Nuprl Lemma : positive-lower-bound_wf
∀[x:{x:ℝ| r0 < x} ]. (positive-lower-bound(x) ∈ {k:ℕ+| (r1/r(k)) < x} )
Proof
Definitions occuring in Statement :
positive-lower-bound: positive-lower-bound(x),
rdiv: (x/y),
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
so_apply: x[s],
positive-lower-bound: positive-lower-bound(x),
pi1: fst(t),
small-reciprocal-real-ext,
has-value: (a)↓,
real: ℝ
Lemmas referenced :
small-reciprocal-real-ext,
all_wf,
exists_wf,
rless_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_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_formula_prop_wf,
int-to-real_wf,
set_wf,
real_wf,
value-type-has-value,
set-value-type,
nat_plus_wf,
less_than_wf,
int-value-type,
rlessw_wf,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
hypothesisEquality,
isectElimination,
because_Cache,
lambdaFormation,
setElimination,
rename,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
addEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 < x\} ]. (positive-lower-bound(x) \mmember{} \{k:\mBbbN{}\msupplus{}| (r1/r(k)) < x\} )
Date html generated:
2016_10_26-AM-09_15_00
Last ObjectModification:
2016_10_12-PM-03_22_09
Theory : reals
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