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(* ========================================================================= *)
(* Rob Arthan's "Descartes's Rule of Signs by an Easy Induction". *)
(* ========================================================================= *)
needs "Multivariate/realanalysis.ml";;
(* ------------------------------------------------------------------------- *)
(* A couple of handy lemmas. *)
(* ------------------------------------------------------------------------- *)
let OPPOSITE_SIGNS = prove
(`!a b:real. a * b < &0 <=> &0 < a /\ b < &0 \/ a < &0 /\ &0 < b`,
REWRITE_TAC[REAL_ARITH `a * b < &0 <=> &0 < a * --b`; REAL_MUL_POS_LT] THEN
REAL_ARITH_TAC);;
let VARIATION_SET_FINITE = prove
(`FINITE s ==> FINITE {p,q | p IN s /\ q IN s /\ P p q}`,
REWRITE_TAC[SET_RULE
`{p,q | p IN s /\ q IN t /\ R p q} =
{p,q | p IN s /\ q IN {q | q IN t /\ R p q}}`] THEN
SIMP_TAC[FINITE_PRODUCT_DEPENDENT; FINITE_RESTRICT]);;
(* ------------------------------------------------------------------------- *)
(* Variation in a sequence of coefficients. *)
(* ------------------------------------------------------------------------- *)
let variation = new_definition
`variation s (a:num->real) =
CARD {(p,q) | p IN s /\ q IN s /\ p < q /\
a(p) * a(q) < &0 /\
!i. i IN s /\ p < i /\ i < q ==> a(i) = &0 }`;;
let VARIATION_EQ = prove
(`!a b s. (!i. i IN s ==> a i = b i) ==> variation s a = variation s b`,
REPEAT STRIP_TAC THEN REWRITE_TAC[variation] THEN AP_TERM_TAC THEN
REWRITE_TAC[EXTENSION; FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN
ASM_MESON_TAC[]);;
let VARIATION_SUBSET = prove
(`!a s t. t SUBSET s /\ (!i. i IN (s DIFF t) ==> a i = &0)
==> variation s a = variation t a`,
REWRITE_TAC[IN_DIFF; SUBSET] THEN REPEAT STRIP_TAC THEN
REWRITE_TAC[variation] THEN AP_TERM_TAC THEN
REWRITE_TAC[EXTENSION; FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN
ASM_MESON_TAC[REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_LT_REFL]);;
let VARIATION_SPLIT = prove
(`!a s n.
FINITE s /\ n IN s /\ ~(a n = &0)
==> variation s a = variation {i | i IN s /\ i <= n} a +
variation {i | i IN s /\ n <= i} a`,
REWRITE_TAC[variation] THEN REPEAT STRIP_TAC THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC CARD_UNION_EQ THEN
ASM_SIMP_TAC[VARIATION_SET_FINITE; FINITE_RESTRICT] THEN
REWRITE_TAC[EXTENSION; FORALL_PAIR_THM] THEN CONJ_TAC THENL
[REWRITE_TAC[IN_INTER; NOT_IN_EMPTY; IN_ELIM_PAIR_THM; IN_NUMSEG] THEN
REWRITE_TAC[IN_ELIM_THM] THEN ARITH_TAC;
REWRITE_TAC[IN_UNION; IN_ELIM_PAIR_THM; IN_NUMSEG] THEN
REPEAT GEN_TAC THEN EQ_TAC THENL
[STRIP_TAC;
STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
MP_TAC(SPEC `n:num` th) THEN ASM_REWRITE_TAC[] THEN ASSUME_TAC th) THEN
SIMP_TAC[TAUT `~(a /\ b) <=> ~b \/ ~a`] THEN MATCH_MP_TAC MONO_OR] THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN_ELIM_THM]) THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
TRY(FIRST_ASSUM MATCH_MP_TAC) THEN
FIRST_ASSUM(fun th -> MP_TAC(SPEC `n:num` th) THEN ASM_REWRITE_TAC[]) THEN
ASM_ARITH_TAC]);;
let VARIATION_SPLIT_NUMSEG = prove
(`!a m n p. n IN m..p /\ ~(a n = &0)
==> variation(m..p) a = variation(m..n) a + variation(n..p) a`,
REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP
(REWRITE_RULE[TAUT `a /\ b /\ c ==> d <=> b /\ c ==> a ==> d`]
VARIATION_SPLIT)) THEN
REWRITE_TAC[FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN
BINOP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_NUMSEG] THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN_NUMSEG]) THEN ASM_ARITH_TAC);;
let VARIATION_1 = prove
(`!a n. variation {n} a = 0`,
REWRITE_TAC[variation; IN_SING] THEN
REWRITE_TAC[ARITH_RULE `p:num = n /\ q = n /\ p < q /\ X <=> F`] THEN
MATCH_MP_TAC(MESON[CARD_CLAUSES] `s = {} ==> CARD s = 0`) THEN
REWRITE_TAC[EXTENSION; FORALL_PAIR_THM; IN_ELIM_PAIR_THM; NOT_IN_EMPTY]);;
let VARIATION_2 = prove
(`!a m n. variation {m,n} a = if a(m) * a(n) < &0 then 1 else 0`,
GEN_TAC THEN MATCH_MP_TAC WLOG_LT THEN REPEAT CONJ_TAC THENL
[REWRITE_TAC[INSERT_AC; VARIATION_1; GSYM REAL_NOT_LE; REAL_LE_SQUARE];
REWRITE_TAC[INSERT_AC; REAL_MUL_SYM];
REPEAT STRIP_TAC THEN REWRITE_TAC[variation; IN_INSERT; NOT_IN_EMPTY] THEN
ONCE_REWRITE_TAC[TAUT
`a /\ b /\ c /\ d /\ e <=> (a /\ b /\ c) /\ d /\ e`] THEN
ASM_SIMP_TAC[ARITH_RULE
`m:num < n
==> ((p = m \/ p = n) /\ (q = m \/ q = n) /\ p < q <=>
p = m /\ q = n)`] THEN
REWRITE_TAC[MESON[] `(p = m /\ q = n) /\ X p q <=>
(p = m /\ q = n) /\ X m n`] THEN
REWRITE_TAC[ARITH_RULE `(i:num = m \/ i = n) /\ m < i /\ i < n <=> F`] THEN
ASM_CASES_TAC `a m * a(n:num) < &0` THEN ASM_REWRITE_TAC[] THENL
[REWRITE_TAC[SET_RULE `{p,q | p = a /\ q = b} = {(a,b)}`] THEN
SIMP_TAC[CARD_CLAUSES; FINITE_RULES; NOT_IN_EMPTY; ARITH];
MATCH_MP_TAC(MESON[CARD_CLAUSES] `s = {} ==> CARD s = 0`) THEN
SIMP_TAC[EXTENSION; FORALL_PAIR_THM; IN_ELIM_PAIR_THM; NOT_IN_EMPTY]]]);;
let VARIATION_3 = prove
(`!a m n p.
