### Nuprl Lemma : div_rem_sum

`∀[a:ℤ]. ∀[n:ℤ-o].  (a = (((a ÷ n) * n) + (a rem n)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  int_nzero: `ℤ-o` uall: `∀[x:A]. B[x]` remainder: `n rem m` divide: `n ÷ m` multiply: `n * m` add: `n + m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` int_nzero: `ℤ-o` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` nequal: `a ≠ b ∈ T ` or: `P ∨ Q` guard: `{T}` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` false: `False` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` 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` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` prop: `ℙ`
Lemmas referenced :  int_nzero_wf eq_int_wf eqtt_to_assert assert_of_eq_int not-equal-2 le_antisymmetry_iff add_functionality_wrt_le zero-add add-zero le-add-cancel condition-implies-le add-commutes istype-void minus-add minus-zero eqff_to_assert set_subtype_base nequal_wf int_subtype_base bool_subtype_base bool_cases_sqequal subtype_base_sq bool_wf iff_transitivity assert_wf bnot_wf not_wf equal-wf-base iff_weakening_uiff assert_of_bnot false_wf
Rules used in proof :  intEquality because_Cache axiomEquality hypothesisEquality thin isectElimination isect_memberEquality sqequalHypSubstitution sqequalRule extract_by_obid hypothesis cut introduction isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution divideRemainderSum setElimination rename natural_numberEquality Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination equalityElimination productElimination independent_isectElimination int_eqReduceTrueSq dependent_functionElimination addEquality independent_functionElimination voidElimination minusEquality applyEquality Error :lambdaEquality_alt,  Error :isect_memberEquality_alt,  Error :universeIsType,  Error :dependent_pairFormation_alt,  equalityTransitivity equalitySymmetry Error :equalityIsType4,  baseApply closedConclusion baseClosed promote_hyp instantiate cumulativity independent_pairFormation Error :functionIsType,  int_eqReduceFalseSq Error :equalityIsType1

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}].    (a  =  (((a  \mdiv{}  n)  *  n)  +  (a  rem  n)))

Date html generated: 2019_06_20-AM-11_23_57
Last ObjectModification: 2018_10_15-PM-00_14_22

Theory : arithmetic

Home Index