Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 1,655 Bytes
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prioritize_real();;
let (TRY_RULE:(thm->thm) -> (thm->thm)) =
fun rl t -> try (rl t) with _ -> t;;
let REAL_MUL_RTIMES =
prove ((`!x a b.
(((~(x=(&0))==>(a*x = b*x)) /\ ~(x=(&0))) ==> (a = b))`),
MESON_TAC[REAL_EQ_MUL_RCANCEL]);;
let GEOMETRIC_SUM = prove(
`!m n x.(~(x=(&1)) ==>
(sum(m,n) (\k.(x pow k)) = ((x pow m) - (x pow (m+n)))/((&1)-x)))`,
let tac1 =
GEN_TAC
THEN INDUCT_TAC
THEN GEN_TAC
THEN DISCH_TAC
THEN (REWRITE_TAC
[sum_DEF;real_pow;ADD_CLAUSES;real_div;REAL_SUB_RDISTRIB;
REAL_SUB_REFL]) in
let tac2 =
(RULE_ASSUM_TAC (TRY_RULE (SPEC (`x:real`))))
THEN (UNDISCH_EL_TAC 1)
THEN (UNDISCH_EL_TAC 0)
THEN (TAUT_TAC (`(A==>(B==>C)) ==> (A ==> ((A==>B) ==>C))`))
THEN (REPEAT DISCH_TAC)
THEN (ASM_REWRITE_TAC[real_div])
THEN (ABBREV_TAC (`a:real = x pow m`))
THEN (ABBREV_TAC (`b:real = x pow (m+n)`)) in
let tac3 =
(MATCH_MP_TAC (SPEC (`&1 - x`) REAL_MUL_RTIMES))
THEN CONJ_TAC
THENL [ALL_TAC; (UNDISCH_TAC (`~(x = (&1))`))
THEN (ACCEPT_TAC (REAL_ARITH (`~(x=(&1)) ==> ~((&1 - x = (&0)))`)))]
THEN (REWRITE_TAC
[GSYM REAL_MUL_ASSOC;REAL_ADD_RDISTRIB;REAL_SUB_RDISTRIB])
THEN (SIMP_TAC[REAL_MUL_LINV])
THEN DISCH_TAC
THEN (REWRITE_TAC
[REAL_SUB_LDISTRIB;REAL_MUL_LID;REAL_MUL_RID;REAL_MUL_ASSOC])
THEN (ACCEPT_TAC (REAL_ARITH (`a - b + b - b*x = a - x*b`))) in
(tac1 THEN tac2 THEN tac3));;
pop_priority();;
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