m < n /\ n < p
==> variation {m,n,p} a = if a(n) = &0 then variation{m,p} a
else variation {m,n} a + variation{n,p} a`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THENL
[MATCH_MP_TAC VARIATION_SUBSET THEN ASM SET_TAC[];
MP_TAC(ISPECL [`a:num->real`; `{m:num,n,p}`; `n:num`] VARIATION_SPLIT) THEN
ASM_SIMP_TAC[FINITE_INSERT; FINITE_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
DISCH_THEN SUBST1_TAC THEN BINOP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INSERT; NOT_IN_EMPTY] THEN
ASM_ARITH_TAC]);;
let VARIATION_OFFSET = prove
(`!p m n a. variation(m+p..n+p) a = variation(m..n) (\i. a(i + p))`,
REPEAT GEN_TAC THEN REWRITE_TAC[variation] THEN
MATCH_MP_TAC BIJECTIONS_CARD_EQ THEN MAP_EVERY EXISTS_TAC
[`\(i:num,j). i - p,j - p`; `\(i:num,j). i + p,j + p`] THEN
REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_PAIR_THM] THEN
SIMP_TAC[VARIATION_SET_FINITE; FINITE_NUMSEG] THEN
REWRITE_TAC[IN_NUMSEG; PAIR_EQ] THEN
REPEAT STRIP_TAC THEN TRY ASM_ARITH_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`y < &0 ==> x = y ==> x < &0`)) THEN
BINOP_TAC THEN AP_TERM_TAC THEN ASM_ARITH_TAC;
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC;
FIRST_X_ASSUM(MP_TAC o SPEC `i - p:num`) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; MATCH_MP_TAC EQ_IMP] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *)
(* The crucial lemma (roughly Lemma 2 in the paper). *)
(* ------------------------------------------------------------------------- *)
let ARTHAN_LEMMA = prove
(`!n a b.
~(a n = &0) /\ (b n = &0) /\ (!m. sum(0..m) a = b m)
==> ?d. ODD d /\ variation (0..n) a = variation (0..n) b + d`,
MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN
DISCH_THEN(LABEL_TAC "*") THEN
REPEAT STRIP_TAC THEN ASM_CASES_TAC `n = 0` THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `0`) THEN
ASM_REWRITE_TAC[SUM_SING_NUMSEG] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE
`~(n = 0) ==> n = 1 \/ 2 <= n`))
THENL
[FIRST_X_ASSUM SUBST_ALL_TAC THEN EXISTS_TAC `1` THEN
CONV_TAC NUM_REDUCE_CONV THEN
CONV_TAC(ONCE_DEPTH_CONV NUMSEG_CONV) THEN
REWRITE_TAC[VARIATION_2; OPPOSITE_SIGNS] THEN
FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `0` th) THEN MP_TAC(SPEC `1` th)) THEN
SIMP_TAC[num_CONV `1`; SUM_CLAUSES_NUMSEG] THEN
CONV_TAC NUM_REDUCE_CONV THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN
ASM_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `?p. 1 < p /\ p <= n /\ ~(a p = &0)` MP_TAC THENL
[ASM_MESON_TAC[ARITH_RULE `2 <= n ==> 1 < n`; LE_REFL];
GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN
REWRITE_TAC[TAUT `a ==> ~(b /\ c /\ ~d) <=> a /\ b /\ c ==> d`] THEN
STRIP_TAC] THEN
REMOVE_THEN "*" (MP_TAC o SPEC `n - 1`) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o SPECL
[`(\i. if i + 1 = 1 then a 0 + a 1 else a(i + 1)):num->real`;
`(\i. b(i + 1)):num->real`]) THEN
ASM_SIMP_TAC[ARITH_RULE `2 <= n ==> ~(n = 1) /\ n - 1 + 1 = n`] THEN
REWRITE_TAC[GSYM(SPEC `1` VARIATION_OFFSET)] THEN ANTS_TAC THENL
[X_GEN_TAC `m:num` THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `sum(0..m+1) a` THEN CONJ_TAC THENL
[SIMP_TAC[SUM_CLAUSES_LEFT; LE_0; ARITH_RULE `0 + 1 <= n + 1`] THEN
CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[REAL_ADD_ASSOC] THEN
AP_TERM_TAC THEN REWRITE_TAC[ARITH_RULE `2 = 1 + 1`; SUM_OFFSET] THEN
MATCH_MP_TAC SUM_EQ_NUMSEG THEN ARITH_TAC;
ASM_REWRITE_TAC[]];
ABBREV_TAC `a':num->real = \m. if m = 1 then a 0 + a 1 else a m` THEN
ASM_SIMP_TAC[ARITH_RULE
`2 <= n ==> n - 1 + 1 = n /\ n - 1 - 1 + 1 = n - 1`] THEN
CONV_TAC NUM_REDUCE_CONV] THEN
SUBGOAL_THEN
`variation (1..n) a' = variation {1,p} a' + variation (p..n) a /\
variation (0..n) a = variation {0,1,p} a + variation (p..n) a`
(CONJUNCTS_THEN SUBST1_TAC)
THENL
[CONJ_TAC THEN MATCH_MP_TAC EQ_TRANS THENL
[EXISTS_TAC `variation(1..p) a' + variation(p..n) a'`;
EXISTS_TAC `variation(0..p) a + variation(p..n) a`] THEN
(CONJ_TAC THENL
[MATCH_MP_TAC VARIATION_SPLIT_NUMSEG THEN EXPAND_TAC "a'" THEN
REWRITE_TAC[IN_NUMSEG] THEN ASM_ARITH_TAC;
BINOP_TAC THENL
[MATCH_MP_TAC VARIATION_SUBSET; MATCH_MP_TAC VARIATION_EQ] THEN
EXPAND_TAC "a'" THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
REWRITE_TAC[IN_NUMSEG] THEN TRY(GEN_TAC THEN ASM_ARITH_TAC) THEN
(CONJ_TAC THENL [ASM_ARITH_TAC; REWRITE_TAC[IN_DIFF]]) THEN
REWRITE_TAC[IN_NUMSEG; IN_INSERT; NOT_IN_EMPTY] THEN
REPEAT STRIP_TAC THEN TRY COND_CASES_TAC THEN
TRY(FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_ARITH_TAC]);
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d:num` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[GSYM INT_OF_NUM_EQ; GSYM INT_OF_NUM_ADD] THEN
DISCH_THEN(ASSUME_TAC o MATCH_MP (INT_ARITH
