### 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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