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Zhangir Azerbayev
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(* ========================================================================= *)
(* Properties of real polynomials (not canonically represented). *)
(* ========================================================================= *)
needs "Library/analysis.ml";;
prioritize_real();;
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:real list->real list->real list`;
"**",`poly_mul:real list->real list->real list`;
"##",`poly_cmul:real->real list->real list`;
"neg",`poly_neg:real list->real list`;
"exp",`poly_exp:real list -> num -> real list`;
"diff",`poly_diff:real list->real list`];;
overload_interface ("divides",`poly_divides:real list->real list->bool`);;
(* ------------------------------------------------------------------------- *)
(* Application of polynomial as a real function. *)
(* ------------------------------------------------------------------------- *)
let poly = new_recursive_definition list_RECURSION
`(poly [] x = &0) /\
(poly (CONS h t) x = h + x * poly t x)`;;
let POLY_CONST = prove
(`!c x. poly [c] x = c`,
REWRITE_TAC[poly] THEN REAL_ARITH_TAC);;
let POLY_X = prove
(`!c x. poly [&0; &1] x = x`,
REWRITE_TAC[poly] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* 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 = (##) (--(&1))`;;
let poly_mul = new_recursive_definition list_RECURSION
`([] ** l2 = []) /\
((CONS h t) ** l2 =
(if t = [] then h ## l2
else (h ## l2) ++ CONS (&0) (t ** l2)))`;;
let poly_exp = new_recursive_definition num_RECURSION
`(p exp 0 = [&1]) /\
(p exp (SUC n) = p ** p exp n)`;;
(* ------------------------------------------------------------------------- *)
(* Differentiation of polynomials (needs an auxiliary function). *)
(* ------------------------------------------------------------------------- *)
let poly_diff_aux = new_recursive_definition list_RECURSION
`(poly_diff_aux n [] = []) /\
(poly_diff_aux n (CONS h t) = CONS (&n * h) (poly_diff_aux (SUC n) t))`;;
let poly_diff = new_definition
`diff l = (if l = [] then [] else (poly_diff_aux 1 (TL l)))`;;
(* ------------------------------------------------------------------------- *)
(* Lengths. *)
(* ------------------------------------------------------------------------- *)
let LENGTH_POLY_DIFF_AUX = prove
(`!l n. LENGTH(poly_diff_aux n l) = LENGTH l`,
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[LENGTH; poly_diff_aux]);;
let LENGTH_POLY_DIFF = prove
(`!l. LENGTH(poly_diff l) = PRE(LENGTH l)`,
LIST_INDUCT_TAC THEN
SIMP_TAC[poly_diff; LENGTH; LENGTH_POLY_DIFF_AUX; NOT_CONS_NIL; TL; PRE]);;
(* ------------------------------------------------------------------------- *)
(* 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:real list`,`p1:real 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; REAL_MUL_LNEG; REAL_MUL_LID]);;
let POLY_MUL_CLAUSES = prove
(`([] ** p2 = []) /\
([h1] ** p2 = h1 ## p2) /\
((CONS h1 (CONS k1 t1)) ** p2 = h1 ## p2 ++ CONS (&0) (CONS k1 t1 ** p2))`,
REWRITE_TAC[poly_mul; NOT_CONS_NIL]);;
let POLY_DIFF_CLAUSES = prove
(`(diff [] = []) /\
(diff [c] = []) /\
(diff (CONS h t) = poly_diff_aux 1 t)`,
REWRITE_TAC[poly_diff; NOT_CONS_NIL; HD; TL; poly_diff_aux]);;
(* ------------------------------------------------------------------------- *)
(* 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; REAL_ADD_LID] THEN
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC[NOT_CONS_NIL; HD; TL; poly; REAL_ADD_RID] THEN
REAL_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
REAL_ARITH_TAC);;
let POLY_NEG = prove
(`!p x. poly (neg p) x = --(poly p x)`,
REWRITE_TAC[poly_neg; POLY_CMUL] THEN
REAL_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; REAL_MUL_LZERO; POLY_CMUL; POLY_ADD] THEN
SPEC_TAC(`h:real`,`h:real`) THEN
SPEC_TAC(`t:real list`,`t:real list`) THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC[poly_mul; POLY_CMUL; POLY_ADD; poly; POLY_CMUL;
REAL_MUL_RZERO; REAL_ADD_RID; NOT_CONS_NIL] THEN
ASM_REWRITE_TAC[POLY_ADD; POLY_CMUL; poly] THEN
REAL_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; real_pow; POLY_MUL] THEN
REWRITE_TAC[poly] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* The derivative is a bit more complicated. *)
(* ------------------------------------------------------------------------- *)
let POLY_DIFF_LEMMA = prove
(`!l n x. ((\x. (x pow (SUC n)) * poly l x) diffl
((x pow n) * poly (poly_diff_aux (SUC n) l) x))(x)`,
LIST_INDUCT_TAC THEN
REWRITE_TAC[poly; poly_diff_aux; REAL_MUL_RZERO; DIFF_CONST] THEN
MAP_EVERY X_GEN_TAC [`n:num`; `x:real`] THEN
REWRITE_TAC[REAL_LDISTRIB; REAL_MUL_ASSOC] THEN
ONCE_REWRITE_TAC[GSYM(ONCE_REWRITE_RULE[REAL_MUL_SYM] (CONJUNCT2 pow))] THEN
POP_ASSUM(MP_TAC o SPECL [`SUC n`; `x:real`]) THEN
SUBGOAL_THEN `(((\x. (x pow (SUC n)) * h)) diffl
((x pow n) * &(SUC n) * h))(x)`
(fun th -> DISCH_THEN(MP_TAC o CONJ th)) THENL
[REWRITE_TAC[REAL_MUL_ASSOC] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
MP_TAC(SPEC `\x. x pow (SUC n)` DIFF_CMUL) THEN BETA_TAC THEN
DISCH_THEN MATCH_MP_TAC THEN
MP_TAC(SPEC `SUC n` DIFF_POW) THEN REWRITE_TAC[SUC_SUB1] THEN
DISCH_THEN(MATCH_ACCEPT_TAC o ONCE_REWRITE_RULE[REAL_MUL_SYM]);
DISCH_THEN(MP_TAC o MATCH_MP DIFF_ADD) THEN BETA_TAC THEN
REWRITE_TAC[REAL_MUL_ASSOC]]);;
let POLY_DIFF = prove
(`!l x. ((\x. poly l x) diffl (poly (diff l) x))(x)`,
LIST_INDUCT_TAC THEN REWRITE_TAC[POLY_DIFF_CLAUSES] THEN
ONCE_REWRITE_TAC[SYM(ETA_CONV `\x. poly l x`)] THEN
REWRITE_TAC[poly; DIFF_CONST] THEN
MAP_EVERY X_GEN_TAC [`x:real`] THEN
MP_TAC(SPECL [`t:(real)list`; `0`; `x:real`] POLY_DIFF_LEMMA) THEN
REWRITE_TAC[SYM(num_CONV `1`)] THEN REWRITE_TAC[pow; REAL_MUL_LID] THEN
REWRITE_TAC[POW_1] THEN
DISCH_THEN(MP_TAC o CONJ (SPECL [`h:real`; `x:real`] DIFF_CONST)) THEN
DISCH_THEN(MP_TAC o MATCH_MP DIFF_ADD) THEN BETA_TAC THEN
REWRITE_TAC[REAL_ADD_LID]);;
(* ------------------------------------------------------------------------- *)
(* Trivial consequences. *)
(* ------------------------------------------------------------------------- *)
let POLY_DIFFERENTIABLE = prove
(`!l x. (\x. poly l x) differentiable x`,
REPEAT GEN_TAC THEN REWRITE_TAC[differentiable] THEN
EXISTS_TAC `poly (diff l) x` THEN
REWRITE_TAC[POLY_DIFF]);;
let POLY_CONT = prove
(`!l x. (\x. poly l x) contl x`,
REPEAT GEN_TAC THEN MATCH_MP_TAC DIFF_CONT THEN
EXISTS_TAC `poly (diff l) x` THEN
MATCH_ACCEPT_TAC POLY_DIFF);;
let POLY_IVT_POS = prove
(`!p a b. a < b /\ poly p a < &0 /\ poly p b > &0
==> ?x. a < x /\ x < b /\ (poly p x = &0)`,
REWRITE_TAC[real_gt] THEN REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`\x. poly p x`; `a:real`; `b:real`; `&0`] IVT) THEN
REWRITE_TAC[POLY_CONT] THEN
EVERY_ASSUM(fun th -> REWRITE_TAC[MATCH_MP REAL_LT_IMP_LE th]) THEN
DISCH_THEN(X_CHOOSE_THEN `x:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `x:real` THEN ASM_REWRITE_TAC[REAL_LT_LE] THEN
CONJ_TAC THEN DISCH_THEN SUBST_ALL_TAC THEN
FIRST_ASSUM SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[REAL_LT_REFL]) THEN
