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|
(* ========================================================================= *)
(* Properties of complex polynomials (not canonically represented). *)
(* ========================================================================= *)
needs "Complex/complexnumbers.ml";;
prioritize_complex();;
parse_as_infix("++",(16,"right"));;
parse_as_infix("**",(20,"right"));;
parse_as_infix("##",(20,"right"));;
parse_as_infix("divides",(14,"right"));;
parse_as_infix("exp",(22,"right"));;
do_list override_interface
["++",`poly_add:complex list->complex list->complex list`;
"**",`poly_mul:complex list->complex list->complex list`;
"##",`poly_cmul:complex->complex list->complex list`;
"neg",`poly_neg:complex list->complex list`;
"divides",`poly_divides:complex list->complex list->bool`;
"exp",`poly_exp:complex list -> num -> complex list`;
"diff",`poly_diff:complex list->complex list`];;
let SIMPLE_COMPLEX_ARITH tm = prove(tm,SIMPLE_COMPLEX_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Polynomials. *)
(* ------------------------------------------------------------------------- *)
let poly = new_recursive_definition list_RECURSION
`(poly [] x = Cx(&0)) /\
(poly (CONS h t) x = h + x * poly t x)`;;
(* ------------------------------------------------------------------------- *)
(* Arithmetic operations on polynomials. *)
(* ------------------------------------------------------------------------- *)
let poly_add = new_recursive_definition list_RECURSION
`([] ++ l2 = l2) /\
((CONS h t) ++ l2 =
(if l2 = [] then CONS h t
else CONS (h + HD l2) (t ++ TL l2)))`;;
let poly_cmul = new_recursive_definition list_RECURSION
`(c ## [] = []) /\
(c ## (CONS h t) = CONS (c * h) (c ## t))`;;
let poly_neg = new_definition
`neg = (##) (--(Cx(&1)))`;;
let poly_mul = new_recursive_definition list_RECURSION
`([] ** l2 = []) /\
((CONS h t) ** l2 =
if t = [] then h ## l2
else (h ## l2) ++ CONS (Cx(&0)) (t ** l2))`;;
let poly_exp = new_recursive_definition num_RECURSION
`(p exp 0 = [Cx(&1)]) /\
(p exp (SUC n) = p ** p exp n)`;;
(* ------------------------------------------------------------------------- *)
(* Useful clausifications. *)
(* ------------------------------------------------------------------------- *)
let POLY_ADD_CLAUSES = prove
(`([] ++ p2 = p2) /\
(p1 ++ [] = p1) /\
((CONS h1 t1) ++ (CONS h2 t2) = CONS (h1 + h2) (t1 ++ t2))`,
REWRITE_TAC[poly_add; NOT_CONS_NIL; HD; TL] THEN
SPEC_TAC(`p1:complex list`,`p1:complex list`) THEN
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[poly_add]);;
let POLY_CMUL_CLAUSES = prove
(`(c ## [] = []) /\
(c ## (CONS h t) = CONS (c * h) (c ## t))`,
REWRITE_TAC[poly_cmul]);;
let POLY_NEG_CLAUSES = prove
(`(neg [] = []) /\
(neg (CONS h t) = CONS (--h) (neg t))`,
REWRITE_TAC[poly_neg; POLY_CMUL_CLAUSES;
COMPLEX_MUL_LNEG; COMPLEX_MUL_LID]);;
let POLY_MUL_CLAUSES = prove
(`([] ** p2 = []) /\
([h1] ** p2 = h1 ## p2) /\
((CONS h1 (CONS k1 t1)) ** p2 =
h1 ## p2 ++ CONS (Cx(&0)) (CONS k1 t1 ** p2))`,
REWRITE_TAC[poly_mul; NOT_CONS_NIL]);;
(* ------------------------------------------------------------------------- *)
(* Various natural consequences of syntactic definitions. *)
(* ------------------------------------------------------------------------- *)
let POLY_ADD = prove
(`!p1 p2 x. poly (p1 ++ p2) x = poly p1 x + poly p2 x`,
LIST_INDUCT_TAC THEN REWRITE_TAC[poly_add; poly; COMPLEX_ADD_LID] THEN
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC[NOT_CONS_NIL; HD; TL; poly; COMPLEX_ADD_RID] THEN
SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_CMUL = prove
(`!p c x. poly (c ## p) x = c * poly p x`,
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[poly; poly_cmul] THEN
SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_NEG = prove
(`!p x. poly (neg p) x = --(poly p x)`,
REWRITE_TAC[poly_neg; POLY_CMUL] THEN
SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_MUL = prove
(`!x p1 p2. poly (p1 ** p2) x = poly p1 x * poly p2 x`,
GEN_TAC THEN LIST_INDUCT_TAC THEN
REWRITE_TAC[poly_mul; poly; COMPLEX_MUL_LZERO; POLY_CMUL; POLY_ADD] THEN
SPEC_TAC(`h:complex`,`h:complex`) THEN
SPEC_TAC(`t:complex list`,`t:complex list`) THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC[poly_mul; POLY_CMUL; POLY_ADD; poly; POLY_CMUL;
COMPLEX_MUL_RZERO; COMPLEX_ADD_RID; NOT_CONS_NIL] THEN
ASM_REWRITE_TAC[POLY_ADD; POLY_CMUL; poly] THEN
SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_EXP = prove
(`!p n x. poly (p exp n) x = (poly p x) pow n`,
GEN_TAC THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[poly_exp; complex_pow; POLY_MUL] THEN
REWRITE_TAC[poly] THEN SIMPLE_COMPLEX_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Lemmas. *)
(* ------------------------------------------------------------------------- *)
let POLY_ADD_RZERO = prove
(`!p. poly (p ++ []) = poly p`,
REWRITE_TAC[FUN_EQ_THM; POLY_ADD; poly; COMPLEX_ADD_RID]);;
let POLY_MUL_ASSOC = prove
(`!p q r. poly (p ** (q ** r)) = poly ((p ** q) ** r)`,
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; COMPLEX_MUL_ASSOC]);;
let POLY_EXP_ADD = prove
(`!d n p. poly(p exp (n + d)) = poly(p exp n ** p exp d)`,
REWRITE_TAC[FUN_EQ_THM; POLY_MUL] THEN
