Nuprl Lemma : rem_bounds_4

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


Proof




Definitions occuring in Statement :  int_lower: {...i} nat: less_than: a < b uall: [x:A]. B[x] le: A ≤ B and: P ∧ Q remainder: rem m minus: -n natural_number: $n
Definitions unfolded in proof :  true: True less_than': less_than'(a;b) top: Top subtype_rel: A ⊆B subtract: m guard: {T} rev_implies:  Q iff: ⇐⇒ Q prop: or: P ∨ Q decidable: Dec(P) uimplies: supposing a rev_uimplies: rev_uimplies(P;Q) uiff: uiff(P;Q) all: x:A. B[x] nequal: a ≠ b ∈  int_lower: {...i} nat: false: False implies:  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 then else fi  assert: b sq_stable: SqStable(P) cand: c∧ B int_nzero: -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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