Nuprl Lemma : qround-eq
∀[r:ℚ]. ∀[k:ℕ+]. (qround(r;k) = (rounded-numerator(r;2 * k)/2 * k) ∈ ℚ)
Proof
Definitions occuring in Statement :
qdiv: (r/s),
qround: qround(r;k),
rounded-numerator: rounded-numerator(r;k),
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
multiply: n * m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
qround: qround(r;k),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
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,
uiff: uiff(P;Q)
Lemmas referenced :
mk-rational-qdiv,
rounded-numerator_wf,
mul_nat_plus,
less_than_wf,
qdiv_wf,
int-subtype-rationals,
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-T-base,
int-equal-in-rationals,
rationals_wf,
not_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
hypothesis,
multiplyEquality,
setElimination,
rename,
because_Cache,
applyEquality,
independent_isectElimination,
intEquality,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
addLevel,
impliesFunctionality,
productElimination,
axiomEquality
Latex:
\mforall{}[r:\mBbbQ{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. (qround(r;k) = (rounded-numerator(r;2 * k)/2 * k))
Date html generated:
2018_05_21-PM-11_47_29
Last ObjectModification:
2017_07_26-PM-06_43_09
Theory : rationals
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