Nuprl Lemma : rdiv-factorial-lemma3
∀x:ℝ. ∀b:ℕ.
(((x * x) ≤ r(b * b)) ⇒ (∀n:ℕ. (((b ÷ 2) ≤ n) ⇒ ((x * x) ≤ (r(((2 * (n + 1)) + 1)!)/r(((2 * n) + 1)!))))))
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],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
le: A ≤ B,
uiff: uiff(P;Q)
Lemmas referenced :
rdiv-factorial-lemma2,
le_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nat_wf,
rleq_wf,
rmul_wf,
int-to-real_wf,
real_wf,
rdiv_wf,
fact_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nat_plus_wf,
rneq-int,
fact-non-zero,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
fact_unroll_1,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
add-subtract-cancel,
le_weakening2,
decidable__lt,
nat_plus_properties,
intformless_wf,
int_formula_prop_less_lemma,
mul_preserves_le,
multiply-is-int-iff,
false_wf,
rleq-int-fractions
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
divideEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
baseClosed,
multiplyEquality,
dependent_set_memberEquality,
addEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
productElimination,
Error :applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
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)) + 1)!)/r(((2 * n) + 1)!))))))
Date html generated:
2016_10_26-AM-09_22_24
Last ObjectModification:
2016_08_25-PM-10_20_49
Theory : reals
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