FIRST_ASSUM CONTR_TAC);;
let POLY_IVT_NEG = prove
(`!p a b. a < b /\ poly p a > &0 /\ poly p b < &0
==> ?x. a < x /\ x < b /\ (poly p x = &0)`,
REPEAT STRIP_TAC THEN MP_TAC(SPEC `poly_neg p` POLY_IVT_POS) THEN
REWRITE_TAC[POLY_NEG;
REAL_ARITH `(--x < &0 <=> x > &0) /\ (--x > &0 <=> x < &0)`] THEN
DISCH_THEN(MP_TAC o SPECL [`a:real`; `b:real`]) THEN
ASM_REWRITE_TAC[REAL_ARITH `(--x = &0) <=> (x = &0)`]);;
let POLY_MVT = prove
(`!p a b. a < b ==>
?x. a < x /\ x < b /\
(poly p b - poly p a = (b - a) * poly (diff p) x)`,
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`poly p`; `a:real`; `b:real`] MVT) THEN
ASM_REWRITE_TAC[CONV_RULE(DEPTH_CONV ETA_CONV) (SPEC_ALL POLY_CONT);
CONV_RULE(DEPTH_CONV ETA_CONV) (SPEC_ALL POLY_DIFFERENTIABLE)] THEN
DISCH_THEN(X_CHOOSE_THEN `l:real` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `x:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `x:real` THEN ASM_REWRITE_TAC[] THEN
AP_TERM_TAC THEN MATCH_MP_TAC DIFF_UNIQ THEN
EXISTS_TAC `poly p` THEN EXISTS_TAC `x:real` THEN
ASM_REWRITE_TAC[CONV_RULE(DEPTH_CONV ETA_CONV) (SPEC_ALL POLY_DIFF)]);;
let POLY_MVT_ADD = prove
(`!p a x. ?y. abs(y) <= abs(x) /\
(poly p (a + x) = poly p a + x * poly (diff p) (a + y))`,
REPEAT GEN_TAC THEN
REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (SPEC `x:real` REAL_LT_NEGTOTAL) THENL
[EXISTS_TAC `&0` THEN
ASM_REWRITE_TAC[REAL_LE_REFL; REAL_ADD_RID; REAL_MUL_LZERO];
MP_TAC(SPECL [`p:real list`; `a:real`; `a + x`] POLY_MVT) THEN
ASM_REWRITE_TAC[REAL_LT_ADDR] THEN
DISCH_THEN(X_CHOOSE_THEN `z:real` MP_TAC) THEN
REWRITE_TAC[REAL_ARITH `(x - y = ((a + b) - a) * z) <=>
(x = y + b * z)`] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[REAL_EQ_ADD_LCANCEL] THEN
EXISTS_TAC `z - a` THEN REWRITE_TAC[REAL_ARITH `x + (y - x) = y`] THEN
MAP_EVERY UNDISCH_TAC [`&0 < x`; `a < z`; `z < a + x`] THEN
REAL_ARITH_TAC;
MP_TAC(SPECL [`p:real list`; `a + x`; `a:real`] POLY_MVT) THEN
ASM_REWRITE_TAC[REAL_ARITH `a + x < a <=> &0 < --x`] THEN
DISCH_THEN(X_CHOOSE_THEN `z:real` MP_TAC) THEN
REWRITE_TAC[REAL_ARITH `(x - y = (a - (a + b)) * z) <=>
(x = y + b * --z)`] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[REAL_EQ_ADD_LCANCEL] THEN
EXISTS_TAC `z - a` THEN REWRITE_TAC[REAL_ARITH `x + (y - x) = y`] THEN
MAP_EVERY UNDISCH_TAC [`&0 < --x`; `a + x < z`; `z < a`] THEN
REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Lemmas. *)
(* ------------------------------------------------------------------------- *)
let POLY_ADD_RZERO = prove
(`!p. poly (p ++ []) = poly p`,
REWRITE_TAC[FUN_EQ_THM; POLY_ADD; poly; REAL_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; REAL_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
REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Lemmas for derivatives. *)
(* ------------------------------------------------------------------------- *)
let POLY_DIFF_AUX_ADD = prove
(`!p1 p2 n. poly (poly_diff_aux n (p1 ++ p2)) =
poly (poly_diff_aux n p1 ++ poly_diff_aux n p2)`,
REPEAT(LIST_INDUCT_TAC THEN REWRITE_TAC[poly_diff_aux; poly_add]) THEN
ASM_REWRITE_TAC[poly_diff_aux; FUN_EQ_THM; poly; NOT_CONS_NIL; HD; TL] THEN
REAL_ARITH_TAC);;
let POLY_DIFF_AUX_CMUL = prove
(`!p c n. poly (poly_diff_aux n (c ## p)) =
poly (c ## poly_diff_aux n p)`,
LIST_INDUCT_TAC THEN
ASM_REWRITE_TAC[FUN_EQ_THM; poly; poly_diff_aux; poly_cmul; REAL_MUL_AC]);;
let POLY_DIFF_AUX_NEG = prove
(`!p n. poly (poly_diff_aux n (neg p)) =
poly (neg (poly_diff_aux n p))`,
REWRITE_TAC[poly_neg; POLY_DIFF_AUX_CMUL]);;
let POLY_DIFF_AUX_MUL_LEMMA = prove
(`!p n. poly (poly_diff_aux (SUC n) p) = poly (poly_diff_aux n p ++ p)`,
LIST_INDUCT_TAC THEN REWRITE_TAC[poly_diff_aux; poly_add; NOT_CONS_NIL] THEN
ASM_REWRITE_TAC[HD; TL; poly; FUN_EQ_THM] THEN
REWRITE_TAC[GSYM REAL_OF_NUM_SUC; REAL_ADD_RDISTRIB; REAL_MUL_LID]);;
(* ------------------------------------------------------------------------- *)
(* Final results for derivatives. *)
(* ------------------------------------------------------------------------- *)
let POLY_DIFF_ADD = prove
(`!p1 p2. poly (diff (p1 ++ p2)) =
poly (diff p1 ++ diff p2)`,
REPEAT LIST_INDUCT_TAC THEN
REWRITE_TAC[poly_add; poly_diff; NOT_CONS_NIL; POLY_ADD_RZERO] THEN
ASM_REWRITE_TAC[HD; TL; POLY_DIFF_AUX_ADD]);;
let POLY_DIFF_CMUL = prove
(`!p c. poly (diff (c ## p)) = poly (c ## diff p)`,
LIST_INDUCT_TAC THEN REWRITE_TAC[poly_diff; poly_cmul] THEN
REWRITE_TAC[NOT_CONS_NIL; HD; TL; POLY_DIFF_AUX_CMUL]);;
let POLY_DIFF_NEG = prove
(`!p. poly (diff (neg p)) = poly (neg (diff p))`,
REWRITE_TAC[poly_neg; POLY_DIFF_CMUL]);;
let POLY_DIFF_MUL_LEMMA = prove
(`!t h. poly (diff (CONS h t)) =
poly (CONS (&0) (diff t) ++ t)`,
REWRITE_TAC[poly_diff; NOT_CONS_NIL] THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[poly_diff_aux; NOT_CONS_NIL; HD; TL] THENL
[REWRITE_TAC[FUN_EQ_THM; poly; poly_add; REAL_MUL_RZERO; REAL_ADD_LID];
REWRITE_TAC[FUN_EQ_THM; poly; POLY_DIFF_AUX_MUL_LEMMA; POLY_ADD] THEN
REAL_ARITH_TAC]);;
let POLY_DIFF_MUL = prove
(`!p1 p2. poly (diff (p1 ** p2)) =
poly (p1 ** diff p2 ++ diff p1 ** p2)`,
LIST_INDUCT_TAC THEN REWRITE_TAC[poly_mul] THENL
[REWRITE_TAC[poly_diff; poly_add; poly_mul]; ALL_TAC] THEN
GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[REWRITE_TAC[POLY_DIFF_CLAUSES] THEN
REWRITE_TAC[poly_add; poly_mul; POLY_ADD_RZERO; POLY_DIFF_CMUL];
ALL_TAC] THEN
REWRITE_TAC[FUN_EQ_THM; POLY_DIFF_ADD; POLY_ADD] THEN
REWRITE_TAC[poly; POLY_ADD; POLY_DIFF_MUL_LEMMA; POLY_MUL] THEN
ASM_REWRITE_TAC[POLY_DIFF_CMUL; POLY_ADD; POLY_MUL] THEN
REAL_ARITH_TAC);;
let POLY_DIFF_EXP = prove
(`!p n. poly (diff (p exp (SUC n))) =
poly ((&(SUC n) ## (p exp n)) ** diff p)`,
GEN_TAC THEN INDUCT_TAC THEN ONCE_REWRITE_TAC[poly_exp] THENL
[REWRITE_TAC[poly_exp; POLY_DIFF_MUL] THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; POLY_ADD; POLY_CMUL] THEN
REWRITE_TAC[poly; POLY_DIFF_CLAUSES; ADD1; ADD_CLAUSES] THEN
REAL_ARITH_TAC;
REWRITE_TAC[POLY_DIFF_MUL] THEN
ASM_REWRITE_TAC[POLY_MUL; POLY_ADD; FUN_EQ_THM; POLY_CMUL] THEN
REWRITE_TAC[poly_exp; POLY_MUL] THEN
REWRITE_TAC[ADD1; GSYM REAL_OF_NUM_ADD] THEN
REAL_ARITH_TAC]);;
let POLY_DIFF_EXP_PRIME = prove
(`!n a. poly (diff ([--a; &1] exp (SUC n))) =
poly (&(SUC n) ## ([--a; &1] exp n))`,
REPEAT GEN_TAC THEN REWRITE_TAC[POLY_DIFF_EXP] THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL] THEN
REWRITE_TAC[poly_diff; poly_diff_aux; TL; NOT_CONS_NIL] THEN
REWRITE_TAC[poly] THEN REAL_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; &1] ** q`,
LIST_INDUCT_TAC THEN REWRITE_TAC[] THENL
[GEN_TAC THEN EXISTS_TAC `[]:real list` THEN
EXISTS_TAC `h:real` THEN
REWRITE_TAC[poly_add; poly_mul; poly_cmul; NOT_CONS_NIL] THEN
REWRITE_TAC[HD; TL; REAL_ADD_RID];
X_GEN_TAC `k:real` THEN POP_ASSUM(STRIP_ASSUME_TAC o SPEC `h:real`) THEN
EXISTS_TAC `CONS (r:real) 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
[REAL_ARITH_TAC; ALL_TAC] THEN
SPEC_TAC(`q:real list`,`q:real list`) THEN
LIST_INDUCT_TAC THEN
REWRITE_TAC[POLY_ADD_CLAUSES; POLY_MUL_CLAUSES; poly_cmul] THEN
REWRITE_TAC[REAL_ADD_RID; REAL_MUL_LID] THEN
REWRITE_TAC[REAL_ADD_AC]]);;
let POLY_LINEAR_DIVIDES = prove
