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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`;; | |