (mod q)`)) THEN | |
| ASM_SIMP_TAC[JACOBI_PRIME] THEN MP_TAC(NUMBER_RULE | |
| `coprime(p:num,a) /\ (t == a) (mod p) /\ p divides t ==> p = 1`) THEN | |
| COND_CASES_TAC THENL [ASM_MESON_TAC[PRIME_1]; DISCH_THEN(K ALL_TAC)] THEN | |
| COND_CASES_TAC THENL [ASM_MESON_TAC[CONG_TRANS; CONG_SYM]; ALL_TAC] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (INTEGER_RULE | |
| `(--a == y) (mod q) ==> (y == a) (mod q) ==> q divides (a - --a)`)) THEN | |
| CONV_TAC INT_REDUCE_CONV THEN | |
| REWRITE_TAC[GSYM num_congruent; GSYM num_divides; INT_OF_NUM_POW] THEN | |
| ASM_SIMP_TAC[CONG_EXP_1] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE_STRONG) THEN | |
| REWRITE_TAC[ARITH_RULE `1 <= q /\ q <= 2 <=> q = 1 \/ q = 2`; ARITH_EQ] THEN | |
| ASM_MESON_TAC[CARMICHAEL_NUMBER_IMP_ODD; NOT_ODD; | |
| ODD_MULT; NUM_ODD_CONV `ODD 2`]);; | |
| let EULER_JACOBI_PSEUDOPRIME_BOUND_PHI_ALT = prove | |
| (`!n. ~(n = 1) /\ ~prime n | |
| ==> 2 * CARD {a | a < n /\ euler_jacobi_pseudoprime a n} <= phi n`, | |
| GEN_TAC THEN ASM_CASES_TAC `n = 0` THEN | |
| ASM_REWRITE_TAC[LT; EMPTY_GSPEC; CARD_CLAUSES; MULT_CLAUSES; LE_0] THEN | |
| STRIP_TAC THEN | |
| MP_TAC(SPEC `n:num` ABSOLUTE_EULER_JACOBI_PSEUDOPRIME) THEN | |
| ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN | |
| DISCH_THEN(X_CHOOSE_THEN `b:num` STRIP_ASSUME_TAC) THEN | |
| MATCH_MP_TAC CARD_SUBSET_HALF_LEMMA THEN | |
| EXISTS_TAC `\a. (a * b) MOD n` THEN | |
| EXISTS_TAC `{a:num | coprime(a,n) /\ a < n}` THEN | |
| SIMP_TAC[PHI_ALT; LE_REFL; SUBSET; FORALL_IN_IMAGE] THEN | |
| REWRITE_TAC[IN_ELIM_THM; IN_DIFF] THEN REPEAT CONJ_TAC THENL | |
| [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{i:num | i < n}` THEN | |
| REWRITE_TAC[FINITE_NUMSEG_LT] THEN SET_TAC[]; | |
| ASM_MESON_TAC[EULER_JACOBI_PSEUDOPRIME_IMP_COPRIME; LT]; | |
| ALL_TAC; | |
| REWRITE_TAC[GSYM CONG] THEN REPEAT STRIP_TAC THEN | |
| MATCH_MP_TAC CONG_IMP_EQ THEN EXISTS_TAC `n:num` THEN | |
| ASM_MESON_TAC[CONG_MULT_RCANCEL]] THEN | |
| X_GEN_TAC `a:num` THEN STRIP_TAC THEN | |
| FIRST_ASSUM(MP_TAC o MATCH_MP EULER_JACOBI_PSEUDOPRIME_IMP_COPRIME) THEN | |
| ASM_REWRITE_TAC[euler_jacobi_pseudoprime; COPRIME_LMOD; COPRIME_LMUL] THEN | |
| STRIP_TAC THEN ASM_REWRITE_TAC[MOD_LT_EQ] THEN | |
| DISCH_THEN(MP_TAC o CONJUNCT2) THEN | |
| REWRITE_TAC[JACOBI_MOD; GSYM INT_OF_NUM_REM; JACOBI_LMUL] THEN | |