(`!a p. (poly p a = &0) <=> (p = []) \/ ?q. p = [--a; &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:real list` THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `r = &0` SUBST_ALL_TAC THENL
[UNDISCH_TAC `poly (CONS h t) a = &0` THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[POLY_ADD; POLY_MUL] THEN
REWRITE_TAC[poly; REAL_MUL_RZERO; REAL_ADD_RID; REAL_MUL_RID] THEN
REWRITE_TAC[REAL_ARITH `--a + a = &0`] THEN REAL_ARITH_TAC;
REWRITE_TAC[poly_mul] THEN REWRITE_TAC[NOT_CONS_NIL] THEN
SPEC_TAC(`q:real list`,`q:real list`) THEN LIST_INDUCT_TAC THENL
[REWRITE_TAC[poly_cmul; poly_add; NOT_CONS_NIL; HD; TL; REAL_ADD_LID];
REWRITE_TAC[poly_cmul; poly_add; NOT_CONS_NIL; HD; TL; REAL_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; REAL_MUL_RZERO; REAL_ADD_RID; REAL_MUL_RID] THEN
REWRITE_TAC[REAL_ARITH `--a + a = &0`] THEN REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Thanks to the finesse of the above, we can use length rather than degree. *)
(* ------------------------------------------------------------------------- *)
let POLY_LENGTH_MUL = prove
(`!q. LENGTH([--a; &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) = &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 = &0` THENL
[UNDISCH_TAC `?a. poly p a = &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:real list` SUBST_ALL_TAC) THEN
FIRST_ASSUM(UNDISCH_TAC o check is_forall o concl) THEN
UNDISCH_TAC `~(poly ([-- a; &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; REAL_MUL_RZERO; FUN_EQ_THM];
DISCH_THEN(K ALL_TAC)] THEN
DISCH_THEN(MP_TAC o SPEC `q:real list`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `i:num->real`) THEN
EXISTS_TAC `\m. if m = SUC n then (a:real) else i m` THEN
REWRITE_TAC[POLY_MUL; LE; REAL_ENTIRE] THEN
X_GEN_TAC `x:real` 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 REAL_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 = &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) = &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) = &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 ==> 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:real->bool` THEN
POP_ASSUM(MP_TAC o SPEC `\z. P z /\ ~(z = (i:num->real) N)`) THEN
DISCH_THEN(X_CHOOSE_TAC `a:real`) THEN
EXISTS_TAC `abs(a) + abs(i(N:num)) + &1` THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[LT] THEN
MP_TAC(REAL_ARITH `!x v. x < abs(v) + abs(x) + &1`) THEN
MP_TAC(REAL_ARITH `!u v x. x < v ==> x < abs(v) + abs(u) + &1`) THEN
MESON_TAC[]);;
let POLY_ROOTS_FINITE = prove
(`!p. ~(poly p = poly []) <=>
?N i. !x. (poly p(x) = &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->real`; `N:num`]
(SPECL [`i:num->real`; `N:num`; `\x. poly p x = &0`] FINITE_LEMMA)) THEN
REWRITE_TAC[] THEN MESON_TAC[REAL_LT_REFL]);;
(* ------------------------------------------------------------------------- *)
(* 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->real`)) THEN
DISCH_THEN(X_CHOOSE_THEN `N1:num` (X_CHOOSE_TAC `i1:num->real`)) THEN
EXISTS_TAC `N1 + N2:num` THEN
EXISTS_TAC `\n:num. if n < N1 then i1(n):real else i2(n - N1)` THEN
X_GEN_TAC `x:real` THEN REWRITE_TAC[REAL_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[REAL_MUL_RZERO; REAL_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[REAL_ARITH `(p + --q = &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
REAL_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; REAL_MUL_RZERO; REAL_ADD_RID;
REAL_OF_NUM_EQ; ARITH; NOT_SUC] THEN
ASM_REWRITE_TAC[POLY_MUL; poly; REAL_ENTIRE] THEN
CONV_TAC TAUT);;
let POLY_PRIME_EQ_0 = prove
(`!a. ~(poly [a ; &1] = poly [])`,
GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; poly] THEN
DISCH_THEN(MP_TAC o SPEC `&1 - a`) THEN
REAL_ARITH_TAC);;
let POLY_EXP_PRIME_EQ_0 = prove
(`!a n. ~(poly ([a ; &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 = &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 = &0` THEN ASM_REWRITE_TAC[] THENL
[REWRITE_TAC[REAL_ADD_LID];
DISCH_THEN(MP_TAC o SPEC `&0`) THEN
POP_ASSUM MP_TAC THEN REAL_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->real`)) THEN
MP_TAC(SPECL [`i:num->real`; `N:num`; `\x. poly t x = &0`] FINITE_LEMMA) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `a:real`) THEN
DISCH_THEN(MP_TAC o SPEC `abs(a) + &1`) THEN
REWRITE_TAC[REAL_ENTIRE; DE_MORGAN_THM] THEN CONJ_TAC THENL
[REAL_ARITH_TAC;
DISCH_THEN(MP_TAC o MATCH_MP (ASSUME `!x. (poly t x = &0) ==> x < a`)) THEN
REAL_ARITH_TAC]);;
let POLY_ZERO = prove
(`!p. (poly p = poly []) <=> ALL (\c. c = &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 REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Useful triviality. *)
(* ------------------------------------------------------------------------- *)
let POLY_DIFF_AUX_ISZERO = prove
(`!p n. ALL (\c. c = &0) (poly_diff_aux (SUC n) p) <=>
ALL (\c. c = &0) p`,
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC
[ALL; poly_diff_aux; REAL_ENTIRE; REAL_OF_NUM_EQ; NOT_SUC]);;
let POLY_DIFF_ISZERO = prove
(`!p. (poly (diff p) = poly []) ==> ?h. poly p = poly [h]`,
REWRITE_TAC[POLY_ZERO] THEN
LIST_INDUCT_TAC THEN ASM_REWRITE_TAC[POLY_DIFF_CLAUSES; ALL] THENL
[EXISTS_TAC `&0` THEN REWRITE_TAC[FUN_EQ_THM; poly] THEN REAL_ARITH_TAC;
REWRITE_TAC[num_CONV `1`; POLY_DIFF_AUX_ISZERO] THEN
REWRITE_TAC[GSYM POLY_ZERO] THEN DISCH_TAC THEN
EXISTS_TAC `h:real` THEN ASM_REWRITE_TAC[poly; FUN_EQ_THM]]);;
let POLY_DIFF_ZERO = prove
(`!p. (poly p = poly []) ==> (poly (diff p) = poly [])`,
REWRITE_TAC[POLY_ZERO] THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[poly_diff; NOT_CONS_NIL] THEN
REWRITE_TAC[ALL; TL] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
SPEC_TAC(`1`,`n:num`) THEN POP_ASSUM_LIST(K ALL_TAC) THEN
SPEC_TAC(`t:real list`,`t:real list`) THEN
LIST_INDUCT_TAC THEN REWRITE_TAC[ALL; poly_diff_aux] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RZERO] THEN
FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]);;
let POLY_DIFF_WELLDEF = prove
(`!p q. (poly p = poly q) ==> (poly (diff p) = poly (diff q))`,
REPEAT STRIP_TAC THEN MP_TAC(SPEC `p ++ neg(q)` POLY_DIFF_ZERO) THEN
REWRITE_TAC[FUN_EQ_THM; POLY_DIFF_ADD; POLY_DIFF_NEG; POLY_ADD] THEN
ASM_REWRITE_TAC[POLY_NEG; poly; REAL_ARITH `a + --a = &0`] THEN
REWRITE_TAC[REAL_ARITH `(a + --b = &0) <=> (a = b)`]);;
(* ------------------------------------------------------------------------- *)
(* Basics of divisibility. *)
(* ------------------------------------------------------------------------- *)
let divides = new_definition
`p1 divides p2 <=> ?q. poly p2 = poly (p1 ** q)`;;
let POLY_PRIMES = prove
(`!a p q. [a; &1] divides (p ** q) <=>
[a; &1] divides p \/ [a; &1] divides q`,
REPEAT GEN_TAC THEN REWRITE_TAC[divides; POLY_MUL; FUN_EQ_THM; poly] THEN
REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_RID; REAL_MUL_RID] THEN EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_THEN `r:real list` (MP_TAC o SPEC `--a`)) THEN
REWRITE_TAC[REAL_ENTIRE; GSYM real_sub; REAL_SUB_REFL; REAL_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[REAL_NEG_NEG] THEN
DISCH_THEN(DISJ_CASES_THEN2 SUBST_ALL_TAC