INDUCT_TAC THEN ASM_REWRITE_TAC[POLY_MUL; ADD_CLAUSES; poly_exp; poly] THEN
SIMPLE_COMPLEX_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Key property that f(a) = 0 ==> (x - a) divides p(x). Very delicate! *)
(* ------------------------------------------------------------------------- *)
let POLY_LINEAR_REM = prove
(`!t h. ?q r. CONS h t = [r] ++ [--a; Cx(&1)] ** q`,
LIST_INDUCT_TAC THEN REWRITE_TAC[] THENL
[GEN_TAC THEN EXISTS_TAC `[]:complex list` THEN
EXISTS_TAC `h:complex` THEN
REWRITE_TAC[poly_add; poly_mul; poly_cmul; NOT_CONS_NIL] THEN
REWRITE_TAC[HD; TL; COMPLEX_ADD_RID];
X_GEN_TAC `k:complex` THEN
POP_ASSUM(STRIP_ASSUME_TAC o SPEC `h:complex`) THEN
EXISTS_TAC `CONS (r:complex) q` THEN EXISTS_TAC `r * a + k` THEN
ASM_REWRITE_TAC[POLY_ADD_CLAUSES; POLY_MUL_CLAUSES; poly_cmul] THEN
REWRITE_TAC[CONS_11] THEN CONJ_TAC THENL
[SIMPLE_COMPLEX_ARITH_TAC; ALL_TAC] THEN
SPEC_TAC(`q:complex list`,`q:complex list`) THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC[POLY_ADD_CLAUSES; POLY_MUL_CLAUSES; poly_cmul] THEN
REWRITE_TAC[COMPLEX_ADD_RID; COMPLEX_MUL_LID] THEN
REWRITE_TAC[COMPLEX_ADD_AC]]);;
let POLY_LINEAR_DIVIDES = prove
(`!a p. (poly p a = Cx(&0)) <=> (p = []) \/ ?q. p = [--a; Cx(&1)] ** q`,
GEN_TAC THEN LIST_INDUCT_TAC THENL
[REWRITE_TAC[poly]; ALL_TAC] THEN
EQ_TAC THEN STRIP_TAC THENL
[DISJ2_TAC THEN STRIP_ASSUME_TAC(SPEC_ALL POLY_LINEAR_REM) THEN
EXISTS_TAC `q:complex list` THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `r = Cx(&0)` SUBST_ALL_TAC THENL
[UNDISCH_TAC `poly (CONS h t) a = Cx(&0)` THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[POLY_ADD; POLY_MUL] THEN
REWRITE_TAC[poly; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID;
COMPLEX_MUL_RID] THEN
REWRITE_TAC[COMPLEX_ADD_LINV] THEN SIMPLE_COMPLEX_ARITH_TAC;
REWRITE_TAC[poly_mul] THEN REWRITE_TAC[NOT_CONS_NIL] THEN
SPEC_TAC(`q:complex list`,`q:complex list`) THEN LIST_INDUCT_TAC THENL
[REWRITE_TAC[poly_cmul; poly_add; NOT_CONS_NIL;
HD; TL; COMPLEX_ADD_LID];
REWRITE_TAC[poly_cmul; poly_add; NOT_CONS_NIL;
HD; TL; COMPLEX_ADD_LID]]];
ASM_REWRITE_TAC[] THEN REWRITE_TAC[poly];
ASM_REWRITE_TAC[] THEN REWRITE_TAC[poly] THEN
REWRITE_TAC[POLY_MUL] THEN REWRITE_TAC[poly] THEN
REWRITE_TAC[poly; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID; COMPLEX_MUL_RID] THEN
REWRITE_TAC[COMPLEX_ADD_LINV] THEN SIMPLE_COMPLEX_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Thanks to the finesse of the above, we can use length rather than degree. *)
(* ------------------------------------------------------------------------- *)
let POLY_LENGTH_MUL = prove
(`!q. LENGTH([--a; Cx(&1)] ** q) = SUC(LENGTH q)`,
let lemma = prove
(`!p h k a. LENGTH (k ## p ++ CONS h (a ## p)) = SUC(LENGTH p)`,
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC[poly_cmul; POLY_ADD_CLAUSES; LENGTH]) in
REWRITE_TAC[poly_mul; NOT_CONS_NIL; lemma]);;
(* ------------------------------------------------------------------------- *)
(* Thus a nontrivial polynomial of degree n has no more than n roots. *)
(* ------------------------------------------------------------------------- *)
let POLY_ROOTS_INDEX_LEMMA = prove
(`!n. !p. ~(poly p = poly []) /\ (LENGTH p = n)
==> ?i. !x. (poly p x = Cx(&0)) ==> ?m. m <= n /\ (x = i m)`,
INDUCT_TAC THENL
[REWRITE_TAC[LENGTH_EQ_NIL] THEN MESON_TAC[];
REPEAT STRIP_TAC THEN ASM_CASES_TAC `?a. poly p a = Cx(&0)` THENL
[UNDISCH_TAC `?a. poly p a = Cx(&0)` THEN
DISCH_THEN(CHOOSE_THEN MP_TAC) THEN
GEN_REWRITE_TAC LAND_CONV [POLY_LINEAR_DIVIDES] THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `q:complex list` SUBST_ALL_TAC) THEN
FIRST_ASSUM(UNDISCH_TAC o check is_forall o concl) THEN
UNDISCH_TAC `~(poly ([-- a; Cx(&1)] ** q) = poly [])` THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[POLY_LENGTH_MUL; SUC_INJ] THEN
DISCH_TAC THEN ASM_CASES_TAC `poly q = poly []` THENL
[ASM_REWRITE_TAC[POLY_MUL; poly; COMPLEX_MUL_RZERO; FUN_EQ_THM];
DISCH_THEN(K ALL_TAC)] THEN
DISCH_THEN(MP_TAC o SPEC `q:complex list`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `i:num->complex`) THEN
EXISTS_TAC `\m. if m = SUC n then a:complex else i m` THEN
REWRITE_TAC[POLY_MUL; LE; COMPLEX_ENTIRE] THEN
X_GEN_TAC `x:complex` THEN DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
[DISCH_THEN(fun th -> EXISTS_TAC `SUC n` THEN MP_TAC th) THEN
REWRITE_TAC[poly] THEN SIMPLE_COMPLEX_ARITH_TAC;
DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `m:num` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `m:num` THEN ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `m:num <= n` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC];
UNDISCH_TAC `~(?a. poly p a = Cx(&0))` THEN
REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_TAC THEN ASM_REWRITE_TAC[]]]);;
let POLY_ROOTS_INDEX_LENGTH = prove
(`!p. ~(poly p = poly [])
==> ?i. !x. (poly p(x) = Cx(&0)) ==> ?n. n <= LENGTH p /\ (x = i n)`,
MESON_TAC[POLY_ROOTS_INDEX_LEMMA]);;
let POLY_ROOTS_FINITE_LEMMA = prove
(`!p. ~(poly p = poly [])
==> ?N i. !x. (poly p(x) = Cx(&0)) ==> ?n:num. n < N /\ (x = i n)`,
MESON_TAC[POLY_ROOTS_INDEX_LENGTH; LT_SUC_LE]);;
let FINITE_LEMMA = prove
(`!i N P. (!x. P x ==> ?n:num. n < N /\ (x = i n))
==> ?a. !x. P x ==> norm(x) < a`,
GEN_TAC THEN ONCE_REWRITE_TAC[RIGHT_IMP_EXISTS_THM] THEN INDUCT_TAC THENL
