### Nuprl Lemma : gcd_exists_n

`∀b:ℕ. ∀a:ℤ.  ∃y:ℤ. GCD(a;b;y)`

Proof

Definitions occuring in Statement :  gcd_p: `GCD(a;b;y)` nat: `ℕ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` guard: `{T}` int_seg: `{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)` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` less_than: `a < b` ge: `i ≥ j ` true: `True` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` nat_plus: `ℕ+` uiff: `uiff(P;Q)` subtract: `n - m`
Lemmas referenced :  gcd_p_shift add_com gcd_p_sym 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 not-lt-2 quot_rem_exists gcd_p_zero iff_weakening_equal true_wf squash_wf int_term_value_add_lemma itermAdd_wf nat_properties nat_wf primrec-wf2 less_than_wf set_wf decidable__lt gcd_p_wf exists_wf guard_wf all_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 int_seg_properties
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin lemma_by_obid sqequalHypSubstitution isectElimination natural_numberEquality because_Cache hypothesisEquality hypothesis 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 introduction addEquality imageElimination imageMemberEquality baseClosed universeEquality independent_functionElimination minusEquality multiplyEquality

Latex:
\mforall{}b:\mBbbN{}.  \mforall{}a:\mBbbZ{}.    \mexists{}y:\mBbbZ{}.  GCD(a;b;y)

Date html generated: 2016_05_14-PM-04_19_03
Last ObjectModification: 2016_01_14-PM-11_41_14

Theory : num_thy_1

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