(X_CHOOSE_THEN `s:real list` SUBST_ALL_TAC)) THENL
[EXISTS_TAC `[]:real list` THEN REWRITE_TAC[poly; REAL_MUL_RZERO];
EXISTS_TAC `s:real list` THEN GEN_TAC THEN
REWRITE_TAC[POLY_MUL; poly] THEN REAL_ARITH_TAC]);
DISCH_THEN(DISJ_CASES_THEN(X_CHOOSE_TAC `s:real list`)) THEN
ASM_REWRITE_TAC[] THENL
[EXISTS_TAC `s ** q`; EXISTS_TAC `p ** s`] THEN
GEN_TAC THEN REWRITE_TAC[POLY_MUL] THEN REAL_ARITH_TAC]);;
let POLY_DIVIDES_REFL = prove
(`!p. p divides p`,
GEN_TAC THEN REWRITE_TAC[divides] THEN EXISTS_TAC `[&1]` THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly] THEN REAL_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:real list` ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t:real list` ASSUME_TAC) THEN
EXISTS_TAC `t ** s` THEN
ASM_REWRITE_TAC[FUN_EQ_THM; POLY_MUL; REAL_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:real list` THEN
REWRITE_TAC[poly_exp; FUN_EQ_THM; POLY_MUL] THEN
REAL_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:real list` ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t:real list` ASSUME_TAC) THEN
EXISTS_TAC `t ++ s` THEN
ASM_REWRITE_TAC[FUN_EQ_THM; POLY_ADD; POLY_MUL] THEN
REAL_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:real list` ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `t:real 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[REAL_ADD_LDISTRIB; REAL_MUL_RNEG] THEN
ASM_REWRITE_TAC[] THEN REAL_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:real 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:real list`) THEN
EXISTS_TAC `s:real list` THEN
X_GEN_TAC `x:real` THEN POP_ASSUM(MP_TAC o SPEC `x:real`) THEN
REAL_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 `[]:real list` THEN
ASM_REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly; REAL_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; &1] exp n) divides p /\
~(([--a; &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; &1] q) /\
~(poly q a = &0)`
MP_TAC THENL
[INDUCT_TAC THENL
[REWRITE_TAC[LENGTH_EQ_NIL] THEN MESON_TAC[]; ALL_TAC] THEN
X_GEN_TAC `p:real list` THEN
ASM_CASES_TAC `poly p a = &0` THENL
[STRIP_TAC THEN UNDISCH_TAC `poly p a = &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:real list` SUBST_ALL_TAC) THEN
UNDISCH_TAC
`!p. (LENGTH p = d) /\ ~(poly p = poly [])
==> ?n q. (p = mulexp (n:num) [--a; &1] q) /\
~(poly q a = &0)` THEN
DISCH_THEN(MP_TAC o SPEC `q:real 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:real list` STRIP_ASSUME_TAC)) THEN
EXISTS_TAC `SUC n` THEN EXISTS_TAC `s:real list` THEN
ASM_REWRITE_TAC[];
STRIP_TAC THEN EXISTS_TAC `0` THEN EXISTS_TAC `p:real 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:real list` STRIP_ASSUME_TAC)) THEN
EXISTS_TAC `n:num` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[divides] THEN CONJ_TAC THENL
[EXISTS_TAC `s:real list` THEN
SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[poly_exp; FUN_EQ_THM; POLY_MUL; poly] THEN
REAL_ARITH_TAC;
DISCH_THEN(X_CHOOSE_THEN `r:real list` MP_TAC) THEN
SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[] THENL
[UNDISCH_TAC `~(poly s a = &0)` THEN CONV_TAC CONTRAPOS_CONV THEN
REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[poly; poly_exp; POLY_MUL] THEN REAL_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 + &1`) THEN
REWRITE_TAC[poly] THEN REAL_ARITH_TAC;
DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN REWRITE_TAC[]]]]]);;
let POLY_ORDER = prove
(`!p a. ~(poly p = poly [])
==> ?!n. ([--a; &1] exp n) divides p /\
~(([--a; &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; &1] exp n) divides p /\
~(([--a; &1] exp (SUC n)) divides p)`;;
let ORDER = prove
(`!p a n. ([--a; &1] exp n) divides p /\
~(([--a; &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 `[]:real list` THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly; REAL_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; &1] exp (order a p)) divides p /\
~(([--a; &1] exp (SUC(order a p))) divides p)`,
MESON_TAC[ORDER]);;
let ORDER_UNIQUE = prove
(`!p a n. ~(poly p = poly []) /\
([--a; &1] exp n) divides p /\
~(([--a; &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 = &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:real list = []` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `q:real list` SUBST_ALL_TAC) THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `a:real` 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:real list` THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly] THEN REAL_ARITH_TAC;
DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `a:real` 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:real list` SUBST1_TAC) THEN
REWRITE_TAC[POLY_MUL; poly] THEN REAL_ARITH_TAC]);;
let ORDER_DIVIDES = prove
(`!p a n. ([--a; &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 `[]:real list` THEN
REWRITE_TAC[FUN_EQ_THM; POLY_MUL; poly; REAL_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; &1] exp (order a p)) ** q)) /\
~([--a; &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:real`) THEN
DISCH_THEN(X_CHOOSE_TAC `q:real list` o REWRITE_RULE[divides]) THEN
EXISTS_TAC `q:real list` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `r: real list` o REWRITE_RULE[divides]) THEN
UNDISCH_TAC `~([-- a; &1] exp SUC (order a p) divides p)` THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[divides] THEN
EXISTS_TAC `r:real list` THEN
ASM_REWRITE_TAC[POLY_MUL; FUN_EQ_THM; poly_exp] THEN
REAL_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:real`
(MATCH_MP ORDER_THM (ASSUME `~(poly p = poly [])`)))) THEN
DISCH_THEN(X_CHOOSE_TAC `r: real list` o REWRITE_RULE[divides]) THEN
MP_TAC(CONJUNCT1 (SPEC `a:real`
(MATCH_MP ORDER_THM (ASSUME `~(poly q = poly [])`)))) THEN
DISCH_THEN(X_CHOOSE_TAC `s: real 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 REAL_ARITH_TAC;
X_CHOOSE_THEN `r: real list` STRIP_ASSUME_TAC
(SPEC `a:real` (MATCH_MP ORDER_DECOMP (ASSUME `~(poly p = poly [])`))) THEN
X_CHOOSE_THEN `s: real list` STRIP_ASSUME_TAC
(SPEC `a:real` (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:real list` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `[--a; &1] divides (r ** s)` MP_TAC THENL
[ALL_TAC; ASM_REWRITE_TAC[POLY_PRIMES]] THEN
REWRITE_TAC[divides] THEN EXISTS_TAC `t:real list` THEN
SUBGOAL_THEN `poly ([-- a; &1] exp (order a p) ** r ** s) =
poly ([-- a; &1] exp (order a p) ** ([-- a; &1] ** t))`
MP_TAC THENL
[ALL_TAC; MESON_TAC[POLY_MUL_LCANCEL; POLY_EXP_PRIME_EQ_0]] THEN
SUBGOAL_THEN `poly ([-- a; &1] exp (order a q) **
[-- a; &1] exp (order a p) ** r ** s) =
poly ([-- a; &1] exp (order a q) **
[-- a; &1] exp (order a p) **
[-- a; &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[REAL_MUL_AC]]);;
let ORDER_DIFF = prove
(`!p a. ~(poly (diff p) = poly []) /\
~(order a p = 0)
==> (order a p = SUC (order a (diff p)))`,
REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
SUBGOAL_THEN `~(poly p = poly [])` MP_TAC THENL
[ASM_MESON_TAC[POLY_DIFF_ZERO]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `q:real list` MP_TAC o
SPEC `a:real` o MATCH_MP ORDER_DECOMP) THEN
