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Text Generation
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language-modeling
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English
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File size: 4,783 Bytes
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needs "Library/transc.ml";;
needs "Examples/sos.ml";;
prioritize_real();;
horizon := 1;;
let rational = new_definition
`rational(r) = ?p q. ~(q = 0) /\ abs(r) = &p/ &q`;;
(* ======== Mizar-style version ============================================ *)
let NSQRT_2_1 = thm `;
!p q. p*p = 2*q*q ==> q = 0
proof
exec MATCH_MP_TAC num_WF;
let p be num;
assume !p'. p' < p ==> !q. p'*p' = 2*q*q ==> q = 0 [1];
let q be num;
assume p*p = 2*q*q [2];
EVEN (p*p) by EVEN_DOUBLE;
EVEN p by EVEN_MULT;
consider p' such that
p = 2*p' [3] by EVEN_EXISTS;
q*q = 2*p'*p' [4] by 2,NUM_RING;
EVEN (q*q) by EVEN_DOUBLE;
EVEN q by EVEN_MULT;
consider q' such that
q = 2*q' [5] by EVEN_EXISTS;
p'*p' = 2*q'*q' [6] by 4,NUM_RING;
assume ~(q = 0) [7];
~(p = 0) by 2,NUM_RING;
p > 0 by ARITH_TAC;
p' < p by 3,ARITH_TAC;
q' = 0 by 1,6;
qed by 5,7,MULT_EQ_0`;;
let SQRT_2_IRRATIONAL_1 = thm `;
~rational(sqrt(&2))
proof
assume rational(sqrt(&2));
set x = abs(sqrt(&2));
consider p q such that
~(q = 0) /\ x = &p/ &q [7] by rational;
~(&q = &0) by REAL_INJ;
x* &q = &p [8] by 7,REAL_DIV_RMUL;
&0 <= &2 by REAL_ARITH_TAC;
sqrt(&2) pow 2 = &2 by SQRT_POW2;
x pow 2 = &2 by REAL_ARITH_TAC;
&p* &p = &2* &q* &q by 8,REAL_RING;
p*p = 2*q*q by 8,REAL_INJ,REAL_OF_NUM_MUL;
qed by 7,NSQRT_2_1`;;
(* ======== "automatically" converted from John's version ================== *)
let NSQRT_2_2 = thm `;
now
now
let p q be num;
assume !m q. m < p ==> m * m = 2 * q * q ==> q = 0 [1];
assume p * p = 2 * q * q [2];
now
let m be num;
assume !m' q. m' < 2 * m ==> m' * m' = 2 * q * q ==> q = 0 [3];
assume (2 * m) * 2 * m = 2 * q * q [4];
(2 * m) * 2 * m = 2 * q * q
==> (q < 2 * m ==> q * q = 2 * m * m ==> m = 0)
==> q = 0
by TIMED_TAC 2 (CONV_TAC SOS_RULE);
(q < 2 * m ==> q * q = 2 * m * m ==> m = 0) ==> q = 0
by POP_ASSUM MP_TAC,4 from -;
thus q = 0 by FIRST_X_ASSUM
(MP_TAC o SPECL [parse_term "q:num"; parse_term "m:num"]),3,4;
end;
(?m. p = 2 * m) ==> q = 0
by DISCH_THEN(X_CHOOSE_THEN (parse_term "m:num") SUBST_ALL_TAC),1,2;
EVEN p ==> q = 0 by REWRITE_TAC[EVEN_EXISTS],1,2;
(EVEN (p * p) <=> EVEN (2 * q * q)) ==> q = 0
by REWRITE_TAC[EVEN_MULT; ARITH],1,2;
thus q = 0 by FIRST_ASSUM(MP_TAC o AP_TERM (parse_term "EVEN")),1,2;
end;
!p q.
(!m q. m < p ==> m * m = 2 * q * q ==> q = 0)
==> p * p = 2 * q * q
==> q = 0 by REPEAT STRIP_TAC;
!p. (!m. m < p ==> (!q. m * m = 2 * q * q ==> q = 0))
==> (!q. p * p = 2 * q * q ==> q = 0)
by REWRITE_TAC[RIGHT_IMP_FORALL_THM];
thus !p q. p * p = 2 * q * q ==> q = 0 by MATCH_MP_TAC num_WF;
end`;;
let SQRT_2_IRRATIONAL_2 = thm `;
now
now
let p q be num;
now
assume ~(q = 0) [1];
~(&2 * &q * &q = &p * &p)
by ASM_MESON_TAC[NSQRT_2_2; REAL_OF_NUM_EQ; REAL_OF_NUM_MUL];
~((\x. x pow 2) (sqrt (&2)) = (\x. x pow 2) (&p / &q))
by ASM_SIMP_TAC[SQRT_POW_2; REAL_POS; REAL_POW_DIV; REAL_POW_2;
REAL_LT_SQUARE; REAL_OF_NUM_EQ; REAL_EQ_RDIV_EQ],1;
thus ~(sqrt (&2) = &p / &q)
by DISCH_THEN(MP_TAC o AP_TERM (parse_term "\\x. x pow 2")),1;
end;
thus ~(~(q = 0) /\ sqrt (&2) = &p / &q)
by DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC);
end;
!p q. ~(~(q = 0) /\ sqrt (&2) = &p / &q) by REPEAT GEN_TAC;
thus ~rational (sqrt (&2))
by SIMP_TAC[rational; real_abs; SQRT_POS_LE; REAL_POS; NOT_EXISTS_THM];
end`;;
(* ======== humanized version of John's version ============================ *)
let NSQRT_2_3 = thm `;
!p q. p*p = 2*q*q ==> q = 0
proof
set P = \p. !q. p*p = 2*q*q ==> q = 0;
now
let p be num;
assume !m. m < p ==> P m [1];
let q be num;
assume p*p = 2*q*q [2];
EVEN(2*q*q) by REWRITE_TAC,EVEN_MULT,ARITH;
EVEN p by 2,EVEN_MULT;
consider m such that p = 2*m [3] by EVEN_EXISTS;
(2*m)*2*m = 2*q*q /\ (q < 2*m /\ q*q = 2*m*m ==> m = 0) ==> q = 0
from TIMED_TAC 2 (CONV_TAC SOS_RULE);
thus q = 0 by 1,2,3;
end;
qed by MATCH_MP_TAC num_WF`;;
let SQRT_2_IRRATIONAL_3 = thm `;
~rational(sqrt(&2))
proof
assume rational(sqrt(&2));
consider p q such that ~(q = 0) /\ sqrt(&2) = &p/ &q [1]
by rational,real_abs,SQRT_POS_LE,REAL_POS;
(&p* &p)/(&q* &q) = &2 [2] by SQRT_POW_2,REAL_POS,REAL_POW_DIV,REAL_POW_2;
&0 < &q* &q by 1,REAL_LT_SQUARE,REAL_OF_NUM_EQ;
&2*(&q* &q) = (&p* &p) by 2,REAL_EQ_RDIV_EQ;
qed by 1,NSQRT_2_3,REAL_OF_NUM_EQ,REAL_OF_NUM_MUL`;;
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