| REWRITE_TAC[GSYM INT_REM_EQ; INT_POW_REM] THEN | |
| REWRITE_TAC[INT_REM_EQ; GSYM INT_OF_NUM_MUL; INT_POW_MUL] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (INTEGER_RULE | |
| `(x * y:int == a * b) (mod n) | |
| ==> (x == a) (mod n) /\ coprime(a,n) ==> (y == b) (mod n)`)) THEN | |
| RULE_ASSUM_TAC(REWRITE_RULE[euler_jacobi_pseudoprime]) THEN | |
| ASM_REWRITE_TAC[INT_COPRIME_LPOW; GSYM num_coprime] THEN | |
| ASM_MESON_TAC[]);; | |
| let EULER_JACOBI_PSEUDOPRIME_BOUND_PHI = prove | |
| (`!n. ~(n = 1) /\ ~prime n | |
| ==> CARD {a | a < n /\ euler_jacobi_pseudoprime a n} <= phi n DIV 2`, | |
| REWRITE_TAC[ARITH_RULE `a <= b DIV 2 <=> 2 * a <= b`] THEN | |
| REWRITE_TAC[EULER_JACOBI_PSEUDOPRIME_BOUND_PHI_ALT]);; | |
| let EULER_JACOBI_PSEUDOPRIME_BOUND_LT = prove | |
| (`!n. ~(n = 0) /\ ~(n = 1) /\ ~prime n | |
| ==> CARD {a | a < n /\ euler_jacobi_pseudoprime a n} < n DIV 2`, | |
| REPEAT STRIP_TAC THEN | |
| MP_TAC(SPEC `n:num` EULER_JACOBI_PSEUDOPRIME_BOUND_PHI_ALT) THEN | |
| ASM_REWRITE_TAC[] THEN | |
| MATCH_MP_TAC(ARITH_RULE `p < n - 1 ==> 2 * a <= p ==> a < n DIV 2`) THEN | |
| MATCH_MP_TAC PHI_LIMIT_COMPOSITE THEN ASM_REWRITE_TAC[]);; | |
| let MILLER_RABIN_IMP_EULER_JACOBI_PSEUDOPRIME = prove | |
| (`!a q. miller_rabin_pseudoprime a q /\ ~(q = 2) | |
| ==> euler_jacobi_pseudoprime a q`, | |
| let lemma0 = prove | |
| (`!x m n (k:A->bool). | |
| n divides m EXP 2 /\ FINITE k | |
| ==> (nproduct k (\i. m * x i + 1) == m * nsum k x + 1) (mod n)`, | |
| REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN | |
| REPEAT GEN_TAC THEN STRIP_TAC THEN | |
| MATCH_MP_TAC FINITE_INDUCT_STRONG THEN | |
| SIMP_TAC[NPRODUCT_CLAUSES; NSUM_CLAUSES; ADD_CLAUSES; MULT_CLAUSES] THEN | |
| REWRITE_TAC[CONG_REFL] THEN MAP_EVERY X_GEN_TAC [`a:A`; `s:A->bool`] THEN | |
| DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (K ALL_TAC)) THEN | |
| POP_ASSUM MP_TAC THEN CONV_TAC NUMBER_RULE) in | |
| let lemma1 = prove | |
| (`!x k m n (s:A->bool). | |
| n divides m EXP 2 /\ FINITE s | |
| ==> (nproduct s (\i. (m * x i + 1) EXP k i) == | |
| m * nsum s (\i. k i * x i) + 1) (mod n)`, | |
| REPEAT STRIP_TAC THEN MATCH_MP_TAC CONG_TRANS THEN | |
| EXISTS_TAC `nproduct s (\i:A. m * k i * x i + 1)` THEN | |
| ASM_SIMP_TAC[lemma0] THEN MATCH_MP_TAC CONG_NPRODUCT THEN | |
| ASM_REWRITE_TAC[] THEN X_GEN_TAC `i:A` THEN DISCH_TAC THEN | |
