### Nuprl Lemma : bezout_ident_n

`∀b:ℕ. ∀a:ℤ.  ∃u,v:ℤ. GCD(a;b;(u * a) + (v * b))`

Proof

Definitions occuring in Statement :  gcd_p: `GCD(a;b;y)` nat: `ℕ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` multiply: `n * m` add: `n + m` int: `ℤ`
Definitions unfolded in proof :  ge: `i ≥ j ` less_than: `a < b` nat: `ℕ` so_apply: `x[s]` so_lambda: `λ2x.t[x]` less_than': `less_than'(a;b)` le: `A ≤ B` subtype_rel: `A ⊆r B` or: `P ∨ Q` decidable: `Dec(P)` prop: `ℙ` top: `Top` not: `¬A` implies: `P `` Q` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` uimplies: `b supposing a` and: `P ∧ Q` lelt: `i ≤ j < k` int_seg: `{i..j-}` guard: `{T}` member: `t ∈ T` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` subtract: `n - m` true: `True` uiff: `uiff(P;Q)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` nat_plus: `ℕ+` squash: `↓T`
Rules used in proof :  multiplyEquality addEquality dependent_set_memberEquality hypothesis_subsumption levelHypothesis equalitySymmetry equalityTransitivity applyEquality addLevel unionElimination computeAll independent_pairFormation sqequalRule voidEquality voidElimination isect_memberEquality dependent_functionElimination intEquality int_eqEquality lambdaEquality dependent_pairFormation independent_isectElimination productElimination rename setElimination hypothesis hypothesisEquality because_Cache natural_numberEquality isectElimination sqequalHypSubstitution extract_by_obid introduction thin cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution Error :dependent_pairFormation_alt,  hyp_replacement applyLambdaEquality Error :universeIsType,  Error :productIsType,  Error :inhabitedIsType,  minusEquality independent_functionElimination Error :isect_memberEquality_alt,  Error :lambdaEquality_alt,  imageElimination imageMemberEquality baseClosed instantiate universeEquality

Latex:
\mforall{}b:\mBbbN{}.  \mforall{}a:\mBbbZ{}.    \mexists{}u,v:\mBbbZ{}.  GCD(a;b;(u  *  a)  +  (v  *  b))

Date html generated: 2019_06_20-PM-02_22_20
Last ObjectModification: 2019_01_11-AM-09_03_44

Theory : num_thy_1

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