### Nuprl Lemma : rem_bounds_1

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

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` le: `A ≤ B` and: `P ∧ Q` remainder: `n rem m` natural_number: `\$n`
Definitions unfolded in proof :  prop: `ℙ` all: `∀x:A. B[x]` uimplies: `b supposing a` guard: `{T}` nequal: `a ≠ b ∈ T ` nat_plus: `ℕ+` nat: `ℕ` false: `False` implies: `P `` 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 ⊆r B` 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` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B` int_nzero: `ℤ-o` subtract: `n - 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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