| ONCE_REWRITE_TAC[MESON[CARD_NUMSEG_1] `a EXP k = a EXP CARD(1..k)`] THEN | |
| SIMP_TAC[GSYM NPRODUCT_CONST; FINITE_NUMSEG] THEN | |
| W(MP_TAC o PART_MATCH (lhand o rator o rand) lemma0 o | |
| lhand o rator o snd) THEN | |
| ASM_SIMP_TAC[FINITE_NUMSEG; NSUM_CONST; CARD_NUMSEG_1]) in | |
| MAP_EVERY X_GEN_TAC [`a:num`; `n:num`] THEN | |
| ASM_CASES_TAC `n = 2` THEN ASM_REWRITE_TAC[] THEN | |
| ASM_CASES_TAC `coprime(a:num,n)` THENL | |
| [ALL_TAC; ASM_MESON_TAC[MILLER_RABIN_PSEUDOPRIME_IMP_COPRIME]] THEN | |
| REWRITE_TAC[miller_rabin_pseudoprime; euler_jacobi_pseudoprime] THEN | |
| ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[ODD] THEN | |
| ASM_CASES_TAC `n = 1` THEN ASM_REWRITE_TAC[INT_CONG_MOD_1; ARITH] THEN | |
| DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN | |
| REWRITE_TAC[num_congruent; GSYM INT_OF_NUM_POW] THEN | |
| ASM_SIMP_TAC[GSYM INT_OF_NUM_SUB; LE_1] THEN | |
| REWRITE_TAC[INTEGER_RULE | |
| `(x:int == n - z) (mod n) <=> (x == --z) (mod n)`] THEN | |
| MP_TAC(SPECL [`n - 1`; `2`] INDEX_DECOMPOSITION) THEN | |
| ASM_REWRITE_TAC[ARITH_EQ; ARITH_RULE `n - 1 = 0 <=> n = 0 \/ n = 1`] THEN | |
| ABBREV_TAC `e = index 2 (n - 1)` THEN | |
| REWRITE_TAC[DIVIDES_2; NOT_EVEN; LEFT_IMP_EXISTS_THM] THEN | |
| X_GEN_TAC `m:num` THEN STRIP_TAC THEN CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN | |
| SUBST1_TAC(SYM(ASSUME `2 EXP e * m = n - 1`)) THEN | |
| SIMP_TAC[DIV_MULT; EXP_EQ_0; ARITH_EQ] THEN | |
| SUBGOAL_THEN `~(e = 0)` ASSUME_TAC THENL | |
| [EXPAND_TAC "e" THEN REWRITE_TAC[INDEX_EQ_0] THEN | |
| ASM_REWRITE_TAC[NOT_EVEN; DIVIDES_2; ODD_SUB] THEN ASM_ARITH_TAC; | |
| ALL_TAC] THEN | |
| SUBGOAL_THEN `(2 EXP e * m) DIV 2 = 2 EXP (e - 1) * m` SUBST1_TAC THENL | |
| [MATCH_MP_TAC DIV_UNIQ THEN EXISTS_TAC `0` THEN | |
| REWRITE_TAC[ARITH_RULE `(ee * m) * 2 + 0 = (2 * ee) * m`] THEN | |
| REWRITE_TAC[ARITH] THEN AP_THM_TAC THEN AP_TERM_TAC THEN | |
| REWRITE_TAC[GSYM(CONJUNCT2 EXP)] THEN AP_TERM_TAC THEN ASM_ARITH_TAC; | |
| ALL_TAC] THEN | |
| DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC MP_TAC) THENL | |
| [ONCE_REWRITE_TAC[MULT_SYM] THEN REWRITE_TAC[GSYM INT_POW_POW] THEN | |
| MATCH_MP_TAC(INTEGER_RULE | |
| `(a:int == &1) (mod n) /\ j = &1 ==> (j == a) (mod n)`) THEN | |
| ASM_SIMP_TAC[INT_CONG_POW_1] THEN | |
| FIRST_ASSUM(fun th -> | |
| GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o RAND_CONV) | |
| [SYM(MATCH_MP PRIME_FACTORIZATION th)]) THEN | |
| ASM_SIMP_TAC[JACOBI_NPRODUCT_RIGHT; FINITE_SPECIAL_DIVISORS] THEN | |
| MATCH_MP_TAC IPRODUCT_EQ_1 THEN X_GEN_TAC `p:num` THEN | |
| REWRITE_TAC[IN_ELIM_THM; JACOBI_REXP] THEN STRIP_TAC THEN | |
| MATCH_MP_TAC(MESON[INT_POW_ONE] `x:int = &1 ==> x pow n = &1`) THEN | |
| MATCH_MP_TAC JACOBI_EQ_1 THEN ASM_SIMP_TAC[EULER_CRITERION] THEN | |
| MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL | |
| [ASM_MESON_TAC[NUMBER_RULE | |
| `coprime(a:num,n) /\ p divides n ==> coprime(a,p)`]; | |
| DISCH_TAC THEN DISJ2_TAC] THEN | |
| REWRITE_TAC[ORDER_DIVIDES] THEN MATCH_MP_TAC DIVIDES_DIVIDES_DIV_IMP THEN | |
| MATCH_MP_TAC DIVIDES_MUL THEN | |
| ASM_REWRITE_TAC[COPRIME_2; DIVIDES_2; EVEN_SUB; ARITH] THEN | |
| CONJ_TAC THENL [ASM_MESON_TAC[NOT_EVEN; divides; EVEN_MULT]; ALL_TAC] THEN | |
| ASM_SIMP_TAC[GSYM PHI_PRIME] THEN CONJ_TAC THENL | |
| [ASM_MESON_TAC[ORDER_DIVIDES_PHI; COPRIME_SYM]; ALL_TAC] THEN | |
| MATCH_MP_TAC(MESON[divides; ODD_MULT] | |
| `!n. ODD n /\ m divides n ==> ODD m`) THEN | |
| EXISTS_TAC `m:num` THEN ASM_REWRITE_TAC[GSYM ORDER_DIVIDES] THEN | |
| REWRITE_TAC[num_congruent; GSYM INT_OF_NUM_POW] THEN | |
| FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INTEGER_RULE | |
| `(x:int == y) (mod n) ==> p divides n ==> (x == y) (mod p)`)) THEN | |
| ASM_REWRITE_TAC[GSYM num_divides]; | |
| ALL_TAC] THEN | |
| DISCH_THEN(X_CHOOSE_THEN `r:num` STRIP_ASSUME_TAC) THEN | |
| SUBGOAL_THEN | |
| `!p. prime p /\ p divides n | |
| ==> ?d. jacobi(a,p) = --(&1) pow d /\ 2 EXP (r + 1) * d + 1 = p` | |
| MP_TAC THENL | |
| [X_GEN_TAC `p:num` THEN STRIP_TAC THEN | |
| FIRST_ASSUM(MP_TAC o SPEC `p:num` o MATCH_MP (NUMBER_RULE | |
| `coprime(a:num,n) ==> !p. p divides n ==> coprime(p,a)`)) THEN | |
| ASM_REWRITE_TAC[] THEN DISCH_TAC THEN | |
| SUBGOAL_THEN `ODD p /\ ~(p = 2)` STRIP_ASSUME_TAC THENL | |
| [ASM_MESON_TAC[PRIME_ODD; ODD_PRIME; divides; ODD_MULT; | |
| NUM_REDUCE_CONV `ODD 2`]; | |
| ALL_TAC] THEN | |
| MP_TAC(SPECL [`order p a`; `2`] INDEX_DECOMPOSITION) THEN | |
| DISCH_THEN(X_CHOOSE_THEN `i:num` MP_TAC) THEN | |
| ABBREV_TAC `j = index 2 (order p a)` THEN | |
| ASM_REWRITE_TAC[ORDER_EQ_0; ARITH_EQ; DIVIDES_2; NOT_EVEN] THEN | |
| STRIP_TAC THEN | |
| SUBGOAL_THEN `j = r + 1` SUBST_ALL_TAC THENL | |
