Nuprl Lemma : small_reciprocal_wf
∀e:{q:ℚ| 0 < q} . (small_reciprocal(e) ∈ {n:ℕ+| (1/n) < e} )
Proof
Definitions occuring in Statement :
small_reciprocal: small_reciprocal(e),
qless: r < s,
qdiv: (r/s),
rationals: ℚ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
small_reciprocal: small_reciprocal(e),
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
uiff: uiff(P;Q),
guard: {T}
Lemmas referenced :
small-reciprocal,
subtype_rel_self,
rationals_wf,
qless_wf,
exists_wf,
nat_plus_wf,
qdiv_wf,
subtype_rel_set,
less_than_wf,
int-subtype-rationals,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-T-base,
int-equal-in-rationals,
not_wf,
isect_wf,
equal_wf,
set_wf,
pi1_wf_top,
uimplies_subtype,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
isectEquality,
natural_numberEquality,
because_Cache,
hypothesisEquality,
lambdaEquality,
intEquality,
independent_isectElimination,
setElimination,
rename,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
baseClosed,
addLevel,
impliesFunctionality,
productElimination,
setEquality,
productEquality,
independent_pairEquality,
dependent_set_memberEquality
Latex:
\mforall{}e:\{q:\mBbbQ{}| 0 < q\} . (small\_reciprocal(e) \mmember{} \{n:\mBbbN{}\msupplus{}| (1/n) < e\} )
Date html generated:
2018_05_22-AM-00_07_56
Last ObjectModification:
2018_05_19-PM-04_05_33
Theory : rationals
Home
Index