SPEC_TAC(`order a p`,`n:num`) THEN
INDUCT_TAC THEN REWRITE_TAC[NOT_SUC; SUC_INJ] THEN
STRIP_TAC THEN MATCH_MP_TAC ORDER_UNIQUE THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `!r. r divides (diff p) <=>
r divides (diff ([-- a; &1] exp SUC n ** q))`
(fun th -> REWRITE_TAC[th]) THENL
[GEN_TAC THEN REWRITE_TAC[divides] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP POLY_DIFF_WELLDEF th]);
ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC[divides; FUN_EQ_THM] THEN
EXISTS_TAC `[--a; &1] ** (diff q) ++ &(SUC n) ## q` THEN
REWRITE_TAC[POLY_DIFF_MUL; POLY_DIFF_EXP_PRIME;
POLY_ADD; POLY_MUL; POLY_CMUL] THEN
REWRITE_TAC[poly_exp; POLY_MUL] THEN REAL_ARITH_TAC;
REWRITE_TAC[FUN_EQ_THM; divides; POLY_DIFF_MUL; POLY_DIFF_EXP_PRIME;
POLY_ADD; POLY_MUL; POLY_CMUL] THEN
DISCH_THEN(X_CHOOSE_THEN `r:real list` ASSUME_TAC) THEN
UNDISCH_TAC `~([-- a; &1] divides q)` THEN
REWRITE_TAC[divides] THEN
EXISTS_TAC `inv(&(SUC n)) ## (r ++ neg(diff q))` THEN
SUBGOAL_THEN
`poly ([--a; &1] exp n ** q) =
poly ([--a; &1] exp n ** ([-- a; &1] ** (inv (&(SUC n)) ##
(r ++ neg (diff q)))))`
MP_TAC THENL
[ALL_TAC; MESON_TAC[POLY_MUL_LCANCEL; POLY_EXP_PRIME_EQ_0]] THEN
REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:real` THEN
SUBGOAL_THEN
`!a b. (&(SUC n) * a = &(SUC n) * b) ==> (a = b)`
MATCH_MP_TAC THENL
[REWRITE_TAC[REAL_EQ_MUL_LCANCEL; REAL_OF_NUM_EQ; NOT_SUC]; ALL_TAC] THEN
REWRITE_TAC[POLY_MUL; POLY_CMUL] THEN
SUBGOAL_THEN `!a b c. &(SUC n) * a * b * inv(&(SUC n)) * c =
a * b * c`
(fun th -> REWRITE_TAC[th]) THENL
[REPEAT GEN_TAC THEN
GEN_REWRITE_TAC LAND_CONV [REAL_MUL_SYM] THEN
REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN AP_TERM_TAC THEN
AP_TERM_TAC THEN
GEN_REWRITE_TAC LAND_CONV [REAL_MUL_SYM] THEN
GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN
REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN AP_TERM_TAC THEN
MATCH_MP_TAC REAL_MUL_RINV THEN
REWRITE_TAC[REAL_OF_NUM_EQ; NOT_SUC]; ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o SPEC `x:real`) THEN
REWRITE_TAC[poly_exp; POLY_MUL; POLY_ADD; POLY_NEG] THEN
REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *)
(* ------------------------------------------------------------------------- *)
let POLY_SQUAREFREE_DECOMP_ORDER = prove
(`!p q d e r s.
~(poly (diff p) = poly []) /\
(poly p = poly (q ** d)) /\
(poly (diff p) = poly (e ** d)) /\
(poly d = poly (r ** p ++ s ** diff p))
==> !a. order a q = (if order a p = 0 then 0 else 1)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `order a p = order a q + order a d` MP_TAC THENL
[MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `order a (q ** d)` THEN
CONJ_TAC THENL
[MATCH_MP_TAC ORDER_POLY THEN ASM_REWRITE_TAC[];
MATCH_MP_TAC ORDER_MUL THEN
FIRST_ASSUM(fun th ->
GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [SYM th]) THEN
ASM_MESON_TAC[POLY_DIFF_ZERO]]; ALL_TAC] THEN
ASM_CASES_TAC `order a p = 0` THEN ASM_REWRITE_TAC[] THENL
[ARITH_TAC; ALL_TAC] THEN
SUBGOAL_THEN `order a (diff p) =
order a e + order a d` MP_TAC THENL
[MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `order a (e ** d)` THEN
CONJ_TAC THENL
[ASM_MESON_TAC[ORDER_POLY]; ASM_MESON_TAC[ORDER_MUL]]; ALL_TAC] THEN
SUBGOAL_THEN `~(poly p = poly [])` ASSUME_TAC THENL
[ASM_MESON_TAC[POLY_DIFF_ZERO]; ALL_TAC] THEN
MP_TAC(SPECL [`p:real list`; `a:real`] ORDER_DIFF) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC(AP_TERM `PRE` th)) THEN
REWRITE_TAC[PRE] THEN DISCH_THEN(ASSUME_TAC o SYM) THEN
SUBGOAL_THEN `order a (diff p) <= order a d` MP_TAC THENL
[SUBGOAL_THEN `([--a; &1] exp (order a (diff p))) divides d`
MP_TAC THENL [ALL_TAC; ASM_MESON_TAC[POLY_ENTIRE; ORDER_DIVIDES]] THEN
SUBGOAL_THEN
`([--a; &1] exp (order a (diff p))) divides p /\
([--a; &1] exp (order a (diff p))) divides (diff p)`
MP_TAC THENL
[REWRITE_TAC[ORDER_DIVIDES; LE_REFL] THEN DISJ2_TAC THEN
REWRITE_TAC[ASSUME `order a (diff p) = PRE (order a p)`] THEN
ARITH_TAC;
DISCH_THEN(CONJUNCTS_THEN MP_TAC) THEN REWRITE_TAC[divides] THEN
REWRITE_TAC[ASSUME `poly d = poly (r ** p ++ s ** diff p)`] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `f:real list` ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `g:real list` ASSUME_TAC) THEN
EXISTS_TAC `r ** g ++ s ** f` THEN ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[FUN_EQ_THM; POLY_MUL; POLY_ADD] THEN ARITH_TAC];
ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* Define being "squarefree" --- NB with respect to real roots only. *)
(* ------------------------------------------------------------------------- *)
let rsquarefree = new_definition
`rsquarefree p <=> ~(poly p = poly []) /\
!a. (order a p = 0) \/ (order a p = 1)`;;
(* ------------------------------------------------------------------------- *)
(* Standard squarefree criterion and rephasing of squarefree decomposition. *)
(* ------------------------------------------------------------------------- *)
let RSQUAREFREE_ROOTS = prove
(`!p. rsquarefree p <=> !a. ~((poly p a = &0) /\ (poly (diff p) a = &0))`,
GEN_TAC THEN REWRITE_TAC[rsquarefree] THEN
ASM_CASES_TAC `poly p = poly []` THEN ASM_REWRITE_TAC[] THENL
[FIRST_ASSUM(SUBST1_TAC o MATCH_MP POLY_DIFF_ZERO) THEN
ASM_REWRITE_TAC[poly; NOT_FORALL_THM];
ASM_CASES_TAC `poly(diff p) = poly []` THEN ASM_REWRITE_TAC[] THENL
[FIRST_ASSUM(X_CHOOSE_THEN `h:real` MP_TAC o
MATCH_MP POLY_DIFF_ISZERO) THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP ORDER_POLY th]) THEN
UNDISCH_TAC `~(poly p = poly [])` THEN ASM_REWRITE_TAC[poly] THEN
REWRITE_TAC[FUN_EQ_THM; poly; REAL_MUL_RZERO; REAL_ADD_RID] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `a:real` THEN DISJ1_TAC THEN
MP_TAC(SPECL [`[h:real]`; `a:real`] ORDER_ROOT) THEN
ASM_REWRITE_TAC[FUN_EQ_THM; poly; REAL_MUL_RZERO; REAL_ADD_RID];
ASM_REWRITE_TAC[ORDER_ROOT; DE_MORGAN_THM; num_CONV `1`] THEN
ASM_MESON_TAC[ORDER_DIFF; SUC_INJ]]]);;
let RSQUAREFREE_DECOMP = prove
(`!p a. rsquarefree p /\ (poly p a = &0)
==> ?q. (poly p = poly ([--a; &1] ** q)) /\
~(poly q a = &0)`,
REPEAT GEN_TAC THEN REWRITE_TAC[rsquarefree] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP ORDER_DECOMP) THEN
DISCH_THEN(X_CHOOSE_THEN `q:real list` MP_TAC o SPEC `a:real`) THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [ORDER_ROOT]) THEN
FIRST_ASSUM(DISJ_CASES_TAC o SPEC `a:real`) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[ARITH] THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
EXISTS_TAC `q:real list` THEN CONJ_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; POLY_MUL] THEN GEN_TAC THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV) [num_CONV `1`] THEN
REWRITE_TAC[poly_exp; POLY_MUL] THEN
REWRITE_TAC[poly] THEN REAL_ARITH_TAC;
DISCH_TAC THEN UNDISCH_TAC `~([-- a; &1] divides q)` THEN
REWRITE_TAC[divides] THEN
UNDISCH_TAC `poly q a = &0` THEN
GEN_REWRITE_TAC LAND_CONV [POLY_LINEAR_DIVIDES] THEN
ASM_CASES_TAC `q:real list = []` THEN ASM_REWRITE_TAC[] THENL
[EXISTS_TAC `[] : real list` THEN REWRITE_TAC[FUN_EQ_THM] THEN
REWRITE_TAC[POLY_MUL; poly; REAL_MUL_RZERO];
MESON_TAC[]]]);;
let POLY_SQUAREFREE_DECOMP = prove
(`!p q d e r s.
