### Nuprl Lemma : absval_div_decreases

`∀[n:{2...}]. ∀[i:ℤ-o].  |i ÷ n| < |i|`

Proof

Definitions occuring in Statement :  absval: `|i|` int_upper: `{i...}` int_nzero: `ℤ-o` less_than: `a < b` uall: `∀[x:A]. B[x]` divide: `n ÷ m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat_plus: `ℕ+` int_upper: `{i...}` int_nzero: `ℤ-o` le: `A ≤ B` and: `P ∧ Q` nequal: `a ≠ b ∈ T ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` prop: `ℙ` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` top: `Top` less_than': `less_than'(a;b)` true: `True` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` nat: `ℕ` less_than: `a < b` squash: `↓T` ge: `i ≥ j ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b`
Lemmas referenced :  absval_div_nat decidable__lt false_wf not-lt-2 not-equal-2 add_functionality_wrt_le add-commutes zero-add le-add-cancel less_than_wf div_rem_sum absval_wf subtype_rel_sets le_wf nequal_wf int_upper_properties int_nzero_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf equal-wf-base int_subtype_base rem_bounds_1 int_nzero_wf member-less_than nat_wf int_upper_wf intformless_wf int_formula_prop_less_lemma decidable__le add-is-int-iff multiply-is-int-iff intformnot_wf int_formula_prop_not_lemma mul_preserves_le int_upper_subtype_nat itermMultiply_wf itermAdd_wf int_term_value_mul_lemma int_term_value_add_lemma decidable__equal_int nat_properties absval_unfold lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf equal-wf-T-base eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot itermMinus_wf int_term_value_minus_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin dependent_set_memberEquality setElimination rename hypothesisEquality hypothesis productElimination dependent_functionElimination natural_numberEquality unionElimination independent_pairFormation lambdaFormation voidElimination independent_functionElimination independent_isectElimination applyEquality lambdaEquality isect_memberEquality voidEquality intEquality because_Cache setEquality applyLambdaEquality dependent_pairFormation int_eqEquality computeAll baseClosed divideEquality equalityTransitivity equalitySymmetry imageElimination pointwiseFunctionality promote_hyp baseApply closedConclusion multiplyEquality minusEquality equalityElimination lessCases sqequalAxiom imageMemberEquality instantiate cumulativity

Latex:
\mforall{}[n:\{2...\}].  \mforall{}[i:\mBbbZ{}\msupminus{}\msupzero{}].    |i  \mdiv{}  n|  <  |i|

Date html generated: 2017_04_14-AM-09_22_50
Last ObjectModification: 2017_02_27-PM-03_58_40

Theory : int_2

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