Nuprl Lemma : gcd_exists_n

b:ℕ. ∀a:ℤ.  ∃y:ℤGCD(a;b;y)


Proof




Definitions occuring in Statement :  gcd_p: GCD(a;b;y) nat: all: x:A. B[x] exists: x:A. B[x] int:
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: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False implies:  Q not: ¬A top: Top prop: decidable: Dec(P) or: P ∨ Q subtype_rel: A ⊆B le: A ≤ B less_than': less_than'(a;b) nat: so_lambda: λ2x.t[x] so_apply: x[s] less_than: a < b ge: i ≥  true: True squash: T iff: ⇐⇒ Q rev_implies:  Q nat_plus: + uiff: uiff(P;Q) subtract: 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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