Nuprl Lemma : rounding-div-property

`∀[a:ℤ]. ∀[n:ℕ+].  ((2 * |(n * [a ÷ n]) - a|) ≤ n)`

Proof

Definitions occuring in Statement :  rounding-div: `[b ÷ m]` absval: `|i|` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` le: `A ≤ B` multiply: `n * m` subtract: `n - m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` rounding-div: `[b ÷ m]` has-value: `(a)↓` uimplies: `b supposing a` nat_plus: `ℕ+` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` implies: `P `` Q` sq_type: `SQType(T)` guard: `{T}` nat: `ℕ` uiff: `uiff(P;Q)` and: `P ∧ Q` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` false: `False` prop: `ℙ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` subtract: `n - m`
Lemmas referenced :  value-type-has-value nat_plus_wf set-value-type less_than_wf int-value-type divrem-sq nat_plus_inc_int_nzero div_rem_sum divide_wfa rem_bounds_absval remainder_wfa subtype_base_sq int_subtype_base nat_wf set_subtype_base le_wf absval-non-neg absval_pos nat_plus_subtype_nat istype-le absval_strict_ubound istype-less_than absval_wf le_witness_for_triv istype-int decidable__lt istype-top istype-void absval_unfold lt_int_wf eqtt_to_assert assert_of_lt_int nat_plus_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermMultiply_wf itermConstant_wf itermVar_wf intformless_wf itermMinus_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_term_value_minus_lemma int_formula_prop_wf eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf itermAdd_wf int_term_value_add_lemma minus-add minus-one-mul mul-commutes add-swap add-associates add-commutes add-mul-special zero-mul add-zero mul-distributes one-mul
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule callbyvalueReduce extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis independent_isectElimination intEquality Error :lambdaEquality_alt,  natural_numberEquality hypothesisEquality Error :inhabitedIsType,  because_Cache applyEquality Error :lambdaFormation_alt,  dependent_functionElimination multiplyEquality instantiate cumulativity equalityTransitivity equalitySymmetry independent_functionElimination Error :dependent_set_memberEquality_alt,  productElimination Error :equalityIstype,  baseApply closedConclusion baseClosed sqequalBase setElimination rename Error :universeIsType,  Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  unionElimination lessCases axiomSqEquality independent_pairFormation voidElimination imageMemberEquality imageElimination minusEquality equalityElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality promote_hyp addEquality

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[n:\mBbbN{}\msupplus{}].    ((2  *  |(n  *  [a  \mdiv{}  n])  -  a|)  \mleq{}  n)

Date html generated: 2019_06_20-PM-01_13_31
Last ObjectModification: 2019_03_06-AM-11_05_59

Theory : int_2

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