Nuprl Lemma : small-reciprocal-proof
∀e:ℚ. ∃m:ℕ+. (1/m) < e supposing 0 < e
Proof
Definitions occuring in Statement :
qless: r < s,
qdiv: (r/s),
rationals: ℚ,
nat_plus: ℕ+,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
subtract: n - m,
less_than': less_than'(a;b),
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
qless_witness,
int-subtype-rationals,
q-elim,
nat_plus_properties,
assert-qeq,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-base,
rationals_wf,
int_subtype_base,
exists_wf,
nat_plus_wf,
qless_wf,
qdiv_wf,
subtype_rel_set,
less_than_wf,
satisfiable-full-omega-tt,
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,
squash_wf,
true_wf,
iff_weakening_equal,
qmul_preserves_qless,
qless-int,
qmul_wf,
qmul_zero_qrng,
qmul-qdiv-cancel,
div_rem_sum,
nequal_wf,
rem_bounds_1,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
le_wf,
div_bounds_1,
decidable__lt,
false_wf,
not-lt-2,
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,
itermMultiply_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
equal_wf,
qmul_one_qrng,
qmul_ac_1_qrng,
qmul-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
applyEquality,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
rename,
dependent_functionElimination,
because_Cache,
productElimination,
setElimination,
addLevel,
impliesFunctionality,
independent_isectElimination,
baseClosed,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
lambdaEquality,
intEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
imageElimination,
imageMemberEquality,
universeEquality,
dependent_set_memberEquality,
unionElimination,
addEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}e:\mBbbQ{}. \mexists{}m:\mBbbN{}\msupplus{}. (1/m) < e supposing 0 < e
Date html generated:
2018_05_22-AM-00_03_17
Last ObjectModification:
2017_07_26-PM-06_51_30
Theory : rationals
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