[REWRITE_TAC[LT] THEN MESON_TAC[]; ALL_TAC] THEN
X_GEN_TAC `P:complex->bool` THEN
POP_ASSUM(MP_TAC o SPEC `\z. P z /\ ~(z = (i:num->complex) N)`) THEN
DISCH_THEN(X_CHOOSE_TAC `a:real`) THEN
EXISTS_TAC `abs(a) + norm(i(N:num)) + &1` THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[LT] THEN
SUBGOAL_THEN `(!x. norm(x) < abs(a) + norm(x) + &1) /\
(!x y. norm(x) < a ==> norm(x) < abs(a) + norm(y) + &1)`
(fun th -> MP_TAC th THEN MESON_TAC[]) THEN
CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
REPEAT GEN_TAC THEN MP_TAC(SPEC `y:complex` COMPLEX_NORM_POS) THEN
REAL_ARITH_TAC);;
let POLY_ROOTS_FINITE = prove
(`!p. ~(poly p = poly []) <=>
?N i. !x. (poly p(x) = Cx(&0)) ==> ?n:num. n < N /\ (x = i n)`,
GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[POLY_ROOTS_FINITE_LEMMA] THEN
REWRITE_TAC[FUN_EQ_THM; LEFT_IMP_EXISTS_THM; NOT_FORALL_THM; poly] THEN
MP_TAC(GENL [`i:num->complex`; `N:num`]
(SPECL [`i:num->complex`; `N:num`; `\x. poly p x = Cx(&0)`]
FINITE_LEMMA)) THEN
REWRITE_TAC[] THEN MESON_TAC[REAL_ARITH `~(abs(x) < x)`; COMPLEX_NORM_CX]);;
(* ------------------------------------------------------------------------- *)
(* Hence get entirety and cancellation for polynomials. *)
(* ------------------------------------------------------------------------- *)
let POLY_ENTIRE_LEMMA = prove
(`!p q. ~(poly p = poly []) /\ ~(poly q = poly [])
==> ~(poly (p ** q) = poly [])`,
REPEAT GEN_TAC THEN REWRITE_TAC[POLY_ROOTS_FINITE] THEN
DISCH_THEN(CONJUNCTS_THEN MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `N2:num` (X_CHOOSE_TAC `i2:num->complex`)) THEN
DISCH_THEN(X_CHOOSE_THEN `N1:num` (X_CHOOSE_TAC `i1:num->complex`)) THEN
EXISTS_TAC `N1 + N2:num` THEN
EXISTS_TAC `\n:num. if n < N1 then i1(n):complex else i2(n - N1)` THEN
X_GEN_TAC `x:complex` THEN REWRITE_TAC[COMPLEX_ENTIRE; POLY_MUL] THEN
DISCH_THEN(DISJ_CASES_THEN (ANTE_RES_THEN (X_CHOOSE_TAC `n:num`))) THENL
[EXISTS_TAC `n:num` THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o CONJUNCT1) THEN ARITH_TAC;
EXISTS_TAC `N1 + n:num` THEN ASM_REWRITE_TAC[LT_ADD_LCANCEL] THEN
REWRITE_TAC[ARITH_RULE `~(m + n < m:num)`] THEN
AP_TERM_TAC THEN ARITH_TAC]);;
let POLY_ENTIRE = prove
(`!p q. (poly (p ** q) = poly []) <=>
(poly p = poly []) \/ (poly q = poly [])`,
REPEAT GEN_TAC THEN EQ_TAC THENL
[MESON_TAC[POLY_ENTIRE_LEMMA];
REWRITE_TAC[FUN_EQ_THM; POLY_MUL] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC[COMPLEX_MUL_RZERO; COMPLEX_MUL_LZERO; poly]]);;
let POLY_MUL_LCANCEL = prove
(`!p q r. (poly (p ** q) = poly (p ** r)) <=>
(poly p = poly []) \/ (poly q = poly r)`,
let lemma1 = prove
(`!p q. (poly (p ++ neg q) = poly []) <=> (poly p = poly q)`,
REWRITE_TAC[FUN_EQ_THM; POLY_ADD; POLY_NEG; poly] THEN
REWRITE_TAC[SIMPLE_COMPLEX_ARITH `(p + --q = Cx(&0)) <=> (p = q)`]) in
let lemma2 = prove
(`!p q r. poly (p ** q ++ neg(p ** r)) = poly (p ** (q ++ neg(r)))`,
REWRITE_TAC[FUN_EQ_THM; POLY_ADD; POLY_NEG; POLY_MUL] THEN
SIMPLE_COMPLEX_ARITH_TAC) in
ONCE_REWRITE_TAC[GSYM lemma1] THEN
REWRITE_TAC[lemma2; POLY_ENTIRE] THEN
REWRITE_TAC[lemma1]);;
let POLY_EXP_EQ_0 = prove
(`!p n. (poly (p exp n) = poly []) <=> (poly p = poly []) /\ ~(n = 0)`,
REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; poly] THEN
REWRITE_TAC[LEFT_AND_FORALL_THM] THEN AP_TERM_TAC THEN ABS_TAC THEN
SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN
REWRITE_TAC[poly_exp; poly; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID;
CX_INJ; REAL_OF_NUM_EQ; ARITH; NOT_SUC] THEN
ASM_REWRITE_TAC[POLY_MUL; poly; COMPLEX_ENTIRE] THEN
CONV_TAC TAUT);;
let POLY_PRIME_EQ_0 = prove
(`!a. ~(poly [a ; Cx(&1)] = poly [])`,
GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; poly] THEN
DISCH_THEN(MP_TAC o SPEC `Cx(&1) - a`) THEN
SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_EXP_PRIME_EQ_0 = prove
(`!a n. ~(poly ([a ; Cx(&1)] exp n) = poly [])`,
MESON_TAC[POLY_EXP_EQ_0; POLY_PRIME_EQ_0]);;
(* ------------------------------------------------------------------------- *)
(* Can also prove a more "constructive" notion of polynomial being trivial. *)
(* ------------------------------------------------------------------------- *)
let POLY_ZERO_LEMMA = prove
(`!h t. (poly (CONS h t) = poly []) ==> (h = Cx(&0)) /\ (poly t = poly [])`,
let lemma = REWRITE_RULE[FUN_EQ_THM; poly] POLY_ROOTS_FINITE in
REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; poly] THEN
ASM_CASES_TAC `h = Cx(&0)` THEN ASM_REWRITE_TAC[] THENL
[REWRITE_TAC[COMPLEX_ADD_LID];
DISCH_THEN(MP_TAC o SPEC `Cx(&0)`) THEN
POP_ASSUM MP_TAC THEN SIMPLE_COMPLEX_ARITH_TAC] THEN
CONV_TAC CONTRAPOS_CONV THEN
DISCH_THEN(MP_TAC o REWRITE_RULE[lemma]) THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` (X_CHOOSE_TAC `i:num->complex`)) THEN
MP_TAC(SPECL
[`i:num->complex`; `N:num`; `\x. poly t x = Cx(&0)`] FINITE_LEMMA) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `a:real`) THEN
DISCH_THEN(MP_TAC o SPEC `Cx(abs(a) + &1)`) THEN
REWRITE_TAC[COMPLEX_ENTIRE; DE_MORGAN_THM] THEN CONJ_TAC THENL
[REWRITE_TAC[CX_INJ] THEN REAL_ARITH_TAC;
DISCH_THEN(MP_TAC o MATCH_MP
(ASSUME `!x. (poly t x = Cx(&0)) ==> norm(x) < a`)) THEN
REWRITE_TAC[COMPLEX_NORM_CX] THEN REAL_ARITH_TAC]);;
let POLY_ZERO = prove