`a + b:int = c + d ==> c = (a + b) - d`)) THEN
REWRITE_TAC[INT_ARITH `a + b:int = c + d <=> d = (a + b) - c`] THEN
ASM_CASES_TAC `a 0 + a 1 = &0` THENL
[SUBGOAL_THEN `!i. 0 < i /\ i < p ==> b i = &0` ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM o SPEC `i:num`) THEN
ASM_SIMP_TAC[SUM_CLAUSES_LEFT; LE_0;
ARITH_RULE `0 < i ==> 0 + 1 <= i`] THEN
CONV_TAC NUM_REDUCE_CONV THEN
ASM_REWRITE_TAC[REAL_ADD_ASSOC; REAL_ADD_LID] THEN
MATCH_MP_TAC SUM_EQ_0_NUMSEG THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `(b:num->real) p = a p` ASSUME_TAC THENL
[FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM th]) THEN
SIMP_TAC[SUM_CLAUSES_RIGHT; ASSUME `1 < p`;
ARITH_RULE `1 < p ==> 0 < p /\ 0 <= p`] THEN
ASM_REWRITE_TAC[REAL_EQ_ADD_RCANCEL_0] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `variation(0..n) b = variation {0,p} b + variation(1..n) b`
SUBST1_TAC THENL
[MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `variation(0..p) b + variation(p..n) b` THEN CONJ_TAC THENL
[MATCH_MP_TAC VARIATION_SPLIT_NUMSEG THEN REWRITE_TAC[IN_NUMSEG] THEN
CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `p:num`) THEN
SIMP_TAC[SUM_CLAUSES_RIGHT; ASSUME `1 < p`;
ARITH_RULE `1 < p ==> 0 < p /\ 0 <= p`] THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`~(ap = &0) ==> s = &0 ==> ~(s + ap = &0)`)) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC;
BINOP_TAC THENL [ALL_TAC; CONV_TAC SYM_CONV] THEN
MATCH_MP_TAC VARIATION_SUBSET THEN
REWRITE_TAC[SUBSET; IN_DIFF; IN_NUMSEG; IN_INSERT; NOT_IN_EMPTY] THEN
(CONJ_TAC THENL [ASM_ARITH_TAC; REPEAT STRIP_TAC]) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC];
ALL_TAC];
SUBGOAL_THEN `variation(0..n) b = variation {0,1} b + variation(1..n) b`
SUBST1_TAC THENL
[MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `variation(0..1) b + variation(1..n) b` THEN CONJ_TAC THENL
[MATCH_MP_TAC VARIATION_SPLIT_NUMSEG THEN REWRITE_TAC[IN_NUMSEG] THEN
CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
FIRST_ASSUM(SUBST1_TAC o SYM o SPEC `1`) THEN
SIMP_TAC[SUM_CLAUSES_NUMSEG; num_CONV `1`] THEN
CONV_TAC NUM_REDUCE_CONV THEN ASM_REWRITE_TAC[];
AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC VARIATION_SUBSET THEN
REWRITE_TAC[SUBSET; IN_DIFF; IN_NUMSEG; IN_INSERT; NOT_IN_EMPTY] THEN
ARITH_TAC];
SUBGOAL_THEN `(b:num->real) 1 = a 0 + a 1` ASSUME_TAC THENL
[FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM th]) THEN
SIMP_TAC[num_CONV `1`; SUM_CLAUSES_NUMSEG] THEN
CONV_TAC NUM_REDUCE_CONV;
ALL_TAC]]] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `0`)) THEN CONV_TAC NUM_REDUCE_CONV THEN
REWRITE_TAC[SUM_SING_NUMSEG] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[GSYM INT_OF_NUM_ADD] THEN
ASM_SIMP_TAC[VARIATION_3; ARITH; OPPOSITE_SIGNS] THEN COND_CASES_TAC THEN
REWRITE_TAC[VARIATION_2; OPPOSITE_SIGNS; REAL_LT_REFL] THEN
EXPAND_TAC "a'" THEN CONV_TAC NUM_REDUCE_CONV THEN
ASM_SIMP_TAC[ARITH_RULE `1 < p ==> ~(p = 1)`; REAL_LT_REFL] THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
CONV_TAC NUM_REDUCE_CONV THEN
CONV_TAC(BINDER_CONV(RAND_CONV(RAND_CONV INT_POLY_CONV))) THEN
REWRITE_TAC[INT_ARITH `x:int = y + --z <=> x + z = y`] THEN
REWRITE_TAC[INT_OF_NUM_ADD; INT_OF_NUM_EQ] THEN
ONCE_REWRITE_TAC[CONJ_SYM] THEN ASM_REWRITE_TAC[UNWIND_THM2] THEN
ASM_REWRITE_TAC[ODD_ADD; ARITH_ODD; ARITH_EVEN] THEN ASM_REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Relate even-ness or oddity of variation to signs of end coefficients. *)
(* ------------------------------------------------------------------------- *)
let VARIATION_OPPOSITE_ENDS = prove
(`!a m n.
m <= n /\ ~(a m = &0) /\ ~(a n = &0)
==> (ODD(variation(m..n) a) <=> a m * a n < &0)`,
REPEAT GEN_TAC THEN WF_INDUCT_TAC `n - m:num` THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `!i:num. m < i /\ i < n ==> a i = &0` THENL
[MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `ODD(variation {m,n} a)` THEN
CONJ_TAC THENL
[AP_TERM_TAC THEN MATCH_MP_TAC VARIATION_SUBSET THEN
ASM_REWRITE_TAC[INSERT_SUBSET; IN_NUMSEG; IN_DIFF; EMPTY_SUBSET] THEN
REWRITE_TAC[LE_REFL; IN_INSERT; NOT_IN_EMPTY] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC;
REWRITE_TAC[VARIATION_2] THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[ARITH]];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
REWRITE_TAC[NOT_IMP] THEN
DISCH_THEN(X_CHOOSE_THEN `p:num` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(SPECL [`n:num`; `p:num`] th) THEN
MP_TAC(SPECL [`p:num`; `m:num`] th)) THEN
ASM_SIMP_TAC[LT_IMP_LE] THEN
REPEAT(ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_TAC]) THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `ODD(variation(m..p) a + variation(p..n) a)` THEN CONJ_TAC THENL