~(poly (diff p) = poly []) /\
(poly p = poly (q ** d)) /\
(poly (diff p) = poly (e ** d)) /\
(poly d = poly (r ** p ++ s ** diff p))
==> rsquarefree q /\ (!a. (poly q a = &0) <=> (poly p a = &0))`,
REPEAT GEN_TAC THEN DISCH_THEN(fun th -> MP_TAC th THEN
ASSUME_TAC(MATCH_MP POLY_SQUAREFREE_DECOMP_ORDER th)) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
SUBGOAL_THEN `~(poly p = poly [])` ASSUME_TAC THENL
[ASM_MESON_TAC[POLY_DIFF_ZERO]; ALL_TAC] THEN
DISCH_THEN(ASSUME_TAC o CONJUNCT1) THEN
UNDISCH_TAC `~(poly p = poly [])` THEN
DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC th) THEN
DISCH_THEN(fun th -> ASM_REWRITE_TAC[] THEN ASSUME_TAC th) THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[POLY_ENTIRE; DE_MORGAN_THM] THEN STRIP_TAC THEN
UNDISCH_TAC `poly p = poly (q ** d)` THEN
DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
ASM_REWRITE_TAC[rsquarefree; ORDER_ROOT] THEN
CONJ_TAC THEN GEN_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[ARITH]);;
(* ------------------------------------------------------------------------- *)
(* Normalization of a polynomial. *)
(* ------------------------------------------------------------------------- *)
let normalize = new_recursive_definition list_RECURSION
`(normalize [] = []) /\
(normalize (CONS h t) =
if normalize t = [] then if h = &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 = &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[REAL_MUL_RZERO; REAL_ADD_LID]);;
(* ------------------------------------------------------------------------- *)
(* 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:real list`,`p:real 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]]);;
(* ------------------------------------------------------------------------- *)
(* Tidier versions of finiteness of roots. *)
(* ------------------------------------------------------------------------- *)
let POLY_ROOTS_FINITE_SET = prove
(`!p. ~(poly p = poly []) ==> FINITE { x | poly p x = &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->real` ASSUME_TAC) THEN
MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `{x:real | ?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:real | ?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:real | ?n. n < SUC N /\ (x = i n)} =
(i N) INSERT {x:real | ?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]]);;
(* ------------------------------------------------------------------------- *)
(* Crude bound for polynomial. *)
(* ------------------------------------------------------------------------- *)
let POLY_MONO = prove
(`!x k p. abs(x) <= k ==> abs(poly p x) <= poly (MAP abs p) k`,
GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
DISCH_TAC THEN LIST_INDUCT_TAC THEN
REWRITE_TAC[poly; REAL_LE_REFL; MAP; REAL_ABS_0] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `abs(h) + abs(x * poly t x)` THEN
REWRITE_TAC[REAL_ABS_TRIANGLE; REAL_LE_LADD] THEN
REWRITE_TAC[REAL_ABS_MUL] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN ASM_REWRITE_TAC[REAL_ABS_POS]);;
(* ------------------------------------------------------------------------- *)
(* Conversions to perform operations if coefficients are rational constants. *)
(* ------------------------------------------------------------------------- *)
let POLY_DIFF_CONV =
let aux_conv0 = GEN_REWRITE_CONV I [CONJUNCT1 poly_diff_aux]
and aux_conv1 = GEN_REWRITE_CONV I [CONJUNCT2 poly_diff_aux]
and diff_conv0 = GEN_REWRITE_CONV I (butlast (CONJUNCTS POLY_DIFF_CLAUSES))
and diff_conv1 = GEN_REWRITE_CONV I [last (CONJUNCTS POLY_DIFF_CLAUSES)] in
let rec POLY_DIFF_AUX_CONV tm =
(aux_conv0 ORELSEC
(aux_conv1 THENC
LAND_CONV REAL_RAT_MUL_CONV THENC
RAND_CONV (LAND_CONV NUM_SUC_CONV THENC POLY_DIFF_AUX_CONV))) tm in
diff_conv0 ORELSEC
(diff_conv1 THENC POLY_DIFF_AUX_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 REAL_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 REAL_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 = &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 REAL_RAT_EQ_CONV))) THENC
norm_conv2)) tm in
POLY_NORMALIZE_CONV;;
(* ------------------------------------------------------------------------- *)
(* Some theorems asserting that operations give non-nil results. *)
(* ------------------------------------------------------------------------- *)
let NOT_POLY_CMUL_NIL = prove
(`!h p. ~(p = []) ==> ~((h ## p) = [])`,
STRIP_TAC THEN LIST_INDUCT_TAC THENL
[SIMP_TAC[]; SIMP_TAC[poly_cmul; NOT_CONS_NIL]]);;
let NOT_POLY_MUL_NIL = prove
(`!p1 p2. ~(p1 = []) /\ ~(p2 = []) ==> ~((p1 ** p2) = [])`,
LIST_INDUCT_TAC THENL
[SIMP_TAC[];
LIST_INDUCT_TAC THENL
[SIMP_TAC[];
SIMP_TAC[poly_mul;NOT_CONS_NIL] THEN
SPEC_TAC (`t:(real)list`,`t:(real)list`) THEN LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_cmul;NOT_CONS_NIL];
SIMP_TAC[poly_cmul;poly_add;NOT_CONS_NIL]]
]
]);;
let NOT_POLY_EXP_NIL = prove
(`!n p . ~(p = []) ==> ~((poly_exp p n) = [])`,
let lem001 = ASSUME `!p . ~(p = []) ==> ~(poly_exp p n = [])` in
let lem002 = SIMP_RULE[NOT_CONS_NIL] (SPEC `CONS (h:real) t` lem001) in
INDUCT_TAC THENL
[SIMP_TAC[poly_exp;NOT_CONS_NIL];
LIST_INDUCT_TAC THENL
[SIMP_TAC[];
SIMP_TAC[lem002;NOT_POLY_MUL_NIL;poly_exp;NOT_CONS_NIL]
]
]);;
let NOT_POLY_EXP_X_NIL = prove
(`!n. ~((poly_exp [&0;&1] n) = [])`,
let lem01 = prove(`~([&0;&1] = [])`,SIMP_TAC[NOT_CONS_NIL]) in
INDUCT_TAC THENL
[SIMP_TAC[poly_exp;NOT_CONS_NIL];
ASM_SIMP_TAC[poly_exp;NOT_POLY_MUL_NIL;lem01]]);;
(* ------------------------------------------------------------------------- *)
(* Some general lemmas. *)
(* ------------------------------------------------------------------------- *)
let POLY_CMUL_LID = prove
(`!p. &1 ## p = p`,
LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_cmul];
ASM_SIMP_TAC[poly_cmul] THEN SIMP_TAC[REAL_ARITH `&1 * h = h`]]);;
let POLY_MUL_LID = prove
