Nuprl Lemma : rational-lower-approx_wf
∀[x:ℕ+ ⟶ ℤ]. ∀[n:ℕ+]. ((below x within 1/n) ∈ ℝ)
Proof
Definitions occuring in Statement :
rational-lower-approx: (below x within 1/n),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rational-lower-approx: (below x within 1/n),
has-value: (a)↓,
uimplies: b supposing a,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
subtype_rel: A ⊆r B
Lemmas referenced :
value-type-has-value,
int-value-type,
subtract_wf,
nat_plus_wf,
mul_nat_plus,
less_than_wf,
int-rdiv_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
nequal_wf,
int-to-real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
baseApply,
closedConclusion,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality
Latex:
\mforall{}[x:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((below x within 1/n) \mmember{} \mBbbR{})
Date html generated:
2016_10_26-AM-09_11_15
Last ObjectModification:
2016_09_05-PM-06_50_10
Theory : reals
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