(`!p. (poly p = poly []) <=> ALL (\c. c = Cx(&0)) p`,
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[ALL] THEN EQ_TAC THENL
[DISCH_THEN(MP_TAC o MATCH_MP POLY_ZERO_LEMMA) THEN ASM_REWRITE_TAC[];
POP_ASSUM(SUBST1_TAC o SYM) THEN STRIP_TAC THEN
ASM_REWRITE_TAC[FUN_EQ_THM; poly] THEN SIMPLE_COMPLEX_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Basics of divisibility. *)
(* ------------------------------------------------------------------------- *)
let divides = new_definition
`p1 divides p2 <=> ?q. poly p2 = poly (p1 ** q)`;;
let POLY_PRIMES = prove
(`!a p q. [a; Cx(&1)] divides (p ** q) <=>
[a; Cx(&1)] divides p \/ [a; Cx(&1)] divides q`,
REPEAT GEN_TAC THEN REWRITE_TAC[divides; POLY_MUL; FUN_EQ_THM; poly] THEN
REWRITE_TAC[COMPLEX_MUL_RZERO; COMPLEX_ADD_RID; COMPLEX_MUL_RID] THEN
EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_THEN `r:complex list` (MP_TAC o SPEC `--a`)) THEN
REWRITE_TAC[COMPLEX_ENTIRE; GSYM complex_sub;
COMPLEX_SUB_REFL; COMPLEX_MUL_LZERO] THEN
DISCH_THEN DISJ_CASES_TAC THENL [DISJ1_TAC; DISJ2_TAC] THEN
(POP_ASSUM(MP_TAC o REWRITE_RULE[POLY_LINEAR_DIVIDES]) THEN
REWRITE_TAC[COMPLEX_NEG_NEG] THEN
DISCH_THEN(DISJ_CASES_THEN2 SUBST_ALL_TAC
(X_CHOOSE_THEN `s:complex list` SUBST_ALL_TAC)) THENL
[EXISTS_TAC `[]:complex list` THEN REWRITE_TAC[poly; COMPLEX_MUL_RZERO];
EXISTS_TAC `s:complex list` THEN GEN_TAC THEN
REWRITE_TAC[POLY_MUL; poly] THEN SIMPLE_COMPLEX_ARITH_TAC]);
DISCH_THEN(DISJ_CASES_THEN(X_CHOOSE_TAC `s:complex list`)) THEN
ASM_REWRITE_TAC[] THENL
[EXISTS_TAC `s ** q`; EXISTS_TAC `p ** s`] THEN
GEN_TAC THEN REWRITE_TAC[POLY_MUL] THEN SIMPLE_COMPLEX_ARITH_TAC]);;
let POLY_DIVIDES_REFL = prove
(`!p. p divides p`,
GEN_TAC THEN REWRITE_TAC[divides] THEN EXISTS_TAC `[Cx(&1)]` THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_DIVIDES_TRANS = prove
(`!p q r. p divides q /\ q divides r ==> p divides r`,
REPEAT GEN_TAC THEN REWRITE_TAC[divides] THEN
DISCH_THEN(CONJUNCTS_THEN MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `s:complex list` ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t:complex list` ASSUME_TAC) THEN
EXISTS_TAC `t ** s` THEN
ASM_REWRITE_TAC[FUN_EQ_THM; POLY_MUL; COMPLEX_MUL_ASSOC]);;
let POLY_DIVIDES_EXP = prove
(`!p m n. m <= n ==> (p exp m) divides (p exp n)`,
REPEAT GEN_TAC THEN REWRITE_TAC[LE_EXISTS] THEN
DISCH_THEN(X_CHOOSE_THEN `d:num` SUBST1_TAC) THEN
SPEC_TAC(`d:num`,`d:num`) THEN INDUCT_TAC THEN
REWRITE_TAC[ADD_CLAUSES; POLY_DIVIDES_REFL] THEN
MATCH_MP_TAC POLY_DIVIDES_TRANS THEN
EXISTS_TAC `p exp (m + d)` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[divides] THEN EXISTS_TAC `p:complex list` THEN
REWRITE_TAC[poly_exp; FUN_EQ_THM; POLY_MUL] THEN
SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_EXP_DIVIDES = prove
(`!p q m n. (p exp n) divides q /\ m <= n ==> (p exp m) divides q`,
MESON_TAC[POLY_DIVIDES_TRANS; POLY_DIVIDES_EXP]);;
let POLY_DIVIDES_ADD = prove
(`!p q r. p divides q /\ p divides r ==> p divides (q ++ r)`,
REPEAT GEN_TAC THEN REWRITE_TAC[divides] THEN
DISCH_THEN(CONJUNCTS_THEN MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `s:complex list` ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t:complex list` ASSUME_TAC) THEN
EXISTS_TAC `t ++ s` THEN
ASM_REWRITE_TAC[FUN_EQ_THM; POLY_ADD; POLY_MUL] THEN
SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_DIVIDES_SUB = prove
(`!p q r. p divides q /\ p divides (q ++ r) ==> p divides r`,
REPEAT GEN_TAC THEN REWRITE_TAC[divides] THEN
DISCH_THEN(CONJUNCTS_THEN MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `s:complex list` ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t:complex list` ASSUME_TAC) THEN
EXISTS_TAC `s ++ neg(t)` THEN
POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
REWRITE_TAC[FUN_EQ_THM; POLY_ADD; POLY_MUL; POLY_NEG] THEN
DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN
REWRITE_TAC[COMPLEX_ADD_LDISTRIB; COMPLEX_MUL_RNEG] THEN
ASM_REWRITE_TAC[] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_DIVIDES_SUB2 = prove
(`!p q r. p divides r /\ p divides (q ++ r) ==> p divides q`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC POLY_DIVIDES_SUB THEN
EXISTS_TAC `r:complex list` THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `p divides (q ++ r)` THEN
REWRITE_TAC[divides; POLY_ADD; FUN_EQ_THM; POLY_MUL] THEN
DISCH_THEN(X_CHOOSE_TAC `s:complex list`) THEN
EXISTS_TAC `s:complex list` THEN
X_GEN_TAC `x:complex` THEN POP_ASSUM(MP_TAC o SPEC `x:complex`) THEN
SIMPLE_COMPLEX_ARITH_TAC);;
let POLY_DIVIDES_ZERO = prove
(`!p q. (poly p = poly []) ==> q divides p`,
REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[divides] THEN
EXISTS_TAC `[]:complex list` THEN
ASM_REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly; COMPLEX_MUL_RZERO]);;
(* ------------------------------------------------------------------------- *)
(* At last, we can consider the order of a root. *)
(* ------------------------------------------------------------------------- *)
let POLY_ORDER_EXISTS = prove
(`!a d. !p. (LENGTH p = d) /\ ~(poly p = poly [])
==> ?n. ([--a; Cx(&1)] exp n) divides p /\
~(([--a; Cx(&1)] exp (SUC n)) divides p)`,
GEN_TAC THEN
(STRIP_ASSUME_TAC o prove_recursive_functions_exist num_RECURSION)
