Nuprl Lemma : rdiv-factorial-lemma2
∀x:ℝ. ∀b:ℕ. (((x * x) ≤ r(b * b)) ⇒ (∀n:ℕ. (((b ÷ 2) ≤ n) ⇒ ((x * x) ≤ (r((2 * (n + 1))!)/r((2 * n)!))))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
divide: n ÷ m,
multiply: n * m,
add: n + m,
natural_number: $n,
fact: (n)!
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
le: A ≤ B
Lemmas referenced :
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
intformeq_wf,
intformless_wf,
mul_preserves_le,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq-int,
req_weakening,
rleq_functionality,
fact-non-zero,
rneq-int,
nat_plus_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
fact_wf,
rdiv_wf,
real_wf,
int-to-real_wf,
rmul_wf,
rleq_wf,
nat_wf,
le_wf,
rdiv-factorial-lemma1,
less_than_wf,
rem_bounds_1,
nequal_wf,
true_wf,
equal_wf,
int_subtype_base,
subtype_base_sq,
div_rem_sum
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
because_Cache,
sqequalRule,
independent_pairFormation,
introduction,
imageMemberEquality,
baseClosed,
divideEquality,
multiplyEquality,
productElimination,
addEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
applyEquality,
imageElimination
Latex:
\mforall{}x:\mBbbR{}. \mforall{}b:\mBbbN{}.
(((x * x) \mleq{} r(b * b)) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. (((b \mdiv{} 2) \mleq{} n) {}\mRightarrow{} ((x * x) \mleq{} (r((2 * (n + 1))!)/r((2 * n)!))))))
Date html generated:
2016_05_18-AM-08_04_46
Last ObjectModification:
2016_01_17-AM-02_19_57
Theory : reals
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