Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 1,845 Bytes
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g `2 <= n /\ n <= 2 ==> f(2,2) + n < f(n,n) + 7`;;
e DISCH_TAC;;
b();;
e(CONV_TAC(REWRITE_CONV[LE_ANTISYM]));;
e(SIMP_TAC[]);;
e(ONCE_REWRITE_TAC[EQ_SYM_EQ]);;
e DISCH_TAC;;
e(ASM_REWRITE_TAC[]);;
e(CONV_TAC ARITH_RULE);;
let trivial = top_thm();;
g `2 <= n /\ n <= 2 ==> f(2,2) + n < f(n,n) + 7`;;
e(CONV_TAC(REWRITE_CONV[LE_ANTISYM]));;
e(SIMP_TAC[]);;
e(ONCE_REWRITE_TAC[EQ_SYM_EQ]);;
e DISCH_TAC;;
e(ASM_REWRITE_TAC[]);;
e(CONV_TAC ARITH_RULE);;
let trivial = top_thm();;
g `2 <= n /\ n <= 2 ==> f(2,2) + n < f(n,n) + 7`;;
e(CONV_TAC(REWRITE_CONV[LE_ANTISYM]) THEN
SIMP_TAC[] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC ARITH_RULE);;
let trivial = top_thm();;
let trivial = prove
(`2 <= n /\ n <= 2 ==> f(2,2) + n < f(n,n) + 7`,
CONV_TAC(REWRITE_CONV[LE_ANTISYM]) THEN
SIMP_TAC[] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC ARITH_RULE);;
let trivial = prove
(`!x y:real. &0 < x * y ==> (&0 < x <=> &0 < y)`,
REPEAT GEN_TAC THEN MP_TAC(SPECL [`--x`; `y:real`] REAL_LE_MUL) THEN
MP_TAC(SPECL [`x:real`; `--y`] REAL_LE_MUL) THEN REAL_ARITH_TAC);;
let trivial = prove
(`!x y:real. &0 < x * y ==> (&0 < x <=> &0 < y)`,
MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THEN REPEAT GEN_TAC THEN
MP_TAC(SPECL [`--x`; `y:real`] REAL_LE_MUL) THEN REAL_ARITH_TAC);;
let SUM_OF_NUMBERS = prove
(`!n. nsum(1..n) (\i. i) = (n * (n + 1)) DIV 2`,
INDUCT_TAC THEN ASM_REWRITE_TAC[NSUM_CLAUSES_NUMSEG] THEN ARITH_TAC);;
let SUM_OF_SQUARES = prove
(`!n. nsum(1..n) (\i. i * i) = (n * (n + 1) * (2 * n + 1)) DIV 6`,
INDUCT_TAC THEN ASM_REWRITE_TAC[NSUM_CLAUSES_NUMSEG] THEN ARITH_TAC);;
let SUM_OF_CUBES = prove
(`!n. nsum(1..n) (\i. i*i*i) = (n * n * (n + 1) * (n + 1)) DIV 4`,
INDUCT_TAC THEN ASM_REWRITE_TAC[NSUM_CLAUSES_NUMSEG] THEN ARITH_TAC);;
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