`(!p q. mulexp 0 p q = q) /\
(!p q n. mulexp (SUC n) p q = p ** (mulexp n p q))` THEN
SUBGOAL_THEN
`!d. !p. (LENGTH p = d) /\ ~(poly p = poly [])
==> ?n q. (p = mulexp (n:num) [--a; Cx(&1)] q) /\
~(poly q a = Cx(&0))`
MP_TAC THENL
[INDUCT_TAC THENL
[REWRITE_TAC[LENGTH_EQ_NIL] THEN MESON_TAC[]; ALL_TAC] THEN
X_GEN_TAC `p:complex list` THEN
ASM_CASES_TAC `poly p a = Cx(&0)` THENL
[STRIP_TAC THEN UNDISCH_TAC `poly p a = Cx(&0)` THEN
DISCH_THEN(MP_TAC o REWRITE_RULE[POLY_LINEAR_DIVIDES]) THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `q:complex list` SUBST_ALL_TAC) THEN
UNDISCH_TAC
`!p. (LENGTH p = d) /\ ~(poly p = poly [])
==> ?n q. (p = mulexp (n:num) [--a; Cx(&1)] q) /\
~(poly q a = Cx(&0))` THEN
DISCH_THEN(MP_TAC o SPEC `q:complex list`) THEN
RULE_ASSUM_TAC(REWRITE_RULE[POLY_LENGTH_MUL; POLY_ENTIRE;
DE_MORGAN_THM; SUC_INJ]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `n:num`
(X_CHOOSE_THEN `s:complex list` STRIP_ASSUME_TAC)) THEN
EXISTS_TAC `SUC n` THEN EXISTS_TAC `s:complex list` THEN
ASM_REWRITE_TAC[];
STRIP_TAC THEN EXISTS_TAC `0` THEN EXISTS_TAC `p:complex list` THEN
ASM_REWRITE_TAC[]];
DISCH_TAC THEN REPEAT GEN_TAC THEN
DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `n:num`
(X_CHOOSE_THEN `s:complex list` STRIP_ASSUME_TAC)) THEN
EXISTS_TAC `n:num` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[divides] THEN CONJ_TAC THENL
[EXISTS_TAC `s:complex list` THEN
SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[poly_exp; FUN_EQ_THM; POLY_MUL; poly] THEN
SIMPLE_COMPLEX_ARITH_TAC;
DISCH_THEN(X_CHOOSE_THEN `r:complex list` MP_TAC) THEN
SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[] THENL
[UNDISCH_TAC `~(poly s a = Cx(&0))` THEN CONV_TAC CONTRAPOS_CONV THEN
REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[poly; poly_exp; POLY_MUL] THEN SIMPLE_COMPLEX_ARITH_TAC;
REWRITE_TAC[] THEN ONCE_ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[poly_exp] THEN
REWRITE_TAC[GSYM POLY_MUL_ASSOC; POLY_MUL_LCANCEL] THEN
REWRITE_TAC[DE_MORGAN_THM] THEN CONJ_TAC THENL
[REWRITE_TAC[FUN_EQ_THM] THEN
DISCH_THEN(MP_TAC o SPEC `a + Cx(&1)`) THEN
REWRITE_TAC[poly] THEN SIMPLE_COMPLEX_ARITH_TAC;
DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN REWRITE_TAC[]]]]]);;
let POLY_ORDER = prove
(`!p a. ~(poly p = poly [])
==> ?!n. ([--a; Cx(&1)] exp n) divides p /\
~(([--a; Cx(&1)] exp (SUC n)) divides p)`,
MESON_TAC[POLY_ORDER_EXISTS; POLY_EXP_DIVIDES; LE_SUC_LT; LT_CASES]);;
(* ------------------------------------------------------------------------- *)
(* Definition of order. *)
(* ------------------------------------------------------------------------- *)
let order = new_definition
`order a p = @n. ([--a; Cx(&1)] exp n) divides p /\
~(([--a; Cx(&1)] exp (SUC n)) divides p)`;;
let ORDER = prove
(`!p a n. ([--a; Cx(&1)] exp n) divides p /\
~(([--a; Cx(&1)] exp (SUC n)) divides p) <=>
(n = order a p) /\
~(poly p = poly [])`,
REPEAT GEN_TAC THEN REWRITE_TAC[order] THEN
EQ_TAC THEN STRIP_TAC THENL
[SUBGOAL_THEN `~(poly p = poly [])` ASSUME_TAC THENL
[FIRST_ASSUM(UNDISCH_TAC o check is_neg o concl) THEN
CONV_TAC CONTRAPOS_CONV THEN REWRITE_TAC[divides] THEN
DISCH_THEN SUBST1_TAC THEN EXISTS_TAC `[]:complex list` THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly; COMPLEX_MUL_RZERO];
ASM_REWRITE_TAC[] THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC SELECT_UNIQUE THEN REWRITE_TAC[]];
ONCE_ASM_REWRITE_TAC[] THEN CONV_TAC SELECT_CONV] THEN
ASM_MESON_TAC[POLY_ORDER]);;
let ORDER_THM = prove
(`!p a. ~(poly p = poly [])
==> ([--a; Cx(&1)] exp (order a p)) divides p /\
~(([--a; Cx(&1)] exp (SUC(order a p))) divides p)`,
MESON_TAC[ORDER]);;
let ORDER_UNIQUE = prove
(`!p a n. ~(poly p = poly []) /\
([--a; Cx(&1)] exp n) divides p /\
~(([--a; Cx(&1)] exp (SUC n)) divides p)
==> (n = order a p)`,
MESON_TAC[ORDER]);;
let ORDER_POLY = prove
(`!p q a. (poly p = poly q) ==> (order a p = order a q)`,
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[order; divides; FUN_EQ_THM; POLY_MUL]);;
let ORDER_ROOT = prove
(`!p a. (poly p a = Cx(&0)) <=> (poly p = poly []) \/ ~(order a p = 0)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `poly p = poly []` THEN
ASM_REWRITE_TAC[poly] THEN EQ_TAC THENL
[DISCH_THEN(MP_TAC o REWRITE_RULE[POLY_LINEAR_DIVIDES]) THEN
ASM_CASES_TAC `p:complex list = []` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `q:complex list` SUBST_ALL_TAC) THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `a:complex` o MATCH_MP ORDER_THM) THEN
ASM_REWRITE_TAC[poly_exp; DE_MORGAN_THM] THEN DISJ2_TAC THEN
REWRITE_TAC[divides] THEN EXISTS_TAC `q:complex list` THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly] THEN SIMPLE_COMPLEX_ARITH_TAC;
DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `a:complex` o MATCH_MP ORDER_THM) THEN
UNDISCH_TAC `~(order a p = 0)` THEN
SPEC_TAC(`order a p`,`n:num`) THEN
INDUCT_TAC THEN ASM_REWRITE_TAC[poly_exp; NOT_SUC] THEN
DISCH_THEN(MP_TAC o CONJUNCT1) THEN REWRITE_TAC[divides] THEN
DISCH_THEN(X_CHOOSE_THEN `s:complex list` SUBST1_TAC) THEN
REWRITE_TAC[POLY_MUL; poly] THEN SIMPLE_COMPLEX_ARITH_TAC]);;
let ORDER_DIVIDES = prove
(`!p a n. ([--a; Cx(&1)] exp n) divides p <=>
(poly p = poly []) \/ n <= order a p`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `poly p = poly []` THEN