| [SUBGOAL_THEN | |
| `2 EXP j * i divides 2 EXP (r + 1) * m /\ | |
| ~(2 EXP j * i divides 2 EXP r * m)` | |
| MP_TAC THENL | |
| [ASM_REWRITE_TAC[GSYM ORDER_DIVIDES] THEN | |
| REWRITE_TAC[EXP_ADD; EXP_1] THEN | |
| REWRITE_TAC[ARITH_RULE `(a * 2) * b = a * b + a * b`] THEN | |
| REWRITE_TAC[EXP_ADD; GSYM EXP_2] THEN | |
| REWRITE_TAC[num_congruent; GSYM INT_OF_NUM_POW] THEN | |
| FIRST_X_ASSUM(MP_TAC o SPEC `&p:int` o MATCH_MP (INTEGER_RULE | |
| `(x:int == y) (mod n) ==> !p. p divides n ==> (x == y) (mod p)`)) THEN | |
| ASM_SIMP_TAC[GSYM num_divides; INTEGER_RULE | |
| `(a:int == -- &1) (mod n) ==> (a pow 2 == &1) (mod n)`] THEN | |
| REWRITE_TAC[TAUT `p ==> ~q <=> ~(p /\ q)`] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP (INTEGER_RULE | |
| `(x:int == -- &1) (mod p) /\ (x == &1) (mod p) | |
| ==> p divides &2`)) THEN | |
| ASM_SIMP_TAC[GSYM num_divides; DIVIDES_PRIME_PRIME; PRIME_2]; | |
| REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(MP_TAC o MATCH_MP (NUMBER_RULE | |
| `e1 * m divides e2 * n | |
| ==> coprime(e1:num,n) /\ coprime(e2,m) | |
| ==> e1 divides e2 /\ m divides n`)) THEN | |
| ASM_REWRITE_TAC[COPRIME_LEXP; COPRIME_2] THEN | |
| ASM_SIMP_TAC[DIVIDES_EXP_LE; LE_REFL] THEN | |
| REWRITE_TAC[ARITH_RULE `j <= r + 1 <=> j = r + 1 \/ j <= r`] THEN | |
| ASM_CASES_TAC `j = r + 1` THEN ASM_REWRITE_TAC[] THEN | |
| STRIP_TAC THEN FIRST_X_ASSUM(fun th -> | |
| GEN_REWRITE_TAC LAND_CONV [GSYM th]) THEN | |
| MATCH_MP_TAC DIVIDES_MUL2 THEN ASM_SIMP_TAC[DIVIDES_EXP_LE; LE_REFL]]; | |
| MP_TAC(SPECL [`a:num`; `p:num`] ORDER_DIVIDES_PHI) THEN | |
| ASM_SIMP_TAC[PHI_PRIME; divides; LEFT_IMP_EXISTS_THM] THEN | |
| X_GEN_TAC `q:num` THEN DISCH_THEN(MP_TAC o MATCH_MP (ARITH_RULE | |
| `p - 1 = d ==> ~(p = 0) ==> d + 1 = p`)) THEN | |
| ANTS_TAC THENL [ASM_MESON_TAC[PRIME_0]; DISCH_TAC] THEN | |
| EXISTS_TAC `i * q:num` THEN ASM_REWRITE_TAC[MULT_ASSOC] THEN | |
| MATCH_MP_TAC INT_CONG_IMP_EQ THEN EXISTS_TAC `&p:int` THEN CONJ_TAC THENL | |
| [MATCH_MP_TAC(INT_ARITH | |
| `&3 <= p /\ abs(x) <= &1 /\ abs(y) <= &1 ==> abs(x - y:int) < p`) THEN | |
| REWRITE_TAC[JACOBI_BOUND; INT_ABS_POW; INT_ABS_NEG] THEN | |
| REWRITE_TAC[INT_ABS_NUM; INT_POW_ONE; INT_LE_REFL] THEN | |
| ASM_MESON_TAC[INT_OF_NUM_LE; ODD_PRIME; divides; ODD_MULT]; | |
| ALL_TAC] THEN | |
| MATCH_MP_TAC INT_CONG_TRANS THEN | |
