Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 1,978 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 |
let SURJECTIVE_IFF_RIGHT_INVERSE = prove
(`(!y. ?x. g x = y) <=> (?f. g o f = I)`,
REWRITE_TAC[FUN_EQ_THM; o_DEF; I_DEF] THEN MESON_TAC[]);;
let INJECTIVE_IFF_LEFT_INVERSE = prove
(`(!x y. f x = f y ==> x = y) <=> (?g. g o f = I)`,
let lemma = MESON[]
`(!x x'. f x = f x' ==> x = x') <=> (!y:B. ?u:A. !x. f x = y ==> u = x)` in
REWRITE_TAC[lemma; FUN_EQ_THM; o_DEF; I_DEF] THEN MESON_TAC[]);;
let cantor = new_definition
`cantor(x,y) = ((x + y) EXP 2 + 3 * x + y) DIV 2`;;
(**** Needs external SDP solver
needs "Examples/sos.ml";;
let CANTOR_LEMMA = prove
(`cantor(x,y) = cantor(x',y') ==> x + y = x' + y'`,
REWRITE_TAC[cantor] THEN CONV_TAC SOS_RULE);;
****)
let CANTOR_LEMMA_LEMMA = prove
(`x + y < x' + y' ==> cantor(x,y) < cantor(x',y')`,
REWRITE_TAC[ARITH_RULE `x + y < z <=> x + y + 1 <= z`] THEN DISCH_TAC THEN
REWRITE_TAC[cantor; ARITH_RULE `3 * x + y = (x + y) + 2 * x`] THEN
MATCH_MP_TAC(ARITH_RULE `x + 2 <= y ==> x DIV 2 < y DIV 2`) THEN
MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `(x + y + 1) EXP 2 + (x + y + 1)` THEN
CONJ_TAC THENL [ARITH_TAC; ALL_TAC] THEN
MATCH_MP_TAC(ARITH_RULE `a:num <= b /\ c <= d ==> a + c <= b + d + e`) THEN
ASM_SIMP_TAC[EXP_2; LE_MULT2]);;
let CANTOR_LEMMA = prove
(`cantor(x,y) = cantor(x',y') ==> x + y = x' + y'`,
MESON_TAC[LT_CASES; LT_REFL; CANTOR_LEMMA_LEMMA]);;
let CANTOR_INJ = prove
(`!w z. cantor w = cantor z ==> w = z`,
REWRITE_TAC[FORALL_PAIR_THM; PAIR_EQ] THEN REPEAT GEN_TAC THEN
DISCH_THEN(fun th -> MP_TAC th THEN ASSUME_TAC(MATCH_MP CANTOR_LEMMA th)) THEN
ASM_REWRITE_TAC[cantor; ARITH_RULE `3 * x + y = (x + y) + 2 * x`] THEN
REWRITE_TAC[ARITH_RULE `(a + b + 2 * x) DIV 2 = (a + b) DIV 2 + x`] THEN
POP_ASSUM MP_TAC THEN ARITH_TAC);;
let CANTOR_THM = prove
(`~(?f:(A->bool)->A. (!x y. f(x) = f(y) ==> x = y))`,
REWRITE_TAC[INJECTIVE_IFF_LEFT_INVERSE; FUN_EQ_THM; I_DEF; o_DEF] THEN
STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `\x:A. ~(g x x)`) THEN
MESON_TAC[]);;
|