Nuprl Lemma : rem_bounds_1

[a:ℕ]. ∀[n:ℕ+].  ((0 ≤ (a rem n)) ∧ rem n < n)


Proof




Definitions occuring in Statement :  nat_plus: + nat: less_than: a < b uall: [x:A]. B[x] le: A ≤ B and: P ∧ Q remainder: rem m natural_number: $n
Definitions unfolded in proof :  prop: all: x:A. B[x] uimplies: supposing a guard: {T} nequal: a ≠ b ∈  nat_plus: + nat: false: False implies:  Q not: ¬A le: A ≤ B and: P ∧ Q member: t ∈ T uall: [x:A]. B[x] or: P ∨ Q decidable: Dec(P) bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) less_than: a < b less_than': less_than'(a;b) top: Top true: True squash: T bfalse: ff exists: x:A. B[x] subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) bnot: ¬bb ifthenelse: if then else fi  assert: b iff: ⇐⇒ Q rev_implies:  Q cand: c∧ B int_nzero: -o subtract: m
Lemmas referenced :  decidable__lt equal_wf less_than_irreflexivity le_weakening less_than_transitivity1 lt_int_wf eqtt_to_assert assert_of_lt_int top_wf istype-void eqff_to_assert set_subtype_base le_wf 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 false_wf eq_int_wf assert_of_eq_int equal-wf-base less_than'_wf iff_weakening_equal subtype_rel_self nequal_wf subtype_rel_sets rem-zero true_wf squash_wf not-lt-2 minus-zero minus-add add-commutes condition-implies-le le-add-cancel zero-add add-zero add-associates add_functionality_wrt_le not-equal-2 decidable__int_equal nat_wf member-less_than nat_plus_wf
Rules used in proof :  isect_memberEquality equalitySymmetry equalityTransitivity axiomEquality intEquality voidElimination independent_functionElimination independent_isectElimination natural_numberEquality lambdaFormation hypothesis rename setElimination remainderEquality isectElimination lemma_by_obid because_Cache hypothesisEquality dependent_functionElimination lambdaEquality independent_pairEquality thin productElimination sqequalHypSubstitution sqequalRule cut introduction isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution unionElimination extract_by_obid independent_pairFormation 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 applyEquality Error :lambdaEquality_alt,  promote_hyp instantiate cumulativity Error :functionIsType,  Error :equalityIsType1,  int_eqReduceTrueSq int_eqReduceFalseSq remainderBounds1 productEquality universeEquality setEquality addLevel minusEquality voidEquality addEquality dependent_set_memberEquality

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



Date html generated: 2019_06_20-AM-11_24_03
Last ObjectModification: 2018_10_15-PM-03_00_39

Theory : arithmetic


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