[AP_TERM_TAC THEN MATCH_MP_TAC VARIATION_SPLIT_NUMSEG THEN
ASM_SIMP_TAC[LT_IMP_LE; IN_NUMSEG];
ASM_REWRITE_TAC[ODD_ADD; OPPOSITE_SIGNS] THEN ASM_REAL_ARITH_TAC]]);;
(* ------------------------------------------------------------------------- *)
(* Polynomial with odd variation has at least one positive root. *)
(* This is the only "analytical" part of the proof. *)
(* ------------------------------------------------------------------------- *)
let REAL_POLYFUN_SGN_AT_INFINITY = prove
(`!a n. ~(a n = &0)
==> ?B. &0 < B /\
!x. B <= abs x
==> real_sgn(sum(0..n) (\i. a i * x pow i)) =
real_sgn(a n * x pow n)`,
let lemma = prove
(`abs(x) < abs(y) ==> real_sgn(x + y) = real_sgn y`,
REWRITE_TAC[real_sgn] THEN REAL_ARITH_TAC) in
REPEAT STRIP_TAC THEN ASM_CASES_TAC `n = 0` THENL
[EXISTS_TAC `&1` THEN ASM_REWRITE_TAC[REAL_LT_01; SUM_SING_NUMSEG];
ALL_TAC] THEN
ABBREV_TAC `B = sum (0..n-1) (\i. abs(a i)) / abs(a n)` THEN
SUBGOAL_THEN `&0 <= B` ASSUME_TAC THENL
[EXPAND_TAC "B" THEN SIMP_TAC[REAL_LE_DIV; REAL_ABS_POS; SUM_POS_LE_NUMSEG];
ALL_TAC] THEN
EXISTS_TAC `&1 + B` THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
X_GEN_TAC `x:real` THEN STRIP_TAC THEN
ASM_SIMP_TAC[SUM_CLAUSES_RIGHT; LE_0; LE_1] THEN MATCH_MP_TAC lemma THEN
MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `sum(0..n-1) (\i. abs(a i)) * abs x pow (n - 1)` THEN
CONJ_TAC THENL
[REWRITE_TAC[GSYM SUM_RMUL] THEN MATCH_MP_TAC SUM_ABS_LE THEN
REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[REAL_ABS_MUL] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS; REAL_ABS_POW] THEN
MATCH_MP_TAC REAL_POW_MONO THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
SUBGOAL_THEN `(x:real) pow n = x * x pow (n - 1)` SUBST1_TAC THENL
[SIMP_TAC[GSYM(CONJUNCT2 real_pow)] THEN AP_TERM_TAC THEN ASM_ARITH_TAC;
REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_POW; REAL_MUL_ASSOC] THEN
MATCH_MP_TAC REAL_LT_RMUL THEN CONJ_TAC THENL
[ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ; GSYM REAL_ABS_NZ] THEN
ASM_REAL_ARITH_TAC;
MATCH_MP_TAC REAL_POW_LT THEN ASM_REAL_ARITH_TAC]]]);;
let REAL_POLYFUN_HAS_POSITIVE_ROOT = prove
(`!a n. a 0 < &0 /\ &0 < a n
==> ?x. &0 < x /\ sum(0..n) (\i. a i * x pow i) = &0`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?x. &0 < x /\ &0 <= sum(0..n) (\i. a i * x pow i)`
STRIP_ASSUME_TAC THENL
[MP_TAC(ISPECL [`a:num->real`; `n:num`] REAL_POLYFUN_SGN_AT_INFINITY) THEN
ASM_SIMP_TAC[REAL_LT_IMP_NZ] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `x:real` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `x:real`)) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
SUBGOAL_THEN `real_sgn(a n * x pow n) = &1` SUBST1_TAC THEN
ASM_SIMP_TAC[REAL_SGN_EQ; REAL_LT_MUL; REAL_POW_LT; real_gt] THEN
REWRITE_TAC[REAL_LT_IMP_LE];
MP_TAC(ISPECL [`\x. sum(0..n) (\i. a i * x pow i)`;
`&0`; `x:real`; `&0`] REAL_IVT_INCREASING) THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE; IN_REAL_INTERVAL;
REAL_POW_ZERO; COND_RAND] THEN
REWRITE_TAC[REAL_MUL_RID; REAL_MUL_RZERO; SUM_DELTA; IN_NUMSEG; LE_0] THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN ANTS_TAC THENL
[MATCH_MP_TAC REAL_CONTINUOUS_ON_SUM THEN
REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC REAL_CONTINUOUS_ON_LMUL THEN
MATCH_MP_TAC REAL_CONTINUOUS_ON_POW THEN
REWRITE_TAC[REAL_CONTINUOUS_ON_ID];
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real` THEN
SIMP_TAC[REAL_LT_LE] THEN ASM_CASES_TAC `y:real = &0` THEN
ASM_SIMP_TAC[REAL_POW_ZERO; COND_RAND; REAL_MUL_RZERO; REAL_MUL_RID] THEN
REWRITE_TAC[SUM_DELTA; IN_NUMSEG; LE_0] THEN ASM_REAL_ARITH_TAC]]);;
let ODD_VARIATION_POSITIVE_ROOT = prove
(`!a n. ODD(variation(0..n) a)
==> ?x. &0 < x /\ sum(0..n) (\i. a i * x pow i) = &0`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?M. !i. i IN 0..n /\ ~(a i = &0) ==> i <= M` MP_TAC THENL
[EXISTS_TAC `n:num` THEN SIMP_TAC[IN_NUMSEG]; ALL_TAC] THEN
SUBGOAL_THEN `?i. i IN 0..n /\ ~(a i = &0)` MP_TAC THENL
[MATCH_MP_TAC(MESON[] `((!i. P i ==> Q i) ==> F) ==> ?i. P i /\ ~Q i`) THEN
DISCH_TAC THEN SUBGOAL_THEN `variation (0..n) a = variation {0} a`
(fun th -> SUBST_ALL_TAC th THEN ASM_MESON_TAC[VARIATION_1; ODD]) THEN
MATCH_MP_TAC VARIATION_SUBSET THEN
ASM_SIMP_TAC[IN_DIFF] THEN REWRITE_TAC[IN_NUMSEG; SING_SUBSET; LE_0];
ALL_TAC] THEN
ONCE_REWRITE_TAC[TAUT `a ==> b ==> c <=> a ==> a /\ b ==> c`] THEN
GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN REWRITE_TAC[num_MAX] THEN
REWRITE_TAC[TAUT `p ==> ~(q /\ r) <=> p /\ q ==> ~r`; IN_NUMSEG] THEN
DISCH_THEN(X_CHOOSE_THEN `p:num` STRIP_ASSUME_TAC) THEN
ONCE_REWRITE_TAC[TAUT `p /\ ~q ==> r <=> p /\ ~r ==> q`] THEN