ASM_REWRITE_TAC[] THENL
[ASM_REWRITE_TAC[divides] THEN EXISTS_TAC `[]:complex list` THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly; COMPLEX_MUL_RZERO];
ASM_MESON_TAC[ORDER_THM; POLY_EXP_DIVIDES; NOT_LE; LE_SUC_LT]]);;
let ORDER_DECOMP = prove
(`!p a. ~(poly p = poly [])
==> ?q. (poly p = poly (([--a; Cx(&1)] exp (order a p)) ** q)) /\
~([--a; Cx(&1)] divides q)`,
REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ORDER_THM) THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC o SPEC `a:complex`) THEN
DISCH_THEN(X_CHOOSE_TAC `q:complex list` o REWRITE_RULE[divides]) THEN
EXISTS_TAC `q:complex list` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `r: complex list` o REWRITE_RULE[divides]) THEN
UNDISCH_TAC `~([-- a; Cx(&1)] exp SUC (order a p) divides p)` THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[divides] THEN
EXISTS_TAC `r:complex list` THEN
ASM_REWRITE_TAC[POLY_MUL; FUN_EQ_THM; poly_exp] THEN
SIMPLE_COMPLEX_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Important composition properties of orders. *)
(* ------------------------------------------------------------------------- *)
let ORDER_MUL = prove
(`!a p q. ~(poly (p ** q) = poly []) ==>
(order a (p ** q) = order a p + order a q)`,
REPEAT GEN_TAC THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
REWRITE_TAC[POLY_ENTIRE; DE_MORGAN_THM] THEN STRIP_TAC THEN
SUBGOAL_THEN `(order a p + order a q = order a (p ** q)) /\
~(poly (p ** q) = poly [])`
MP_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
REWRITE_TAC[GSYM ORDER] THEN CONJ_TAC THENL
[MP_TAC(CONJUNCT1 (SPEC `a:complex`
(MATCH_MP ORDER_THM (ASSUME `~(poly p = poly [])`)))) THEN
DISCH_THEN(X_CHOOSE_TAC `r: complex list` o REWRITE_RULE[divides]) THEN
MP_TAC(CONJUNCT1 (SPEC `a:complex`
(MATCH_MP ORDER_THM (ASSUME `~(poly q = poly [])`)))) THEN
DISCH_THEN(X_CHOOSE_TAC `s: complex list` o REWRITE_RULE[divides]) THEN
REWRITE_TAC[divides; FUN_EQ_THM] THEN EXISTS_TAC `s ** r` THEN
ASM_REWRITE_TAC[POLY_MUL; POLY_EXP_ADD] THEN SIMPLE_COMPLEX_ARITH_TAC;
X_CHOOSE_THEN `r: complex list` STRIP_ASSUME_TAC
(SPEC `a:complex` (MATCH_MP ORDER_DECOMP
(ASSUME `~(poly p = poly [])`))) THEN
X_CHOOSE_THEN `s: complex list` STRIP_ASSUME_TAC
(SPEC `a:complex` (MATCH_MP ORDER_DECOMP
(ASSUME `~(poly q = poly [])`))) THEN
ASM_REWRITE_TAC[divides; FUN_EQ_THM; POLY_EXP_ADD; POLY_MUL; poly_exp] THEN
DISCH_THEN(X_CHOOSE_THEN `t:complex list` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `[--a; Cx(&1)] divides (r ** s)` MP_TAC THENL
[ALL_TAC; ASM_REWRITE_TAC[POLY_PRIMES]] THEN
REWRITE_TAC[divides] THEN EXISTS_TAC `t:complex list` THEN
SUBGOAL_THEN `poly ([-- a; Cx(&1)] exp (order a p) ** r ** s) =
poly ([-- a; Cx(&1)] exp (order a p) **
([-- a; Cx(&1)] ** t))`
MP_TAC THENL
[ALL_TAC; MESON_TAC[POLY_MUL_LCANCEL; POLY_EXP_PRIME_EQ_0]] THEN
SUBGOAL_THEN `poly ([-- a; Cx(&1)] exp (order a q) **
[-- a; Cx(&1)] exp (order a p) ** r ** s) =
poly ([-- a; Cx(&1)] exp (order a q) **
[-- a; Cx(&1)] exp (order a p) **
[-- a; Cx(&1)] ** t)`
MP_TAC THENL
[ALL_TAC; MESON_TAC[POLY_MUL_LCANCEL; POLY_EXP_PRIME_EQ_0]] THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; POLY_ADD] THEN
FIRST_ASSUM(UNDISCH_TAC o check is_forall o concl) THEN
REWRITE_TAC[COMPLEX_MUL_AC]]);;
(* ------------------------------------------------------------------------- *)
(* Normalization of a polynomial. *)
(* ------------------------------------------------------------------------- *)
let normalize = new_recursive_definition list_RECURSION
`(normalize [] = []) /\
(normalize (CONS h t) =
if normalize t = [] then if h = Cx(&0) then [] else [h]
else CONS h (normalize t))`;;
let POLY_NORMALIZE = prove
(`!p. poly (normalize p) = poly p`,
LIST_INDUCT_TAC THEN REWRITE_TAC[normalize; poly] THEN
ASM_CASES_TAC `h = Cx(&0)` THEN ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[poly; FUN_EQ_THM] THEN
UNDISCH_TAC `poly (normalize t) = poly t` THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[poly] THEN
REWRITE_TAC[COMPLEX_MUL_RZERO; COMPLEX_ADD_LID]);;
let LENGTH_NORMALIZE_LE = prove
(`!p. LENGTH(normalize p) <= LENGTH p`,
LIST_INDUCT_TAC THEN REWRITE_TAC[LENGTH; normalize; LE_REFL] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[LENGTH; LE_SUC] THEN
COND_CASES_TAC THEN REWRITE_TAC[LENGTH] THEN ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* The degree of a polynomial. *)
(* ------------------------------------------------------------------------- *)
let degree = new_definition
`degree p = PRE(LENGTH(normalize p))`;;
let DEGREE_ZERO = prove
(`!p. (poly p = poly []) ==> (degree p = 0)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[degree] THEN
SUBGOAL_THEN `normalize p = []` SUBST1_TAC THENL
[POP_ASSUM MP_TAC THEN SPEC_TAC(`p:complex list`,`p:complex list`) THEN
REWRITE_TAC[POLY_ZERO] THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[normalize; ALL] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `normalize t = []` (fun th -> REWRITE_TAC[th]) THEN
FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
REWRITE_TAC[LENGTH; PRE]]);;
(* ------------------------------------------------------------------------- *)
(* Show that the degree is welldefined. *)