(`!p. [&1] ** p = p`,
LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_mul;poly_cmul];
ONCE_REWRITE_TAC[poly_mul] THEN SIMP_TAC[POLY_CMUL_LID]]);;
let POLY_MUL_RID = prove
(`!p. p ** [&1] = p`,
LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_mul];
ASM_CASES_TAC `t:(real)list = []` THEN
ASM_SIMP_TAC[poly_mul;poly_cmul;poly_add;NOT_CONS_NIL;HD;TL;
REAL_ARITH `h + (real_of_num 0) = h`;REAL_ARITH `h * (real_of_num 1) = h`]
]);;
let POLY_ADD_SYM = prove
(`!x y . x ++ y = y ++ x`,
let lem1 = ASSUME `!y . t ++ y = y ++ t` in
let lem2 = SPEC `t':(real)list` lem1 in
LIST_INDUCT_TAC THENL
[LIST_INDUCT_TAC THENL [SIMP_TAC[poly_add]; SIMP_TAC[poly_add]];
LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_add];
SIMP_TAC[POLY_ADD_CLAUSES] THEN
ONCE_REWRITE_TAC[lem2] THEN
SIMP_TAC[SPECL [`h:real`;`h':real`] REAL_ADD_SYM]
]
]);;
let POLY_ADD_ASSOC = prove
(`!x y z . x ++ (y ++ z) = (x ++ y) ++ z`,
let lem1 = ASSUME `!y z. t ++ y ++ z = (t ++ y) ++ z` in
let lem2 = SPECL [`t':(real)list`;`t'':(real)list`] lem1 in
LIST_INDUCT_TAC THENL
[SIMP_TAC[POLY_ADD_CLAUSES];
LIST_INDUCT_TAC THENL
[SIMP_TAC[POLY_ADD_CLAUSES];
LIST_INDUCT_TAC THENL
[SIMP_TAC[POLY_ADD_CLAUSES];
SIMP_TAC[POLY_ADD_CLAUSES] THEN
SIMP_TAC[REAL_ADD_ASSOC] THEN
SIMP_TAC[lem2]
]
]
]);;
(* ------------------------------------------------------------------------- *)
(* Heads and tails resulting from operations. *)
(* ------------------------------------------------------------------------- *)
let TL_POLY_MUL_X = prove
(`!p. TL ([&0;&1] ** p) = p`,
LIST_INDUCT_TAC THENL
[ONCE_REWRITE_TAC[poly_mul] THEN
SIMP_TAC[NOT_CONS_NIL;poly_cmul;poly_add;TL;poly_mul];
ONCE_REWRITE_TAC[poly_mul] THEN SIMP_TAC[NOT_CONS_NIL] THEN
ONCE_REWRITE_TAC[poly_cmul] THEN ONCE_REWRITE_TAC[poly_add] THEN
SIMP_TAC[NOT_CONS_NIL] THEN SIMP_TAC[TL;POLY_MUL_LID] THEN
SPEC_TAC (`h:real`,`h:real`) THEN
SPEC_TAC (`t:(real)list`,`t:(real)list`) THEN
LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_cmul;poly_add];
ASM_SIMP_TAC[poly_cmul;poly_add;NOT_CONS_NIL;HD;TL;
REAL_ARITH `(&0) * h + h' = h'`]
]
]);;
let HD_POLY_MUL_X = prove
(`!p. HD ([&0;&1] ** p) = &0`,
LIST_INDUCT_TAC THEN
SIMP_TAC[poly_mul;NOT_CONS_NIL;poly_cmul;poly_add;HD;
REAL_ARITH `&0 * h + &0 = &0`]);;
let TL_POLY_EXP_X_SUC = prove
(`!n . TL (poly_exp [&0;&1] (SUC n)) = poly_exp [&0;&1] n`,
SIMP_TAC[poly_exp;TL_POLY_MUL_X]);;
let HD_POLY_EXP_X_SUC = prove
(`!n . HD (poly_exp [&0;&1] (SUC n)) = &0`,
INDUCT_TAC THENL
[SIMP_TAC[poly_exp;poly_add;HD;TL;poly_cmul;poly_mul;NOT_CONS_NIL;
REAL_ARITH `&0 * &1 + &0 = &0`];
SIMP_TAC[poly_exp;HD_POLY_MUL_X]]);;
let HD_POLY_ADD = prove
(`!p1 p2. ~(p1 = []) /\ ~(p2 = []) ==> HD (p1 ++ p2) = (HD p1) + (HD p2)`,
LIST_INDUCT_TAC THENL
[SIMP_TAC[];
LIST_INDUCT_TAC THENL
[SIMP_TAC[];
SIMP_TAC[NOT_CONS_NIL;poly_add] THEN
ONCE_REWRITE_TAC[ISPECL [`h':real`;`t':(real)list`] NOT_CONS_NIL] THEN
SIMP_TAC[HD]
]
]);;
let HD_POLY_CMUL = prove
(`!x p . ~(p = []) ==> HD (x ## p) = x * (HD p)`,
STRIP_TAC THEN LIST_INDUCT_TAC THENL
[SIMP_TAC[]; SIMP_TAC[NOT_CONS_NIL;poly_cmul;HD]]);;
let TL_POLY_CMUL = prove
(`!x p . ~(p = []) ==> TL (x ## p) = x ## (TL p)`,
STRIP_TAC THEN LIST_INDUCT_TAC THENL
[SIMP_TAC[]; SIMP_TAC[NOT_CONS_NIL;poly_cmul;TL]]);;
let HD_POLY_MUL = prove
(`!p1 p2 . ~(p1 = []) /\ ~(p2 = []) ==> HD (p1 ** p2) = (HD p1) * (HD p2)`,
LIST_INDUCT_TAC THENL
[SIMP_TAC[];
LIST_INDUCT_TAC THENL
[SIMP_TAC[];
SIMP_TAC[NOT_CONS_NIL;poly_mul] THEN
ASM_CASES_TAC `(t:(real)list) = []` THENL
[ASM_SIMP_TAC[poly_cmul;HD];
ASM_SIMP_TAC[poly_cmul;poly_add;NOT_CONS_NIL;HD] THEN REAL_ARITH_TAC
]
]
]);;
let HD_POLY_EXP = prove
(`!n p . ~(p = []) ==> HD (poly_exp p n) = (HD p) pow n`,
INDUCT_TAC THENL
[SIMP_TAC[poly_exp] THEN LIST_INDUCT_TAC THENL
[SIMP_TAC[]; SIMP_TAC[HD;pow]];
SIMP_TAC[poly_exp] THEN LIST_INDUCT_TAC THENL
[SIMP_TAC[];
SIMP_TAC[HD;GSYM pow;NOT_CONS_NIL;poly_mul] THEN
ASM_CASES_TAC `(t:(real)list) = []` THENL
[ASM_SIMP_TAC[HD_POLY_CMUL;NOT_POLY_CMUL_NIL;NOT_POLY_EXP_NIL;
NOT_CONS_NIL;HD;GSYM pow];
ASM_SIMP_TAC[NOT_POLY_CMUL_NIL;NOT_POLY_EXP_NIL;NOT_CONS_NIL;
HD_POLY_ADD;HD;HD_POLY_CMUL;GSYM pow] THEN
REAL_ARITH_TAC]
]
]);;
(* ------------------------------------------------------------------------- *)
(* Additional general lemmas. *)
(* ------------------------------------------------------------------------- *)
let POLY_ADD_IDENT = prove
(`neutral (++) = []`,
let l1 = ASSUME `!x. (!y. x ++ y = y /\ y ++ x = y)
==> (!y. (CONS h t) ++ y = y /\ y ++ (CONS h t) = y)` in
let l2 = SPEC `[]:(real)list` l1 in
let l3 = SIMP_RULE[POLY_ADD_CLAUSES] l2 in
let l4 = SPEC `[]:(real)list` l3 in
let l5 = CONJUNCT1 l4 in
let l6 = SIMP_RULE[POLY_ADD_CLAUSES;NOT_CONS_NIL] l5 in
let l7 = NOT_INTRO (DISCH_ALL l6) in
ONCE_REWRITE_TAC[neutral] THEN SELECT_ELIM_TAC THEN LIST_INDUCT_TAC THENL
[SIMP_TAC[];SIMP_TAC[l7]]);;
let POLY_ADD_NEUTRAL = prove
(`!x. neutral (++) ++ x = x`,
SIMP_TAC[POLY_ADD_IDENT;POLY_ADD_CLAUSES]);;
let MONOIDAL_POLY_ADD = prove
(`monoidal poly_add`,
let lem1 = CONJ POLY_ADD_SYM (CONJ POLY_ADD_ASSOC POLY_ADD_NEUTRAL) in
ONCE_REWRITE_TAC[monoidal] THEN ACCEPT_TAC lem1);;
let POLY_DIFF_AUX_ADD_LEMMA = prove
(`!t1 t2 n. poly_diff_aux n (t1 ++ t2) =
(poly_diff_aux n t1) ++ (poly_diff_aux n t2)`,
let lem = REAL_ARITH `!n h h'. (&n * h) + (&n * h') = &n * (h + h')` in
LIST_INDUCT_TAC THEN SIMP_TAC[POLY_ADD_CLAUSES;poly_diff_aux] THEN
LIST_INDUCT_TAC THEN SIMP_TAC[POLY_ADD_CLAUSES;poly_diff_aux] THEN
STRIP_TAC THEN
ONCE_REWRITE_TAC[POLY_ADD_CLAUSES] THEN
ONCE_REWRITE_TAC[poly_diff_aux] THEN
ONCE_REWRITE_TAC[POLY_ADD_CLAUSES] THEN
ONCE_REWRITE_TAC[lem] THEN
ASM_SIMP_TAC[]);;
let POLYDIFF_ADD = prove
(`!p1 p2. (poly_diff (p1 ++ p2)) = (poly_diff p1 ++ poly_diff p2)`,
let lem1 = prove
(`!h0 t0 h1 t1. ~(((CONS h0 t0) ++ (CONS h1 t1)) = [])`,
SIMP_TAC[POLY_ADD_CLAUSES;NOT_CONS_NIL]) in
let lem2 = prove
(`!h0 t0 h1 t1.