DISCH_THEN(X_CHOOSE_THEN `q:num` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `p:num <= q` ASSUME_TAC THENL
[ASM_MESON_TAC[NOT_LT]; ALL_TAC] THEN
SUBGOAL_THEN `(a:num->real) p * a q < &0` ASSUME_TAC THENL
[ASM_SIMP_TAC[GSYM VARIATION_OPPOSITE_ENDS] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[]
`ODD p ==> p = q ==> ODD q`)) THEN
MATCH_MP_TAC VARIATION_SUBSET THEN
REWRITE_TAC[SUBSET_NUMSEG; IN_NUMSEG; IN_DIFF; DE_MORGAN_THM] THEN
CONJ_TAC THENL [ASM_ARITH_TAC; REPEAT STRIP_TAC] THEN
FIRST_X_ASSUM(fun th -> MATCH_MP_TAC th THEN ASM_ARITH_TAC);
ALL_TAC] THEN
MP_TAC(ISPECL [`\i. (a:num->real)(p + i) / a q`; `q - p:num`]
REAL_POLYFUN_HAS_POSITIVE_ROOT) THEN
ASM_SIMP_TAC[ADD_CLAUSES; ARITH_RULE `p:num <= q ==> p + q - p = q`] THEN
ANTS_TAC THENL
[REWRITE_TAC[real_div; OPPOSITE_SIGNS; REAL_MUL_POS_LT] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPPOSITE_SIGNS]) THEN
REWRITE_TAC[REAL_ARITH `x < &0 <=> &0 < --x`; GSYM REAL_INV_NEG] THEN
REWRITE_TAC[REAL_LT_INV_EQ] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real` THEN
MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_RING
`!a. y = a * x ==> x = &0 ==> y = &0`) THEN
EXISTS_TAC `(a:num->real) q * x pow p` THEN
REWRITE_TAC[GSYM SUM_LMUL; REAL_ARITH
`(aq * xp) * api / aq * xi:real = (aq / aq) * api * (xp * xi)`] THEN
ASM_CASES_TAC `(a:num->real) q = &0` THENL
[ASM_MESON_TAC[REAL_MUL_LZERO; REAL_LT_REFL]; ALL_TAC] THEN
ASM_SIMP_TAC[GSYM REAL_POW_ADD; REAL_DIV_REFL; REAL_MUL_LID] THEN
ONCE_REWRITE_TAC[ADD_SYM] THEN MP_TAC(SPEC `p:num` SUM_OFFSET) THEN
DISCH_THEN(fun th -> REWRITE_TAC[GSYM th]) THEN
MATCH_MP_TAC SUM_SUPERSET THEN
REWRITE_TAC[SUBSET_NUMSEG; IN_NUMSEG; IN_DIFF; DE_MORGAN_THM] THEN
CONJ_TAC THENL [ASM_ARITH_TAC; REPEAT STRIP_TAC] THEN
REWRITE_TAC[REAL_ENTIRE] THEN DISJ1_TAC THEN
FIRST_X_ASSUM(fun th -> MATCH_MP_TAC th THEN ASM_ARITH_TAC));;
(* ------------------------------------------------------------------------- *)
(* Define root multiplicities. *)
(* ------------------------------------------------------------------------- *)
let multiplicity = new_definition
`multiplicity f r =
@k. ?a n. ~(sum(0..n) (\i. a i * r pow i) = &0) /\
!x. f(x) = (x - r) pow k * sum(0..n) (\i. a i * x pow i)`;;
let MULTIPLICITY_UNIQUE = prove
(`!f a r b m k.
(!x. f(x) = (x - r) pow k * sum(0..m) (\j. b j * x pow j)) /\
~(sum(0..m) (\j. b j * r pow j) = &0)
==> k = multiplicity f r`,
let lemma = prove
(`!i j f g. f real_continuous_on (:real) /\ g real_continuous_on (:real) /\
~(f r = &0) /\ ~(g r = &0)
==> (!x. (x - r) pow i * f(x) = (x - r) pow j * g(x))
==> j = i`,
MATCH_MP_TAC WLOG_LT THEN
REWRITE_TAC[] THEN CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(TAUT `F ==> p`) THEN
MP_TAC(ISPECL [`atreal r`; `f:real->real`;
`(f:real->real) r`; `&0`]
REALLIM_UNIQUE) THEN
ASM_REWRITE_TAC[TRIVIAL_LIMIT_ATREAL] THEN CONJ_TAC THENL
[REWRITE_TAC[GSYM REAL_CONTINUOUS_ATREAL] THEN
ASM_MESON_TAC[REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT; REAL_OPEN_UNIV;
IN_UNIV];
MATCH_MP_TAC REALLIM_TRANSFORM_EVENTUALLY THEN
EXISTS_TAC `\x:real. (x - r) pow (j - i) * g x` THEN
REWRITE_TAC[] THEN CONJ_TAC THENL
[REWRITE_TAC[EVENTUALLY_ATREAL] THEN EXISTS_TAC `&1` THEN
REWRITE_TAC[REAL_LT_01; REAL_ARITH `&0 < abs(x - r) <=> ~(x = r)`] THEN
X_GEN_TAC `x:real` THEN STRIP_TAC THEN MATCH_MP_TAC(REAL_RING
`!a. a * x = a * y /\ ~(a = &0) ==> x = y`) THEN
EXISTS_TAC `(x - r:real) pow i` THEN
ASM_REWRITE_TAC[REAL_MUL_ASSOC; GSYM REAL_POW_ADD; REAL_POW_EQ_0] THEN
ASM_SIMP_TAC[REAL_SUB_0; ARITH_RULE `i:num < j ==> i + j - i = j`];
SUBST1_TAC(REAL_ARITH `&0 = &0 * (g:real->real) r`) THEN
MATCH_MP_TAC REALLIM_MUL THEN CONJ_TAC THENL
[REWRITE_TAC[] THEN MATCH_MP_TAC REALLIM_NULL_POW THEN
REWRITE_TAC[GSYM REALLIM_NULL; REALLIM_ATREAL_ID] THEN ASM_ARITH_TAC;
REWRITE_TAC[GSYM REAL_CONTINUOUS_ATREAL] THEN
ASM_MESON_TAC[REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT;
REAL_OPEN_UNIV; IN_UNIV]]]]) in
REPEAT STRIP_TAC THEN REWRITE_TAC[multiplicity] THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC SELECT_UNIQUE THEN
X_GEN_TAC `j:num` THEN EQ_TAC THEN ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM] THENL
[REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MATCH_MP_TAC lemma THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN
MATCH_MP_TAC REAL_CONTINUOUS_ON_SUM THEN
REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC REAL_CONTINUOUS_ON_LMUL THEN
MATCH_MP_TAC REAL_CONTINUOUS_ON_POW THEN
REWRITE_TAC[REAL_CONTINUOUS_ON_ID];
DISCH_THEN SUBST1_TAC THEN
MAP_EVERY EXISTS_TAC [`b:num->real`; `m:num`] THEN ASM_REWRITE_TAC[]]);;
let MULTIPLICITY_WORKS = prove
(`!r n a.
(?i. i IN 0..n /\ ~(a i = &0))
==> ?b m.