(* ------------------------------------------------------------------------- *)
let POLY_CONS_EQ = prove
(`(poly (CONS h1 t1) = poly (CONS h2 t2)) <=>
(h1 = h2) /\ (poly t1 = poly t2)`,
REWRITE_TAC[FUN_EQ_THM] THEN EQ_TAC THENL [ALL_TAC; SIMP_TAC[poly]] THEN
ONCE_REWRITE_TAC[SIMPLE_COMPLEX_ARITH `(a = b) <=> (a + --b = Cx(&0))`] THEN
REWRITE_TAC[GSYM POLY_NEG; GSYM POLY_ADD] THEN DISCH_TAC THEN
SUBGOAL_THEN `poly (CONS h1 t1 ++ neg(CONS h2 t2)) = poly []` MP_TAC THENL
[ASM_REWRITE_TAC[poly; FUN_EQ_THM]; ALL_TAC] THEN
REWRITE_TAC[poly_neg; poly_cmul; poly_add; NOT_CONS_NIL; HD; TL] THEN
DISCH_THEN(MP_TAC o MATCH_MP POLY_ZERO_LEMMA) THEN
SIMP_TAC[poly; COMPLEX_MUL_LNEG; COMPLEX_MUL_LID]);;
let POLY_NORMALIZE_ZERO = prove
(`!p. (poly p = poly []) <=> (normalize p = [])`,
REWRITE_TAC[POLY_ZERO] THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[ALL; normalize] THEN
ASM_CASES_TAC `normalize t = []` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[NOT_CONS_NIL] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[NOT_CONS_NIL]);;
let POLY_NORMALIZE_EQ_LEMMA = prove
(`!p q. (poly p = poly q) ==> (normalize p = normalize q)`,
LIST_INDUCT_TAC THENL
[MESON_TAC[POLY_NORMALIZE_ZERO]; ALL_TAC] THEN
LIST_INDUCT_TAC THENL
[MESON_TAC[POLY_NORMALIZE_ZERO]; ALL_TAC] THEN
REWRITE_TAC[POLY_CONS_EQ] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[normalize] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `t':complex list`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN SUBST1_TAC THEN REFL_TAC);;
let POLY_NORMALIZE_EQ = prove
(`!p q. (poly p = poly q) <=> (normalize p = normalize q)`,
MESON_TAC[POLY_NORMALIZE_EQ_LEMMA; POLY_NORMALIZE]);;
let DEGREE_WELLDEF = prove
(`!p q. (poly p = poly q) ==> (degree p = degree q)`,
SIMP_TAC[degree; POLY_NORMALIZE_EQ]);;
(* ------------------------------------------------------------------------- *)
(* Degree of a product with a power of linear terms. *)
(* ------------------------------------------------------------------------- *)
let NORMALIZE_EQ = prove
(`!p. ~(LAST p = Cx(&0)) ==> (normalize p = p)`,
MATCH_MP_TAC list_INDUCT THEN REWRITE_TAC[NOT_CONS_NIL] THEN
REWRITE_TAC[normalize; LAST] THEN REPEAT GEN_TAC THEN
REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[normalize]));;
let NORMAL_DEGREE = prove
(`!p. ~(LAST p = Cx(&0)) ==> (degree p = LENGTH p - 1)`,
SIMP_TAC[degree; NORMALIZE_EQ] THEN REPEAT STRIP_TAC THEN ARITH_TAC);;
let LAST_LINEAR_MUL_LEMMA = prove
(`!p a b x.
LAST(a ## p ++ CONS x (b ## p)) = if p = [] then x else b * LAST(p)`,
LIST_INDUCT_TAC THEN
REWRITE_TAC[poly_cmul; poly_add; LAST; HD; TL; NOT_CONS_NIL] THEN
REPEAT GEN_TAC THEN
SUBGOAL_THEN `~(a ## t ++ CONS (b * h) (b ## t) = [])`
ASSUME_TAC THENL
[SPEC_TAC(`t:complex list`,`t:complex list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[poly_cmul; poly_add; NOT_CONS_NIL];
ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[]);;
let LAST_LINEAR_MUL = prove
(`!p. ~(p = []) ==> (LAST([a; Cx(&1)] ** p) = LAST p)`,
SIMP_TAC[poly_mul; NOT_CONS_NIL; LAST_LINEAR_MUL_LEMMA; COMPLEX_MUL_LID]);;
let NORMAL_NORMALIZE = prove
(`!p. ~(normalize p = []) ==> ~(LAST(normalize p) = Cx(&0))`,
LIST_INDUCT_TAC THEN REWRITE_TAC[normalize] THEN
POP_ASSUM MP_TAC THEN ASM_CASES_TAC `normalize t = []` THEN
ASM_REWRITE_TAC[LAST; NOT_CONS_NIL] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[LAST]);;
let LINEAR_MUL_DEGREE = prove
(`!p a. ~(poly p = poly []) ==> (degree([a; Cx(&1)] ** p) = degree(p) + 1)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `degree([a; Cx(&1)] ** normalize p) = degree(normalize p) + 1`
MP_TAC THENL
[FIRST_ASSUM(ASSUME_TAC o
GEN_REWRITE_RULE RAND_CONV [POLY_NORMALIZE_ZERO]) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP NORMAL_NORMALIZE) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP LAST_LINEAR_MUL) THEN
SIMP_TAC[NORMAL_DEGREE] THEN REPEAT STRIP_TAC THEN
SUBST1_TAC(SYM(SPEC `a:complex` COMPLEX_NEG_NEG)) THEN
REWRITE_TAC[POLY_LENGTH_MUL] THEN
UNDISCH_TAC `~(normalize p = [])` THEN
SPEC_TAC(`normalize p`,`p:complex list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[NOT_CONS_NIL; LENGTH] THEN ARITH_TAC;
MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN
TRY(AP_THM_TAC THEN AP_TERM_TAC) THEN MATCH_MP_TAC DEGREE_WELLDEF THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; POLY_NORMALIZE]]);;
let LINEAR_POW_MUL_DEGREE = prove
(`!n a p. degree([a; Cx(&1)] exp n ** p) =
if poly p = poly [] then 0 else degree p + n`,
INDUCT_TAC THEN REWRITE_TAC[poly_exp] THENL
[GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `degree(p)` THEN CONJ_TAC THENL
[MATCH_MP_TAC DEGREE_WELLDEF THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly; COMPLEX_MUL_RZERO;
COMPLEX_ADD_RID; COMPLEX_MUL_LID];
MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `degree []` THEN CONJ_TAC THENL
[MATCH_MP_TAC DEGREE_WELLDEF THEN ASM_REWRITE_TAC[];
REWRITE_TAC[degree; LENGTH; normalize; ARITH]]];
REWRITE_TAC[ADD_CLAUSES] THEN MATCH_MP_TAC DEGREE_WELLDEF THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly; COMPLEX_MUL_RZERO;
COMPLEX_ADD_RID; COMPLEX_MUL_LID]];
ALL_TAC] THEN