(TL ((CONS h0 t0) ++ (CONS h1 t1))
= (TL (CONS h0 t0)) ++ (TL (CONS h1 t1)))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[poly_add] THEN
ONCE_REWRITE_TAC[NOT_CONS_NIL] THEN REWRITE_TAC[TL]
THEN SIMP_TAC[]) in
REPEAT LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_add;poly_diff];
SIMP_TAC[poly_add;poly_diff];
SIMP_TAC[poly_add;poly_diff;POLY_ADD_CLAUSES];
SIMP_TAC[poly_diff] THEN
ONCE_REWRITE_TAC[lem1;NOT_CONS_NIL] THEN
SIMP_TAC[lem2;POLY_DIFF_AUX_ADD_LEMMA]
]);;
let POLY_DIFF_AUX_POLY_CMUL = prove
(`!p c n. poly_diff_aux n (c ## p) = c ## (poly_diff_aux n p)`,
let lem01 = ASSUME
`!c n. poly_diff_aux n (c ## t) = c ## poly_diff_aux n t` in
let lem02 = SPECL [`c:real`;`SUC n`] lem01 in
LIST_INDUCT_TAC THEN STRIP_TAC THEN STRIP_TAC THEN
SIMP_TAC[poly_cmul;poly_diff_aux;lem02;
REAL_ARITH `(a:real) * b * c = b * a * c`]);;
let POLY_CMUL_POLY_DIFF = prove
(`!p c. poly_diff (c ## p) = c ## (poly_diff p)`,
LIST_INDUCT_TAC THEN
SIMP_TAC[poly_diff;POLY_DIFF_AUX_POLY_CMUL;TL_POLY_CMUL;
poly_cmul;NOT_CONS_NIL]);;
(* ------------------------------------------------------------------------- *)
(* Theorems about the lengths of lists from the polynomial operations. *)
(* ------------------------------------------------------------------------- *)
let POLY_CMUL_LENGTH = prove
(`!c p. LENGTH (c ## p) = LENGTH p`,
STRIP_TAC THEN LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_cmul];
SIMP_TAC[poly_cmul] THEN ASM_SIMP_TAC[LENGTH]
]);;
let POLY_ADD_LENGTH = prove
(`!p q. LENGTH (p ++ q) = MAX (LENGTH p) (LENGTH q)`,
LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_add;LENGTH] THEN ARITH_TAC;
LIST_INDUCT_TAC THENL
[SIMP_TAC[poly_add;LENGTH] THEN ARITH_TAC;
SIMP_TAC[poly_add;LENGTH] THEN
ONCE_REWRITE_TAC[NOT_CONS_NIL] THEN SIMP_TAC[HD;TL;LENGTH] THEN
ASM_SIMP_TAC[] THEN
ONCE_REWRITE_TAC[ARITH_RULE `MAX x y = if (x > y) then x else y`] THEN
ASM_CASES_TAC `LENGTH (t:(real)list) > LENGTH (t':(real)list)` THENL
[ASM_SIMP_TAC[ARITH_RULE `x > y ==> (SUC x) > (SUC y)`];
ASM_SIMP_TAC[ARITH_RULE `~(x > y) ==> ~((SUC x) > (SUC y))`]]
]
]);;
let POLY_MUL_LENGTH = prove
(`!p h t. LENGTH (p ** (CONS h t)) >= LENGTH p`,
let lemma01 = ASSUME `!h t'. LENGTH (t ** CONS h t') >= LENGTH t` in
let lemma02 = SPECL [`h':real`;`t':(real)list`] lemma01 in
let lemma03 = ONCE_REWRITE_RULE[ARITH_RULE `(x:num) >= y <=> SUC x >= SUC y`]
lemma02 in
let lemma05 = ARITH_RULE `(y:num) >= z ==> (x + (y - x) >= z) ` in
let lemma06 = SPECL [`SUC (LENGTH (t ** (CONS (h':real) t')))`;
`LENGTH (h ## (CONS h' t'))`;
`SUC (LENGTH (t:(real)list))`] (GEN_ALL lemma05) in
let lemma07 = MATCH_MP (lemma06) (lemma03) in
LIST_INDUCT_TAC THENL
[SIMP_TAC[POLY_MUL_CLAUSES] THEN ARITH_TAC;
SIMP_TAC[poly_mul] THEN ASM_CASES_TAC `(t:(real)list) = []` THENL
[ASM_SIMP_TAC[POLY_CMUL_LENGTH;LENGTH] THEN ARITH_TAC;
ASM_SIMP_TAC[POLY_ADD_LENGTH;LENGTH;lemma07;
ARITH_RULE `!x y. (MAX x y) = x + (y - x)`]
]
]);;
let POLY_EXP_X_REC = prove
(`!n. poly_exp [&0;&1] (SUC n) = CONS (&0) (poly_exp [&0;&1] n)`,
let lem01 = MATCH_MP CONS_HD_TL (SPEC `(SUC n)` NOT_POLY_EXP_X_NIL) in
let lem02 = ONCE_REWRITE_RULE[HD_POLY_EXP_X_SUC; TL_POLY_EXP_X_SUC] lem01 in
ACCEPT_TAC (GEN_ALL lem02));;
let POLY_MUL_LENGTH2 = prove
(`!q p. ~(q = []) ==> LENGTH (p ** q) >= LENGTH p`,
LIST_INDUCT_TAC THEN SIMP_TAC[NOT_CONS_NIL; POLY_MUL_LENGTH]);;
let POLY_EXP_X_LENGTH = prove
(`!n. LENGTH (poly_exp [&0;&1] n) = SUC n`,
INDUCT_TAC THEN
ASM_SIMP_TAC[poly_exp;LENGTH; POLY_EXP_X_REC;
ARITH_RULE `(SUC x) = (SUC y) <=> x = y`]);;
(* ------------------------------------------------------------------------- *)
(* Expansion of a polynomial as a power sum. *)
(* ------------------------------------------------------------------------- *)
let POLY_SUM_EQUIV = prove
(`!p x.
~(p = []) ==>
poly p x = sum (0..(PRE (LENGTH p))) (\i. (EL i p)*(x pow i))`,
let lem000 = ARITH_RULE `0 <= 0 + 1 /\ 0 <= (LENGTH (t:(real)list))` in
let lem001 = SPECL
[`f:num->real`;`0`;`0`;`LENGTH (t:(real)list)`]
SUM_COMBINE_R in
let lem002 = MP lem001 lem000 in
let lem003 = SPECL
[`f:num->real`;`1`;`LENGTH (t:(real)list)`]
SUM_OFFSET_0 in
let lem004 = ASSUME `~((t:(real)list) = [])` in
let lem005 = ONCE_REWRITE_RULE[GSYM LENGTH_EQ_NIL] lem004 in
let lem006 = ONCE_REWRITE_RULE[ARITH_RULE `~(x = 0) <=> (1 <= x)`] lem005 in
let lem007 = MP lem003 lem006 in
let lem017 = ARITH_RULE `1 <= (LENGTH (t:(real)list))
==> ((LENGTH t) - 1 = PRE (LENGTH t))` in
let lem018 = MP lem017 lem006 in
LIST_INDUCT_TAC THENL
[ SIMP_TAC[NOT_CONS_NIL]
;
ASM_CASES_TAC `(t:(real)list) = []` THENL
[
ASM_SIMP_TAC[POLY_CONST;LENGTH;PRE]
THEN ONCE_REWRITE_TAC[NUMSEG_CONV `0..0`]
THEN ONCE_REWRITE_TAC[SUM_SING]
THEN BETA_TAC
THEN ONCE_REWRITE_TAC[EL]
THEN ONCE_REWRITE_TAC[HD]
THEN REAL_ARITH_TAC
;
ASM_SIMP_TAC[POLY_CONST;LENGTH;PRE]
THEN ONCE_REWRITE_TAC[poly]
THEN ONCE_REWRITE_TAC[GSYM lem002]
THEN ONCE_REWRITE_TAC[ARITH_RULE `0 + 1 = 1`]
THEN ONCE_REWRITE_TAC[NUMSEG_CONV `0..0`]
THEN ONCE_REWRITE_TAC[SUM_SING]
THEN BETA_TAC
THEN SIMP_TAC[EL;HD]
THEN ONCE_REWRITE_TAC[lem007]
THEN BETA_TAC
THEN ONCE_REWRITE_TAC[GSYM ADD1]
THEN SIMP_TAC[EL;TL]
THEN ONCE_REWRITE_TAC[real_pow]
THEN ONCE_REWRITE_TAC[REAL_MUL_RID]
THEN ONCE_REWRITE_TAC[REAL_ARITH `(A:real) * B * C = B * (A * C)`]
THEN ONCE_REWRITE_TAC[NSUM_LMUL]
THEN ONCE_REWRITE_TAC[SUM_LMUL]
THEN ASM_SIMP_TAC[]
THEN SIMP_TAC[NOT_CONS_NIL]
THEN ONCE_REWRITE_TAC[lem018]
THEN SIMP_TAC[]
]]);;
let ITERATE_RADD_POLYADD = prove
(`!n x f. iterate (+) (0..n) (\i.poly (f i) x) =
poly (iterate (++) (0..n) f) x`,
INDUCT_TAC THEN
ASM_SIMP_TAC[ITERATE_CLAUSES_NUMSEG; MONOIDAL_REAL_ADD; MONOIDAL_POLY_ADD;
LE_0; POLY_ADD]);;
(* ------------------------------------------------------------------------- *)
(* Now we're finished with polynomials... *)
(* ------------------------------------------------------------------------- *)
do_list reduce_interface
["divides",`poly_divides:real list->real list->bool`;
"exp",`poly_exp:real list -> num -> real list`;
"diff",`poly_diff:real list->real list`];;
unparse_as_infix "exp";;