~(sum(0..m) (\i. b i * r pow i) = &0) /\
!x. sum(0..n) (\i. a i * x pow i) =
(x - r) pow multiplicity (\x. sum(0..n) (\i. a i * x pow i)) r *
sum(0..m) (\i. b i * x pow i)`,
REWRITE_TAC[multiplicity] THEN CONV_TAC(ONCE_DEPTH_CONV SELECT_CONV) THEN
GEN_TAC THEN MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN
DISCH_TAC THEN X_GEN_TAC `a:num->real` THEN
ASM_CASES_TAC `(a:num->real) n = &0` THENL
[ASM_CASES_TAC `n = 0` THEN
ASM_REWRITE_TAC[NUMSEG_SING; IN_SING; UNWIND_THM2]
THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC `a:num->real`) THEN
ASM_SIMP_TAC[SUM_CLAUSES_RIGHT; LE_0; LE_1] THEN
REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_RID] THEN
DISCH_THEN MATCH_MP_TAC THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `i:num` MP_TAC) THEN
REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN
EXISTS_TAC `i:num` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `i:num = n` THENL [ASM_MESON_TAC[]; ASM_ARITH_TAC];
ALL_TAC] THEN
DISCH_THEN(K ALL_TAC) THEN
ASM_CASES_TAC `sum(0..n) (\i. a i * r pow i) = &0` THENL
[ASM_CASES_TAC `n = 0` THENL
[UNDISCH_TAC `sum (0..n) (\i. a i * r pow i) = &0` THEN
ASM_REWRITE_TAC[NUMSEG_SING; IN_SING; UNWIND_THM2; SUM_SING] THEN
REWRITE_TAC[real_pow; REAL_MUL_RID] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
MP_TAC(GEN `x:real` (ISPECL [`a:num->real`; `x:real`; `r:real`; `n:num`]
REAL_SUB_POLYFUN)) THEN ASM_SIMP_TAC[LE_1; REAL_SUB_RZERO] THEN
ABBREV_TAC `b j = sum (j + 1..n) (\i. a i * r pow (i - j - 1))` THEN
DISCH_THEN(K ALL_TAC) THEN
FIRST_X_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE I [GSYM FUN_EQ_THM]) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC `b:num->real`) THEN ANTS_TAC THENL
[EXISTS_TAC `n - 1` THEN REWRITE_TAC[IN_NUMSEG; LE_REFL; LE_0] THEN
EXPAND_TAC "b" THEN REWRITE_TAC[] THEN
ASM_SIMP_TAC[SUB_ADD; LE_1; SUM_SING_NUMSEG; real_pow; REAL_MUL_RID;
ARITH_RULE `n - (n - 1) - 1 = 0`];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `k:num` (fun th ->
EXISTS_TAC `SUC k` THEN MP_TAC th)) THEN
REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
STRIP_TAC THEN ASM_REWRITE_TAC[real_pow; GSYM REAL_MUL_ASSOC];
MAP_EVERY EXISTS_TAC [`0`; `a:num->real`; `n:num`] THEN
ASM_REWRITE_TAC[real_pow; REAL_MUL_LID]]);;
let MULTIPLICITY_OTHER_ROOT = prove
(`!a n r s.
~(r = s) /\ (?i. i IN 0..n /\ ~(a i = &0))
==> multiplicity (\x. (x - r) pow m * sum(0..n) (\i. a i * x pow i)) s =
multiplicity (\x. sum(0..n) (\i. a i * x pow i)) s`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC MULTIPLICITY_UNIQUE THEN
REWRITE_TAC[] THEN
MP_TAC(ISPECL [`s:real`; `n:num`; `a:num->real`]
MULTIPLICITY_WORKS) THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`c:num->real`; `p:num`] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (ASSUME_TAC o GSYM)) THEN
SUBGOAL_THEN
`?b q. !x. sum(0..q) (\j. b j * x pow j) =
(x - r) pow m * sum (0..p) (\i. c i * x pow i)`
MP_TAC THENL
[ALL_TAC;
REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
STRIP_TAC THEN
ASM_REWRITE_TAC[REAL_RING `r * x = s * r * y <=> r = &0 \/ s * y = x`] THEN
ASM_REWRITE_TAC[REAL_ENTIRE; REAL_POW_EQ_0; REAL_SUB_0]] THEN
MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`c:num->real`; `p:num`; `m:num`] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN INDUCT_TAC THEN REPEAT GEN_TAC THENL
[MAP_EVERY EXISTS_TAC [`c:num->real`; `p:num`] THEN
ASM_REWRITE_TAC[real_pow; REAL_MUL_LID];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`p:num`; `c:num->real`]) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:num->real`; `n:num`] THEN
DISCH_THEN(ASSUME_TAC o GSYM) THEN
ASM_REWRITE_TAC[real_pow; GSYM REAL_MUL_ASSOC] THEN
EXISTS_TAC `\i. (if 0 < i then a(i - 1) else &0) -
(if i <= n then r * a i else &0)` THEN
EXISTS_TAC `n + 1` THEN
REWRITE_TAC[REAL_SUB_RDISTRIB; SUM_SUB_NUMSEG] THEN X_GEN_TAC `x:real` THEN
BINOP_TAC THENL
[MP_TAC(ARITH_RULE `0 <= n + 1`) THEN SIMP_TAC[SUM_CLAUSES_LEFT] THEN
DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[SUM_OFFSET; LT_REFL] THEN
REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_LID; ARITH_RULE `0 < i + 1`] THEN
REWRITE_TAC[GSYM SUM_LMUL; ADD_SUB; REAL_POW_ADD; REAL_POW_1];
SIMP_TAC[SUM_CLAUSES_RIGHT; LE_0; ARITH_RULE `0 < n + 1`] THEN
REWRITE_TAC[ADD_SUB; ARITH_RULE `~(n + 1 <= n)`] THEN
SIMP_TAC[REAL_MUL_LZERO; REAL_ADD_RID; GSYM SUM_LMUL]] THEN
MATCH_MP_TAC SUM_EQ_NUMSEG THEN REWRITE_TAC[REAL_MUL_AC]);;
(* ------------------------------------------------------------------------- *)
(* The main lemmas to be applied iteratively. *)
(* ------------------------------------------------------------------------- *)
let VARIATION_POSITIVE_ROOT_FACTOR = prove
(`!a n r.
