### Nuprl Lemma : div_absval_bound

`∀[M:ℕ+]. ∀[z:ℤ]. ∀[n:ℕ].  |z ÷ M| ≤ n supposing |z| ≤ (n * M)`

Proof

Definitions occuring in Statement :  absval: `|i|` nat_plus: `ℕ+` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` divide: `n ÷ m` multiply: `n * m` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` implies: `P `` Q` guard: `{T}` nat: `ℕ` nat_plus: `ℕ+` ge: `i ≥ j ` subtype_rel: `A ⊆r B` le: `A ≤ B` and: `P ∧ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` prop: `ℙ` uiff: `uiff(P;Q)` less_than': `less_than'(a;b)` absval: `|i|` nequal: `a ≠ b ∈ T ` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` int_lower: `{...i}` gt: `i > j`
Lemmas referenced :  decidable__equal_int subtype_base_sq int_subtype_base nat_properties nat_plus_properties decidable__le absval_wf full-omega-unsat intformand_wf intformnot_wf intformle_wf itermVar_wf itermConstant_wf itermMultiply_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_constant_lemma int_term_value_mul_lemma int_formula_prop_wf absval_le_zero zero-div-rem nat_plus_inc_int_nzero istype-false intformeq_wf int_formula_prop_eq_lemma mul_preserves_le nat_plus_subtype_nat le_witness_for_triv istype-le istype-nat nat_plus_wf mul_cancel_in_le intformless_wf int_formula_prop_less_lemma equal_wf absval_mul iff_weakening_equal absval_pos div_rem_sum div_rem_sum2 rem_bounds_1 absval_unfold subtract_wf lt_int_wf eqtt_to_assert assert_of_lt_int istype-top itermSubtract_wf int_term_value_subtract_lemma eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf less_than_wf istype-less_than itermMinus_wf int_term_value_minus_lemma rem_bounds_2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution dependent_functionElimination thin because_Cache natural_numberEquality hypothesis unionElimination instantiate isectElimination cumulativity intEquality independent_isectElimination independent_functionElimination equalityTransitivity equalitySymmetry hypothesisEquality setElimination rename applyEquality sqequalRule productElimination approximateComputation Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :universeIsType,  Error :lambdaFormation_alt,  minusEquality Error :inhabitedIsType,  multiplyEquality Error :isectIsTypeImplies,  divideEquality Error :equalityIstype,  baseClosed sqequalBase imageElimination imageMemberEquality Error :dependent_set_memberEquality_alt,  remainderEquality equalityElimination lessCases axiomSqEquality closedConclusion promote_hyp

Latex:
\mforall{}[M:\mBbbN{}\msupplus{}].  \mforall{}[z:\mBbbZ{}].  \mforall{}[n:\mBbbN{}].    |z  \mdiv{}  M|  \mleq{}  n  supposing  |z|  \mleq{}  (n  *  M)

Date html generated: 2019_06_20-PM-01_18_54
Last ObjectModification: 2019_02_12-PM-02_04_40

Theory : int_2

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