REPEAT GEN_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `degree([a; Cx (&1)] exp n ** ([a; Cx (&1)] ** p))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC DEGREE_WELLDEF THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; COMPLEX_MUL_AC]; ALL_TAC] THEN
ASM_REWRITE_TAC[POLY_ENTIRE] THEN
ASM_CASES_TAC `poly p = poly []` THEN ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[LINEAR_MUL_DEGREE] THEN
SUBGOAL_THEN `~(poly [a; Cx (&1)] = poly [])`
(fun th -> REWRITE_TAC[th] THEN ARITH_TAC) THEN
REWRITE_TAC[POLY_NORMALIZE_EQ] THEN
REWRITE_TAC[normalize; CX_INJ; REAL_OF_NUM_EQ; ARITH; NOT_CONS_NIL]);;
(* ------------------------------------------------------------------------- *)
(* Show that the order of a root (or nonroot!) is bounded by degree. *)
(* ------------------------------------------------------------------------- *)
let ORDER_DEGREE = prove
(`!a p. ~(poly p = poly []) ==> order a p <= degree p`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `a:complex` o MATCH_MP ORDER_THM) THEN
DISCH_THEN(MP_TAC o REWRITE_RULE[divides] o CONJUNCT1) THEN
DISCH_THEN(X_CHOOSE_THEN `q:complex list` ASSUME_TAC) THEN
FIRST_ASSUM(SUBST1_TAC o MATCH_MP DEGREE_WELLDEF) THEN
ASM_REWRITE_TAC[LINEAR_POW_MUL_DEGREE] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FUN_EQ_THM]) THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[POLY_MUL] THENL
[UNDISCH_TAC `~(poly p = poly [])` THEN
SIMP_TAC[FUN_EQ_THM; POLY_MUL; poly; COMPLEX_MUL_RZERO];
DISCH_TAC THEN ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Tidier versions of finiteness of roots. *)
(* ------------------------------------------------------------------------- *)
let POLY_ROOTS_FINITE_SET = prove
(`!p. ~(poly p = poly []) ==> FINITE { x | poly p x = Cx(&0)}`,
GEN_TAC THEN REWRITE_TAC[POLY_ROOTS_FINITE] THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `i:num->complex` ASSUME_TAC) THEN
MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `{x:complex | ?n:num. n < N /\ (x = i n)}` THEN
CONJ_TAC THENL
[SPEC_TAC(`N:num`,`N:num`) THEN POP_ASSUM_LIST(K ALL_TAC) THEN
INDUCT_TAC THENL
[SUBGOAL_THEN `{x:complex | ?n. n < 0 /\ (x = i n)} = {}`
(fun th -> REWRITE_TAC[th; FINITE_RULES]) THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; LT];
SUBGOAL_THEN `{x:complex | ?n. n < SUC N /\ (x = i n)} =
(i N) INSERT {x:complex | ?n. n < N /\ (x = i n)}`
SUBST1_TAC THENL
[REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INSERT; LT] THEN MESON_TAC[];
MATCH_MP_TAC(CONJUNCT2 FINITE_RULES) THEN ASM_REWRITE_TAC[]]];
ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM]]);;
(* ------------------------------------------------------------------------- *)
(* Conversions to perform operations if coefficients are rational constants. *)
(* ------------------------------------------------------------------------- *)
let COMPLEX_RAT_MUL_CONV =
GEN_REWRITE_CONV I [GSYM CX_MUL] THENC RAND_CONV REAL_RAT_MUL_CONV;;
let COMPLEX_RAT_ADD_CONV =
GEN_REWRITE_CONV I [GSYM CX_ADD] THENC RAND_CONV REAL_RAT_ADD_CONV;;
let COMPLEX_RAT_EQ_CONV =
GEN_REWRITE_CONV I [CX_INJ] THENC REAL_RAT_EQ_CONV;;
let POLY_CMUL_CONV =
let cmul_conv0 = GEN_REWRITE_CONV I [CONJUNCT1 poly_cmul]
and cmul_conv1 = GEN_REWRITE_CONV I [CONJUNCT2 poly_cmul] in
let rec POLY_CMUL_CONV tm =
(cmul_conv0 ORELSEC
(cmul_conv1 THENC
LAND_CONV COMPLEX_RAT_MUL_CONV THENC
RAND_CONV POLY_CMUL_CONV)) tm in
POLY_CMUL_CONV;;
let POLY_ADD_CONV =
let add_conv0 = GEN_REWRITE_CONV I (butlast (CONJUNCTS POLY_ADD_CLAUSES))
and add_conv1 = GEN_REWRITE_CONV I [last (CONJUNCTS POLY_ADD_CLAUSES)] in
let rec POLY_ADD_CONV tm =
(add_conv0 ORELSEC
(add_conv1 THENC
LAND_CONV COMPLEX_RAT_ADD_CONV THENC
RAND_CONV POLY_ADD_CONV)) tm in
POLY_ADD_CONV;;
let POLY_MUL_CONV =
let mul_conv0 = GEN_REWRITE_CONV I [CONJUNCT1 POLY_MUL_CLAUSES]
and mul_conv1 = GEN_REWRITE_CONV I [CONJUNCT1(CONJUNCT2 POLY_MUL_CLAUSES)]
and mul_conv2 = GEN_REWRITE_CONV I [CONJUNCT2(CONJUNCT2 POLY_MUL_CLAUSES)] in
let rec POLY_MUL_CONV tm =
(mul_conv0 ORELSEC
(mul_conv1 THENC POLY_CMUL_CONV) ORELSEC
(mul_conv2 THENC
LAND_CONV POLY_CMUL_CONV THENC
RAND_CONV(RAND_CONV POLY_MUL_CONV) THENC
POLY_ADD_CONV)) tm in
POLY_MUL_CONV;;
let POLY_NORMALIZE_CONV =
let pth = prove
(`normalize (CONS h t) =
(\n. if n = [] then if h = Cx(&0) then [] else [h] else CONS h n)
(normalize t)`,
REWRITE_TAC[normalize]) in
let norm_conv0 = GEN_REWRITE_CONV I [CONJUNCT1 normalize]
and norm_conv1 = GEN_REWRITE_CONV I [pth]
and norm_conv2 = GEN_REWRITE_CONV DEPTH_CONV
[COND_CLAUSES; NOT_CONS_NIL; EQT_INTRO(SPEC_ALL EQ_REFL)] in
let rec POLY_NORMALIZE_CONV tm =
(norm_conv0 ORELSEC
(norm_conv1 THENC
RAND_CONV POLY_NORMALIZE_CONV THENC
BETA_CONV THENC
RATOR_CONV(RAND_CONV(RATOR_CONV(LAND_CONV COMPLEX_RAT_EQ_CONV))) THENC
norm_conv2)) tm in
POLY_NORMALIZE_CONV;;
(* ------------------------------------------------------------------------- *)
(* Now we're finished with polynomials... *)
(* ------------------------------------------------------------------------- *)
(************** keep this for now
do_list reduce_interface
["divides",`poly_divides:complex list->complex list->bool`;
"exp",`poly_exp:complex list -> num -> complex list`;
"diff",`poly_diff:complex list->complex list`];;
unparse_as_infix "exp";;
****************)
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