### Nuprl Lemma : quot_rem_exists_n

`∀a:ℕ. ∀b:ℕ+.  ∃q:ℕ. ∃r:ℕb. (a = ((q * b) + r) ∈ ℤ)`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat_plus: `ℕ+` nat: `ℕ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` multiply: `n * m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` guard: `{T}` int_seg: `{i..j-}` nat_plus: `ℕ+` nat: `ℕ` ge: `i ≥ j ` lelt: `i ≤ j < k` and: `P ∧ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  int_term_value_mul_lemma itermMultiply_wf int_term_value_add_lemma itermAdd_wf primrec-wf2 less_than_wf set_wf decidable__lt all_wf equal_wf exists_wf le_wf int_formula_prop_eq_lemma int_term_value_subtract_lemma int_formula_prop_not_lemma intformeq_wf itermSubtract_wf intformnot_wf decidable__le lelt_wf false_wf int_seg_subtype subtract_wf decidable__equal_int int_seg_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_and_lemma intformle_wf itermConstant_wf itermVar_wf intformless_wf intformand_wf satisfiable-full-omega-tt nat_properties nat_plus_properties int_seg_properties nat_wf nat_plus_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid hypothesis thin sqequalHypSubstitution isectElimination natural_numberEquality because_Cache hypothesisEquality setElimination rename productElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll unionElimination addLevel applyEquality equalityTransitivity equalitySymmetry setEquality levelHypothesis hypothesis_subsumption dependent_set_memberEquality addEquality multiplyEquality functionEquality introduction independent_functionElimination

Latex:
\mforall{}a:\mBbbN{}.  \mforall{}b:\mBbbN{}\msupplus{}.    \mexists{}q:\mBbbN{}.  \mexists{}r:\mBbbN{}b.  (a  =  ((q  *  b)  +  r))

Date html generated: 2016_05_14-PM-04_18_53
Last ObjectModification: 2016_01_14-PM-11_42_27

Theory : num_thy_1

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