| EXISTS_TAC `(&a:int) pow ((p - 1) DIV 2)` THEN | |
| ASM_SIMP_TAC[JACOBI_EULER] THEN | |
| SUBGOAL_THEN `(-- &1:int) pow (i * q) = -- &1 pow q` | |
| SUBST1_TAC THENL | |
| [ASM_REWRITE_TAC[INT_POW_NEG; INT_POW_ONE; GSYM NOT_ODD] THEN | |
| ASM_REWRITE_TAC[ODD_MULT]; | |
| ALL_TAC] THEN | |
| SUBGOAL_THEN `(p - 1) DIV 2 = order p a DIV 2 * q` SUBST1_TAC THENL | |
| [FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP (ARITH_RULE | |
| `a + 1 = p ==> p - 1 = a`)) THEN | |
| SUBST1_TAC(SYM(ASSUME `2 EXP (r + 1) * i = order p a`)) THEN | |
| REWRITE_TAC[GSYM ADD1; EXP; GSYM MULT_ASSOC] THEN | |
| SIMP_TAC[DIV_MULT; ARITH_EQ] THEN REWRITE_TAC[MULT_AC]; | |
| REWRITE_TAC[GSYM INT_POW_POW]] THEN | |
| MATCH_MP_TAC INT_CONG_POW THEN REWRITE_TAC[INTEGER_RULE | |
| `(a:int == -- &1) (mod p) <=> p divides (a + &1)`] THEN | |
| MP_TAC(SPECL [`p:num`; `1`; `a EXP (order p a DIV 2)`] | |
| CONG_SQUARE_1_PRIME_POWER) THEN | |
| ASM_REWRITE_TAC[EXP_1; INT_OF_NUM_ADD; INT_OF_NUM_POW] THEN | |
| REWRITE_TAC[CONG_MINUS1; EXP_EXP; GSYM num_divides] THEN | |
| REWRITE_TAC[ORDER_DIVIDES] THEN MATCH_MP_TAC(TAUT | |
| `~r /\ ~q /\ p ==> (p <=> q \/ r \/ s) ==> s`) THEN | |
| CONJ_TAC THENL [ASM_MESON_TAC[PRIME_0]; ALL_TAC] THEN | |
| SUBST1_TAC(SYM(ASSUME `2 EXP (r + 1) * i = order p a`)) THEN | |
| REWRITE_TAC[GSYM ADD1; EXP; GSYM MULT_ASSOC] THEN | |
| SIMP_TAC[DIV_MULT; ARITH_EQ] THEN CONJ_TAC THENL | |
| [ALL_TAC; REWRITE_TAC[MULT_AC; DIVIDES_REFL]] THEN | |
| DISCH_THEN(MP_TAC o MATCH_MP DIVIDES_LE) THEN | |
| REWRITE_TAC[ARITH_RULE `2 * n <= n <=> n = 0`] THEN | |
| REWRITE_TAC[EXP_EQ_0; MULT_EQ_0; ARITH_EQ] THEN ASM_MESON_TAC[ODD]]; | |
| REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN | |
| X_GEN_TAC `d:num->num` THEN | |
| REWRITE_TAC[TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN | |
| REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN | |
| SUBGOAL_THEN | |
| `(nsum {p | prime p /\ p divides n} (\p. index p n * d p) == | |
| 2 EXP (e - r - 1) * m) (mod 2)` | |
| ASSUME_TAC THENL | |
| [MATCH_MP_TAC(NUMBER_RULE | |
| `!a. (a * x == a * y) (mod (a * n)) /\ ~(a = 0) | |
| ==> (x == y) (mod n)`) THEN | |
| EXISTS_TAC `2 EXP (r + 1)` THEN | |
| REWRITE_TAC[EXP_EQ_0; ARITH_EQ; MULT_ASSOC; GSYM EXP_ADD] THEN | |
| ASM_SIMP_TAC[ARITH_RULE `r < e ==> (r + 1) + e - r - 1 = e`] THEN | |
| MATCH_MP_TAC(NUMBER_RULE | |
