Nuprl Lemma : rationals-dense-in-interval
∀I:Interval. dense-in-interval(I;λr.∃z:ℤ. ∃d:ℕ+. (r = (r(z)/r(d))))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
interval: Interval,
rdiv: (x/y),
req: x = y,
int-to-real: r(n),
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
lambda: λx.A[x],
int: ℤ
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
dense-in-interval: dense-in-interval(I;X),
all: ∀x:A. B[x],
cand: A c∧ B,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
sq_exists: ∃x:{A| B[x]},
rless: x < y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat_plus: ℕ+,
and: P ∧ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
interval_wf,
i-member_wf,
real_wf,
set_wf,
rless_wf,
nat_plus_wf,
exists_wf,
req_wf,
equal_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
rneq-int,
req_weakening,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
rationals-dense
Rules used in proof :
lambdaEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
unionElimination,
natural_numberEquality,
independent_functionElimination,
inrFormation,
independent_isectElimination,
because_Cache,
dependent_pairFormation,
productElimination,
dependent_set_memberEquality,
dependent_functionElimination
Latex:
\mforall{}I:Interval. dense-in-interval(I;\mlambda{}r.\mexists{}z:\mBbbZ{}. \mexists{}d:\mBbbN{}\msupplus{}. (r = (r(z)/r(d))))
Date html generated:
2017_10_03-AM-10_20_06
Last ObjectModification:
2017_07_31-PM-00_07_13
Theory : reals
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