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(* ------------------------------------------------------------------------- *)
(* From Multivariate/misc.ml *)
(* ------------------------------------------------------------------------- *)
prioritize_real();;
let REAL_POW_LBOUND = prove
(`!x n. &0 <= x ==> &1 + &n * x <= (&1 + x) pow n`,
GEN_TAC THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN DISCH_TAC THEN
INDUCT_TAC THEN
REWRITE_TAC[real_pow; REAL_MUL_LZERO; REAL_ADD_RID; REAL_LE_REFL] THEN
REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(&1 + x) * (&1 + &n * x)` THEN
ASM_SIMP_TAC[REAL_LE_LMUL; REAL_ARITH `&0 <= x ==> &0 <= &1 + x`] THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS; REAL_ARITH
`&1 + (n + &1) * x <= (&1 + x) * (&1 + n * x) <=> &0 <= n * x * x`]);;
let REAL_ARCH_POW = prove
(`!x y. &1 < x ==> ?n. y < x pow n`,
REPEAT STRIP_TAC THEN
MP_TAC(SPEC `x - &1` REAL_ARCH) THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN
DISCH_THEN(MP_TAC o SPEC `y:real`) THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LTE_TRANS THEN
EXISTS_TAC `&1 + &n * (x - &1)` THEN
ASM_SIMP_TAC[REAL_ARITH `x < y ==> x < &1 + y`] THEN
ASM_MESON_TAC[REAL_POW_LBOUND; REAL_SUB_ADD2; REAL_ARITH
`&1 < x ==> &0 <= x - &1`]);;
let ABS_CASES = thm `;
!x. x = &0 \/ &0 < abs(x)`;;
let LL = REAL_ARITH `&1 < k ==> &0 < k`;;
(* ------------------------------------------------------------------------- *)
(* Miz3 solutions to IMO problem from ICMS 2006. *)
(* ------------------------------------------------------------------------- *)
horizon := 0;;
let IMO_1 = thm `;
!k. &1 < k ==> &0 < k [LL] by REAL_ARITH;
now
let f g be real->real;
let x be real;
assume !x y. f (x + y) + f (x - y) = &2 * f x * g y [1];
assume ~(!x. f x = &0) [2];
assume !x. abs (f x) <= &1 [3];
now
let k be real;
assume sup (IMAGE (\x. abs (f x)) (:real)) = k [4];
~(IMAGE (\x. abs (f x)) (:real) = {}) /\ (?b. !x. abs (f x) <= b) [5]
by ASM SET_TAC[],-,3;
now
assume !x. abs (f x) <= k [6];
assume !b. (!x. abs (f x) <= b) ==> k <= b [7];
now
let y be real;
assume &1 < abs (g y) [8];
!x. abs (f x) <= k / abs (g y) [9]
by ASM_MESON_TAC[REAL_LE_RDIV_EQ; REAL_ABS_MUL; LL;
REAL_ARITH (parse_term
"u + v = &2 * z /\\ abs u <= k /\\ abs v <= k ==> abs z <= k")
],-,1,6;
~(k <= k / abs (g y))
by TIMED_TAC 2
(ASM_MESON_TAC[REAL_NOT_LE; REAL_LT_LDIV_EQ; REAL_LT_LMUL;
REAL_MUL_RID; LL; REAL_ARITH (parse_term
"~(z = &0) /\\ abs z <= k ==> &0 < k")
]),LL,2,6,8;
(!x. abs (f x) <= k / abs (g y)) /\ ~(k <= k / abs (g y))
by CONJ_TAC,-,9;
((!x. abs (f x) <= k / abs (g y)) ==> k <= k / abs (g y)) ==> F
by SIMP_TAC[NOT_IMP; NOT_FORALL_THM],-;
thus F by FIRST_X_ASSUM(MP_TAC o
SPEC (parse_term "k / abs(g(y:real))")),-,7;
end;
~(?y. &1 < abs (g y)) by STRIP_TAC,-;
thus !y. abs (g y) <= &1
by SIMP_TAC[GSYM REAL_NOT_LT; GSYM NOT_EXISTS_THM],-;
end;
(!x. abs (f x) <= k) /\ (!b. (!x. abs (f x) <= b) ==> k <= b)
