### Nuprl Lemma : rem_bounds_4

`∀[a:ℕ]. ∀[n:{...-1}].  ((0 ≤ (a rem n)) ∧ a rem n < -n)`

Proof

Definitions occuring in Statement :  int_lower: `{...i}` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` le: `A ≤ B` and: `P ∧ Q` remainder: `n rem m` minus: `-n` natural_number: `\$n`
Definitions unfolded in proof :  true: `True` less_than': `less_than'(a;b)` top: `Top` subtype_rel: `A ⊆r B` subtract: `n - m` guard: `{T}` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` prop: `ℙ` or: `P ∨ Q` decidable: `Dec(P)` uimplies: `b supposing a` rev_uimplies: `rev_uimplies(P;Q)` uiff: `uiff(P;Q)` all: `∀x:A. B[x]` nequal: `a ≠ b ∈ T ` int_lower: `{...i}` nat: `ℕ` false: `False` implies: `P `` Q` not: `¬A` le: `A ≤ B` and: `P ∧ Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` less_than: `a < b` squash: `↓T` bfalse: `ff` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` sq_stable: `SqStable(P)` cand: `A c∧ B` int_nzero: `ℤ-o`
Lemmas referenced :  nat_wf int_lower_wf member-less_than or_wf le-add-cancel add_functionality_wrt_le minus-zero minus-add zero-add add-swap add-commutes add-associates condition-implies-le not-le-2 false_wf le_wf decidable__le not-equal-2 less_than'_wf decidable__lt lt_int_wf eqtt_to_assert assert_of_lt_int top_wf istype-void eqff_to_assert set_subtype_base int_subtype_base bool_subtype_base bool_cases_sqequal subtype_base_sq bool_wf iff_transitivity assert_wf bnot_wf not_wf less_than_wf iff_weakening_uiff assert_of_bnot eq_int_wf assert_of_eq_int le_antisymmetry_iff sq_stable_from_decidable add-zero le-add-cancel-alt equal-wf-base istype-top istype-int istype-assert istype-less_than not-lt-2 iff_weakening_equal subtype_rel_self nequal_wf subtype_rel_sets rem-zero true_wf squash_wf decidable__int_equal add-inverse le_reflexive add_functionality_wrt_lt
Rules used in proof :  equalitySymmetry equalityTransitivity axiomEquality orFunctionality addLevel independent_functionElimination minusEquality intEquality voidEquality isect_memberEquality applyEquality inrFormation voidElimination lambdaFormation independent_pairFormation inlFormation unionElimination addEquality independent_isectElimination natural_numberEquality hypothesis rename setElimination remainderEquality isectElimination extract_by_obid because_Cache hypothesisEquality dependent_functionElimination lambdaEquality independent_pairEquality thin productElimination sqequalHypSubstitution sqequalRule cut introduction isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution lessCases Error :remPositive,  Error :inhabitedIsType,  Error :lambdaFormation_alt,  equalityElimination Error :isect_memberFormation_alt,  axiomSqEquality Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  Error :universeIsType,  imageMemberEquality baseClosed imageElimination Error :dependent_pairFormation_alt,  Error :equalityIsType4,  baseApply closedConclusion Error :lambdaEquality_alt,  promote_hyp instantiate cumulativity Error :functionIsType,  Error :equalityIsType1,  int_eqReduceTrueSq int_eqReduceFalseSq remainderBounds4 productEquality universeEquality setEquality dependent_set_memberEquality

Latex:
\mforall{}[a:\mBbbN{}].  \mforall{}[n:\{...-1\}].    ((0  \mleq{}  (a  rem  n))  \mwedge{}  a  rem  n  <  -n)

Date html generated: 2019_06_20-AM-11_24_15
Last ObjectModification: 2018_10_16-PM-03_24_54

Theory : arithmetic

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