Nuprl Lemma : rational-truncate
∀q:ℚ. ∀e:{e:ℚ| 0 < e} . (∃q':ℚ [(|q - q'| ≤ e)])
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
rationals: ℚ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
squash: ↓T,
true: True,
pi1: fst(t),
cand: A c∧ B,
pi2: snd(t),
nat_plus: ℕ+,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
mk-rational: mk-rational(a;b),
sq_stable: SqStable(P),
rev_implies: P ⇐ Q,
qsub: r - s
Lemmas referenced :
set_wf,
rationals_wf,
qless_wf,
int-subtype-rationals,
q_less_wf,
bool_wf,
eqtt_to_assert,
assert-q_less-eq,
iff_weakening_equal,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_wf,
squash_wf,
true_wf,
rational-truncate1,
all_wf,
sq_exists_wf,
nat_plus_wf,
qle_wf,
qabs_wf,
qsub_wf,
qdiv_wf,
subtype_rel_set,
less_than_wf,
int_nzero-rational,
subtype_rel_sets,
nequal_wf,
nat_plus_properties,
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-base,
int_subtype_base,
qmul_wf,
subtype_rel_transitivity,
int_nzero_wf,
mk-rational_wf,
integer-part_wf,
sq_stable_from_decidable,
decidable__qle,
qdiv-int-elim,
fractional-part_wf,
integer-fractional-parts,
mon_assoc_q,
qadd_ac_1_q,
qadd_wf,
qinverse_q,
mon_ident_q,
qless_complement_qorder,
qabs-of-nonneg,
qless_transitivity_2_qorder,
qle_weakening_lt_qorder
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_set_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
hypothesisEquality,
setElimination,
rename,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
voidElimination,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
setEquality,
productEquality,
intEquality,
applyLambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
functionExtensionality,
dependent_set_memberEquality,
independent_pairEquality,
minusEquality
Latex:
\mforall{}q:\mBbbQ{}. \mforall{}e:\{e:\mBbbQ{}| 0 < e\} . (\mexists{}q':\mBbbQ{} [(|q - q'| \mleq{} e)])
Date html generated:
2018_05_22-AM-00_31_22
Last ObjectModification:
2017_07_26-PM-06_58_59
Theory : rationals
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