Nuprl Lemma : small-reciprocal-real
∀x:{x:ℝ| r0 < x} . ∃k:ℕ+. ((r1/r(k)) < x)
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
so_apply: x[s],
rless: x < y,
sq_exists: ∃x:{A| B[x]},
uimplies: b supposing a,
nat_plus: ℕ+,
int-to-real: r(n),
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
rational-approx: (x within 1/n),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
rneq: x ≠ y,
guard: {T},
sq_stable: SqStable(P),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rge: x ≥ y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ
Lemmas referenced :
set_wf,
nat_plus_wf,
less_than_wf,
int-to-real_wf,
real_wf,
rlessw_wf,
rless_wf,
subtype_rel_self,
equal_wf,
set-value-type,
int-value-type,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
rational-approx-property2,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
rdiv_wf,
rless-int,
sq_stable__rless,
rsub_wf,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
nequal_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd-preserves-rless,
radd_wf,
rinv_wf2,
equal-wf-T-base,
rmul_wf,
rless_functionality,
radd_functionality,
rinv-as-rdiv,
req_weakening,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
mul_bounds_1b,
radd-int-fractions,
int-rdiv-req,
mul_nat_plus,
rless-int-fractions,
one-mul,
mul-commutes,
mul-swap,
mul-distributes,
mul-associates,
mul-distributes-right,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
multiply_functionality_wrt_le,
le_weakening,
less_than_functionality,
le_wf,
nat_plus_subtype_nat,
mul_bounds_1a
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
addEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
dependent_functionElimination,
dependent_set_memberEquality,
cutEval,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
intEquality,
unionElimination,
imageElimination,
productElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
because_Cache,
minusEquality,
inrFormation,
imageMemberEquality,
baseClosed,
multiplyEquality,
baseApply,
closedConclusion,
addLevel,
levelHypothesis
Latex:
\mforall{}x:\{x:\mBbbR{}| r0 < x\} . \mexists{}k:\mBbbN{}\msupplus{}. ((r1/r(k)) < x)
Date html generated:
2017_10_03-AM-08_51_03
Last ObjectModification:
2017_07_28-AM-07_34_22
Theory : reals
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