| `(x + 1 == y + 1) (mod n) ==> (y == x) (mod n)`) THEN | |
| ASM_SIMP_TAC[ARITH_RULE `~(n = 0) ==> n - 1 + 1 = n`] THEN | |
| REWRITE_TAC[ONCE_REWRITE_RULE[MULT_SYM] (GSYM(CONJUNCT2 EXP))] THEN | |
| REWRITE_TAC[ARITH_RULE `SUC(r + 1) = r + 2`] THEN | |
| MP_TAC(ISPECL | |
| [`d:num->num`; `\p. index p n`; `2 EXP (r + 1)`; `2 EXP (r + 2)`; | |
| `{p | prime p /\ p divides n}`] lemma1) THEN | |
| SIMP_TAC[EXP_EXP; DIVIDES_EXP_LE; LE_REFL] THEN | |
| REWRITE_TAC[ARITH_RULE `r + 2 <= (r + 1) * 2`] THEN | |
| ASM_SIMP_TAC[FINITE_SPECIAL_DIVISORS; PRIME_FACTORIZATION]; | |
| ALL_TAC] THEN | |
| FIRST_ASSUM(fun th -> | |
| GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV o RAND_CONV o RAND_CONV) | |
| [SYM(MATCH_MP PRIME_FACTORIZATION th)]) THEN | |
| ASM_SIMP_TAC[JACOBI_NPRODUCT_RIGHT; FINITE_SPECIAL_DIVISORS] THEN | |
| ASM_SIMP_TAC[JACOBI_REXP] THEN | |
| REWRITE_TAC[ONCE_REWRITE_RULE[MULT_SYM] INT_POW_POW] THEN | |
| ASM_SIMP_TAC[GSYM INT_POW_NSUM; FINITE_SPECIAL_DIVISORS] THEN | |
| FIRST_ASSUM(MP_TAC o MATCH_MP CONG_DIVIDES) THEN REWRITE_TAC[DIVIDES_2] THEN | |
| DISCH_THEN(SUBST1_TAC o MATCH_MP (METIS[INT_POW_NEG; INT_POW_ONE] | |
| `(EVEN m <=> EVEN n) ==> (--(&1:int) pow m = --(&1) pow n)`)) THEN | |
| FIRST_ASSUM(SUBST1_TAC o MATCH_MP (ARITH_RULE | |
| `r < e ==> e - 1 = r + (e - r - 1)`)) THEN | |
| REWRITE_TAC[EXP_ADD; ARITH_RULE `(a * b) * m:num = b * a * m`] THEN | |
| ONCE_REWRITE_TAC[MULT_SYM] THEN ONCE_REWRITE_TAC[GSYM INT_POW_POW] THEN | |
| MATCH_MP_TAC INT_CONG_POW THEN ONCE_REWRITE_TAC[INT_CONG_SYM] THEN | |
| ASM_REWRITE_TAC[INT_POW_NEG; INT_POW_ONE; GSYM NOT_ODD]);; | |
| let MILLER_RABIN_EQ_EULER_JACOBI_PSEUDOPRIME = prove | |
| (`!a n. ~(n = 2) /\ (?p k. prime p /\ ODD p /\ p EXP k = n) \/ | |
| (n == 3) (mod 4) | |
| ==> (miller_rabin_pseudoprime a n <=> euler_jacobi_pseudoprime a n)`, | |
| REPEAT GEN_TAC THEN ASM_CASES_TAC `n = 2` THEN ASM_REWRITE_TAC[] THENL | |
| [REWRITE_TAC[CONG] THEN CONV_TAC NUM_REDUCE_CONV; DISCH_TAC] THEN | |
| MP_TAC(SPECL [`a:num`; `n:num`] MILLER_RABIN_EQ_WEAK_EULER_PSEUDOPRIME) THEN | |
| ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT | |
| `(q ==> r) /\ (p ==> q) | |
| ==> (p <=> r) ==> (p <=> q)`) THEN | |
| REWRITE_TAC[EULER_JACOBI_IMP_WEAK_EULER_PSEUDOPRIME] THEN | |
| ASM_SIMP_TAC[ MILLER_RABIN_IMP_EULER_JACOBI_PSEUDOPRIME]);; | |