~(a n = &0) /\ &0 < r /\ sum(0..n) (\i. a i * r pow i) = &0
==> ?b. ~(b(n - 1) = &0) /\
(!x. sum(0..n) (\i. a i * x pow i) =
(x - r) * sum(0..n-1) (\i. b i * x pow i)) /\
?d. ODD d /\ variation(0..n) a = variation(0..n-1) b + d`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `n = 0` THENL
[ASM_SIMP_TAC[SUM_CLAUSES_NUMSEG; real_pow; REAL_MUL_RID] THEN MESON_TAC[];
STRIP_TAC] THEN
ABBREV_TAC `b = \j. sum (j + 1..n) (\i. a i * r pow (i - j - 1))` THEN
EXISTS_TAC `b:num->real` THEN REPEAT CONJ_TAC THENL
[EXPAND_TAC "b" THEN REWRITE_TAC[] THEN ASM_SIMP_TAC[SUB_ADD; LE_1] THEN
ASM_SIMP_TAC[SUM_SING_NUMSEG; ARITH_RULE `n - (n - 1) - 1 = 0`] THEN
ASM_REWRITE_TAC[real_pow; REAL_MUL_RID];
MP_TAC(GEN `x:real` (SPECL [`a:num->real`; `x:real`; `r:real`; `n:num`]
REAL_SUB_POLYFUN)) THEN
ASM_SIMP_TAC[LE_1; REAL_SUB_RZERO] THEN DISCH_THEN(K ALL_TAC) THEN
EXPAND_TAC "b" THEN REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `(b:num->real) n = &0` ASSUME_TAC THENL
[EXPAND_TAC "b" THEN REWRITE_TAC[] THEN MATCH_MP_TAC SUM_EQ_0_NUMSEG THEN
ARITH_TAC;
ALL_TAC] THEN
MP_TAC(ISPECL
[`n:num`; `\i. if i <= n then a i * (r:real) pow i else &0`;
`\i. if i <= n then --b i * (r:real) pow (i + 1) else &0`]
ARTHAN_LEMMA) THEN
ASM_SIMP_TAC[REAL_ENTIRE; REAL_POW_EQ_0; REAL_LT_IMP_NZ; REAL_NEG_0;
LE_REFL] THEN
ANTS_TAC THENL
[X_GEN_TAC `j:num` THEN EXPAND_TAC "b" THEN REWRITE_TAC[] THEN
ASM_CASES_TAC `j:num <= n` THEN ASM_REWRITE_TAC[] THENL
[SUBGOAL_THEN `!i:num. i <= j ==> i <= n` MP_TAC THENL
[ASM_ARITH_TAC; SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC)] THEN
REWRITE_TAC[REAL_ARITH `a:real = --b * c <=> a + b * c = &0`] THEN
REWRITE_TAC[GSYM SUM_RMUL; GSYM REAL_POW_ADD; GSYM REAL_MUL_ASSOC] THEN
SIMP_TAC[ARITH_RULE `j + 1 <= k ==> k - j - 1 + j + 1 = k`] THEN
ASM_SIMP_TAC[SUM_COMBINE_R; LE_0];
REWRITE_TAC[GSYM SUM_RESTRICT_SET; IN_NUMSEG] THEN
ASM_SIMP_TAC[ARITH_RULE
`~(j <= n) ==> ((0 <= i /\ i <= j) /\ i <= n <=> 0 <= i /\ i <= n)`] THEN
ASM_REWRITE_TAC[GSYM numseg]];
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:num` THEN
MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN MATCH_MP_TAC(ARITH_RULE
`x':num = x /\ y' = y ==> x' = y' + d ==> x = y + d`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `variation(0..n) (\i. a i * r pow i)` THEN CONJ_TAC THENL
[MATCH_MP_TAC VARIATION_EQ THEN SIMP_TAC[IN_NUMSEG];
ALL_TAC];
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `variation(0..n) (\i. --b i * r pow (i + 1))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC VARIATION_EQ THEN SIMP_TAC[IN_NUMSEG];
ALL_TAC] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `variation(0..n-1) (\i. --b i * r pow (i + 1))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC VARIATION_SUBSET THEN
REWRITE_TAC[SUBSET_NUMSEG; IN_DIFF; IN_NUMSEG] THEN
CONJ_TAC THENL [ARITH_TAC; X_GEN_TAC `i:num` THEN STRIP_TAC] THEN
SUBGOAL_THEN `i:num = n` SUBST_ALL_TAC THENL
[ASM_ARITH_TAC; ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC];
ALL_TAC]] THEN
REWRITE_TAC[variation] THEN
ONCE_REWRITE_TAC[REAL_ARITH
`(a * x) * (b * x'):real = (x * x') * a * b`] THEN
SIMP_TAC[NOT_IMP; GSYM CONJ_ASSOC; GSYM REAL_POW_ADD;
REAL_ARITH `--x * --y:real = x * y`] THEN
ONCE_REWRITE_TAC[REAL_ARITH `x * y < &0 <=> &0 < x * --y`] THEN
ASM_SIMP_TAC[REAL_LT_MUL_EQ; REAL_POW_LT] THEN
ASM_SIMP_TAC[REAL_MUL_RNEG; REAL_ENTIRE; REAL_NEG_EQ_0; REAL_POW_EQ_0] THEN
ASM_SIMP_TAC[REAL_LT_IMP_NZ]]);;
let VARIATION_POSITIVE_ROOT_MULTIPLE_FACTOR = prove
(`!r n a.
~(a n = &0) /\ &0 < r /\ sum(0..n) (\i. a i * r pow i) = &0
==> ?b k m. 0 < k /\ m < n /\ ~(b m = &0) /\
(!x. sum(0..n) (\i. a i * x pow i) =
(x - r) pow k * sum(0..m) (\i. b i * x pow i)) /\
~(sum(0..m) (\j. b j * r pow j) = &0) /\
?d. EVEN d /\ variation(0..n) a = variation(0..m) b + k + d`,
GEN_TAC THEN MATCH_MP_TAC num_WF THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN X_GEN_TAC `a:num->real` THEN
ASM_CASES_TAC `n = 0` THENL
[ASM_SIMP_TAC[SUM_CLAUSES_NUMSEG; real_pow; REAL_MUL_RID] THEN MESON_TAC[];
STRIP_TAC] THEN
MP_TAC(ISPECL [`a:num->real`; `n:num`; `r:real`]
VARIATION_POSITIVE_ROOT_FACTOR) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `c:num->real` MP_TAC) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `d:num` STRIP_ASSUME_TAC) THEN
ASM_CASES_TAC `sum(0..n-1) (\i. c i * r pow i) = &0` THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `c:num->real`)] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:num->real` THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN
DISCH_THEN(X_CHOOSE_THEN `k:num` MP_TAC) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `e:num` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `SUC k` THEN ASM_REWRITE_TAC[real_pow; REAL_MUL_ASSOC] THEN
REPEAT(CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN
REWRITE_TAC[ADD1; ADD_ASSOC] THEN EXISTS_TAC `d - 1 + e`;
MAP_EVERY EXISTS_TAC [`c:num->real`; `1`; `n - 1`] THEN
ASM_REWRITE_TAC[REAL_POW_1] THEN
REPEAT(CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN
EXISTS_TAC `d - 1`] THEN
UNDISCH_TAC `ODD d` THEN GEN_REWRITE_TAC LAND_CONV [ODD_EXISTS] THEN
DISCH_THEN(X_CHOOSE_THEN `p:num` SUBST1_TAC) THEN
ASM_REWRITE_TAC[SUC_SUB1; EVEN_ADD; EVEN_MULT; ARITH] THEN ARITH_TAC);;
let VARIATION_POSITIVE_ROOT_MULTIPLICITY_FACTOR = prove
(`!r n a.