==> (!y. abs (g y) <= &1) by STRIP_TAC,-;
(~(IMAGE (\x. abs (f x)) (:real) = {}) /\ (?b. !x. abs (f x) <= b)
==> (!x. abs (f x) <= k) /\ (!b. (!x. abs (f x) <= b) ==> k <= b))
==> (!y. abs (g y) <= &1) by ANTS_TAC,-,5;
(~(IMAGE (\x. abs (f x)) (:real) = {}) /\
(?b. !x. x IN IMAGE (\x. abs (f x)) (:real) ==> x <= b)
==> (!x. x IN IMAGE (\x. abs (f x)) (:real)
==> x <= sup (IMAGE (\x. abs (f x)) (:real))) /\
(!b. (!x. x IN IMAGE (\x. abs (f x)) (:real) ==> x <= b)
==> sup (IMAGE (\x. abs (f x)) (:real)) <= b))
==> (!y. abs (g y) <= &1)
by ASM_SIMP_TAC[FORALL_IN_IMAGE; EXISTS_IN_IMAGE; IN_UNIV],-,4;
thus !y. abs (g y) <= &1
by MP_TAC(SPEC (parse_term "IMAGE (\\x. abs(f(x))) (:real)") SUP),-;
end;
!y. abs (g y) <= &1
by ABBREV_TAC (parse_term "k = sup (IMAGE (\\x. abs(f(x))) (:real))"),-;
thus abs (g x) <= &1
by SPEC_TAC ((parse_term "x:real"),(parse_term "y:real")),-;
end;
thus !f g. (!x y. f(x + y) + f(x - y) = &2 * f(x) * g(y)) /\
~(!x. f(x) = &0) /\ (!x. abs(f(x)) <= &1)
==> !x. abs(g(x)) <= &1 by REPEAT STRIP_TAC,-`;;
horizon := 1;;
let IMO_2 = thm `;
let f g be real->real;
assume !x y. f (x + y) + f (x - y) = &2 * f x * g y [1];
assume ~(!x. f x = &0) [2];
assume !x. abs (f x) <= &1 [3];
thus !x. abs (g x) <= &1
proof set s = IMAGE (\x. abs (f x)) (:real);
~(s = {}) [4] by SET_TAC;
!b. (!y. y IN s ==> y <= b) <=> (!x. abs (f x) <= b) by IN_IMAGE,IN_UNIV;
set k = sup s;
(!x. abs (f x) <= k) /\ !b. (!x. abs (f x) <= b) ==> k <= b [5] by 3,4,SUP;
assume ~thesis;
consider y such that &1 < abs (g y) [6] by REAL_NOT_LT;
&0 < abs (g y) [7] by REAL_ARITH;
!x. abs (f x) <= k / abs (g y) [8]
proof let x be real;
abs (f (x + y)) <= k /\ abs (f (x - y)) <= k /\
f (x + y) + f (x - y) = &2 * f x * g y by 1,5;
abs (f x * g y) <= k by REAL_ARITH;
qed by 7,REAL_ABS_MUL,REAL_LE_RDIV_EQ;
consider x such that &0 < abs (f x) /\ abs (f x) <= k by 2,5,ABS_CASES;
&0 < k by REAL_ARITH;
k / abs (g y) < k by 6,7,REAL_LT_LMUL,REAL_MUL_RID,REAL_LT_LDIV_EQ;
qed by 5,8,REAL_NOT_LE`;;
let IMO_3 = thm `;
let f g be real->real;
assume !x y. f (x + y) + f (x - y) = &2 * f x * g y [1];
assume ~(!x. f x = &0) [2];
assume !x. abs (f x) <= &1 [3];
thus !x. abs (g x) <= &1
proof
now [4] let y be real;
!x. abs (f x * g y pow 0) <= &1 [5] by 3,real_pow,REAL_MUL_RID;
now let l be num;
assume !x. abs (f x * g y pow l) <= &1;
let x be real;
abs (f (x + y) * g y pow l) <= &1 /\
abs (f (x - y) * g y pow l) <= &1;
abs ((f (x + y) + f (x - y)) * g y pow l) <= &2 by REAL_ARITH;
abs ((&2 * f x * g y) * g y pow l) <= &2 by 1;
abs (f x * g y * g y pow l) <= &1 by REAL_ARITH;
thus abs (f x * g y pow SUC l) <= &1 by real_pow,REAL_MUL_RID;
end;
thus !l x. abs (f x * g y pow l) <= &1 by INDUCT_TAC,5;
end;
!x y. ~(x = &0) /\ &1 < abs(y) ==> ?n. &1 < abs(y pow n * x)
by SIMP_TAC,REAL_ABS_MUL,REAL_ABS_POW,GSYM REAL_LT_LDIV_EQ,
GSYM REAL_ABS_NZ,REAL_ARCH_POW;
qed by 2,4,REAL_NOT_LE,REAL_MUL_SYM`;;
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