Nuprl Lemma : fractional-part-rep
∀r:{r:ℚ| (0 ≤ r) ∧ r < 1} . ∃a,b:ℕ. ((0 ≤ a) ∧ a < b ∧ (r = (a/b) ∈ ℚ))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
qdiv: (r/s),
rationals: ℚ,
nat: ℕ,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
uiff: uiff(P;Q),
uimplies: b supposing a,
nat: ℕ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q
Lemmas referenced :
set_wf,
rationals_wf,
qle_wf,
int-subtype-rationals,
qless_wf,
squash_wf,
sq_stable__and,
sq_stable_from_decidable,
decidable__qle,
decidable__qless,
qless_witness,
q-elim,
nat_plus_properties,
assert-qeq,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-base,
int_subtype_base,
exists_wf,
nat_wf,
le_wf,
less_than_wf,
equal_wf,
qdiv_wf,
subtype_rel_set,
int_nzero-rational,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformle_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_le_lemma,
int_formula_prop_wf,
nequal_wf,
equal-wf-T-base,
decidable__le,
qmul_preserves_qle2,
qle-int,
qle_witness,
qmul_wf,
qmul_preserves_qless,
qless-int,
true_wf,
qmul_zero_qrng,
qmul-qdiv-cancel,
iff_weakening_equal,
qmul_one_qrng,
equal-wf-base-T,
qmul-preserves-eq,
intformnot_wf,
int_formula_prop_not_lemma,
int-equal-in-rationals,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
natural_numberEquality,
applyEquality,
hypothesisEquality,
because_Cache,
isect_memberEquality,
independent_functionElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
addLevel,
impliesFunctionality,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
intEquality,
dependent_set_memberEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
baseApply,
closedConclusion,
equalityTransitivity,
unionElimination,
isect_memberFormation,
universeEquality,
minusEquality
Latex:
\mforall{}r:\{r:\mBbbQ{}| (0 \mleq{} r) \mwedge{} r < 1\} . \mexists{}a,b:\mBbbN{}. ((0 \mleq{} a) \mwedge{} a < b \mwedge{} (r = (a/b)))
Date html generated:
2018_05_22-AM-00_32_29
Last ObjectModification:
2017_07_26-PM-06_59_11
Theory : rationals
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