### Nuprl Lemma : quot_rem_exists

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

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat_plus: `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` decidable: `Dec(P)` or: `P ∨ Q` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` exists: `∃x:A. B[x]` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` and: `P ∧ Q` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` guard: `{T}` ge: `i ≥ j ` subtract: `n - m`
Lemmas referenced :  nat_plus_wf istype-int decidable__le quot_rem_exists_n le_wf int_seg_wf int_subtype_base nat_plus_properties full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermMinus_wf itermVar_wf int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_minus_lemma int_term_value_var_lemma int_formula_prop_wf decidable__equal_int istype-false int_seg_properties nat_properties decidable__lt intformless_wf intformeq_wf int_formula_prop_less_lemma int_formula_prop_eq_lemma less_than_wf itermAdd_wf itermMultiply_wf int_term_value_add_lemma int_term_value_mul_lemma subtract_wf itermSubtract_wf int_term_value_subtract_lemma minus-add mul-distributes-right add-associates minus-one-mul mul-associates mul-commutes add-swap add-commutes add-mul-special zero-mul zero-add
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  Error :universeIsType,  cut introduction extract_by_obid hypothesis sqequalHypSubstitution dependent_functionElimination thin natural_numberEquality hypothesisEquality unionElimination Error :dependent_set_memberEquality_alt,  isectElimination productElimination Error :dependent_pairFormation_alt,  setElimination rename sqequalRule Error :productIsType,  Error :equalityIsType4,  Error :inhabitedIsType,  applyEquality addEquality multiplyEquality because_Cache minusEquality independent_isectElimination approximateComputation independent_functionElimination Error :lambdaEquality_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation equalityTransitivity equalitySymmetry

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

Date html generated: 2019_06_20-PM-02_22_13
Last ObjectModification: 2018_10_05-PM-05_45_41

Theory : num_thy_1

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