~(a n = &0) /\ &0 < r /\ sum(0..n) (\i. a i * r pow i) = &0
==> ?b m. m < n /\ ~(b m = &0) /\
(!x. sum(0..n) (\i. a i * x pow i) =
(x - r) pow
(multiplicity (\x. sum(0..n) (\i. a i * x pow i)) r) *
sum(0..m) (\i. b i * x pow i)) /\
~(sum(0..m) (\j. b j * r pow j) = &0) /\
?d. EVEN d /\
variation(0..n) a = variation(0..m) b +
multiplicity (\x. sum(0..n) (\i. a i * x pow i)) r + d`,
REPEAT GEN_TAC THEN
DISCH_THEN(MP_TAC o MATCH_MP VARIATION_POSITIVE_ROOT_MULTIPLE_FACTOR) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:num->real` THEN
DISCH_THEN(X_CHOOSE_THEN `k:num` MP_TAC) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:num` THEN
DISCH_TAC THEN
SUBGOAL_THEN `multiplicity (\x. sum(0..n) (\i. a i * x pow i)) r = k`
(fun th -> ASM_REWRITE_TAC[th]) THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC MULTIPLICITY_UNIQUE THEN
MAP_EVERY EXISTS_TAC [`b:num->real`; `m:num`] THEN ASM_REWRITE_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Hence the main theorem. *)
(* ------------------------------------------------------------------------- *)
let DESCARTES_RULE_OF_SIGNS = prove
(`!f a n. f = (\x. sum(0..n) (\i. a i * x pow i)) /\
(?i. i IN 0..n /\ ~(a i = &0))
==> ?d. EVEN d /\
variation(0..n) a =
nsum {r | &0 < r /\ f(r) = &0} (\r. multiplicity f r) + d`,
REPEAT GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
DISCH_THEN(fun th -> REWRITE_TAC[th]) THEN
MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`a:num->real`; `n:num`] THEN
MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
X_GEN_TAC `a:num->real` THEN ASM_CASES_TAC `(a:num->real) n = &0` THENL
[ASM_CASES_TAC `n = 0` THEN
ASM_REWRITE_TAC[NUMSEG_SING; IN_SING; UNWIND_THM2]
THENL [ASM_MESON_TAC[]; DISCH_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN ANTS_TAC THENL
[ASM_ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `a:num->real`)] THEN
ANTS_TAC THENL
[ASM_MESON_TAC[IN_NUMSEG; ARITH_RULE `i <= n ==> i <= n - 1 \/ i = n`];
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:num` THEN
ASM_SIMP_TAC[LE_0; LE_1; SUM_CLAUSES_RIGHT] THEN
REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_RID] THEN
DISCH_THEN(SUBST1_TAC o SYM o CONJUNCT2) THEN
MATCH_MP_TAC VARIATION_SUBSET THEN
REWRITE_TAC[SUBSET_NUMSEG; IN_DIFF; IN_NUMSEG] THEN
CONJ_TAC THENL [ASM_ARITH_TAC; X_GEN_TAC `i:num` THEN STRIP_TAC] THEN
SUBGOAL_THEN `i:num = n` (fun th -> ASM_REWRITE_TAC[th]) THEN
ASM_ARITH_TAC];
DISCH_THEN(K ALL_TAC)] THEN
ASM_CASES_TAC `{r | &0 < r /\ sum(0..n) (\i. a i * r pow i) = &0} = {}` THENL
[ASM_REWRITE_TAC[NSUM_CLAUSES; ADD_CLAUSES] THEN
ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[UNWIND_THM1] THEN
ONCE_REWRITE_TAC[GSYM NOT_ODD] THEN
DISCH_THEN(MP_TAC o MATCH_MP ODD_VARIATION_POSITIVE_ROOT) THEN
ASM SET_TAC[];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `r:real` THEN STRIP_TAC THEN
MP_TAC(ISPECL [`r:real`; `n:num`; `a:num->real`]
VARIATION_POSITIVE_ROOT_MULTIPLICITY_FACTOR) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`b:num->real`; `m:num`] THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `m:num`) THEN
ANTS_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC `b:num->real`) THEN ANTS_TAC THENL
[EXISTS_TAC `m:num` THEN ASM_REWRITE_TAC[IN_NUMSEG; LE_REFL; LE_0];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:num`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `d2:num`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
EXISTS_TAC `d1 + d2:num` THEN
CONJ_TAC THENL [ASM_REWRITE_TAC[EVEN_ADD]; ALL_TAC] THEN
MATCH_MP_TAC(ARITH_RULE
`x + y = z ==> (x + d1) + (y + d2):num = z + d1 + d2`) THEN
SUBGOAL_THEN
`{r | &0 < r /\ sum(0..n) (\i. a i * r pow i) = &0} =
r INSERT {r | &0 < r /\ sum(0..m) (\i. b i * r pow i) = &0}`
SUBST1_TAC THENL
[MATCH_MP_TAC(SET_RULE `x IN s /\ s DELETE x = t ==> s = x INSERT t`) THEN
CONJ_TAC THENL
[REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[];
ONCE_ASM_REWRITE_TAC[] THEN
REWRITE_TAC[REAL_ENTIRE; REAL_POW_EQ_0; REAL_SUB_0] THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_DELETE] THEN
X_GEN_TAC `s:real` THEN
FIRST_X_ASSUM(K ALL_TAC o SPEC_VAR) THEN
ASM_CASES_TAC `s:real = r` THEN ASM_REWRITE_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN
`FINITE {r | &0 < r /\ sum(0..m) (\i. b i * r pow i) = &0}`
MP_TAC THENL
[MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `{r | sum(0..m) (\i. b i * r pow i) = &0}` THEN
SIMP_TAC[SUBSET; IN_ELIM_THM; REAL_POLYFUN_FINITE_ROOTS] THEN
EXISTS_TAC `m:num` THEN ASM_REWRITE_TAC[IN_NUMSEG; LE_0; LE_REFL];
SIMP_TAC[NSUM_CLAUSES; IN_ELIM_THM] THEN DISCH_TAC] THEN
FIRST_X_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE I [GSYM FUN_EQ_THM]) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(ARITH_RULE `s1:num = s2 ==> s1 + m = m + s2`) THEN
MATCH_MP_TAC NSUM_EQ THEN
X_GEN_TAC `s:real` THEN REWRITE_TAC[IN_ELIM_THM] THEN STRIP_TAC THEN
FIRST_X_ASSUM(fun t -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [t]) THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC MULTIPLICITY_OTHER_ROOT THEN
REWRITE_TAC[MESON[] `(?i. P i /\ Q i) <=> ~(!i. P i ==> ~Q i)`] THEN
REPEAT STRIP_TAC THEN
UNDISCH_TAC `~(sum (0..m) (\j. b j * r pow j) = &0)` THEN ASM_SIMP_TAC[] THEN
REWRITE_TAC[REAL_MUL_LZERO; SUM_0]);;
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