### Nuprl Lemma : gcd-non-neg

`∀[y,x:ℕ].  (0 ≤ gcd(x;y))`

Proof

Definitions occuring in Statement :  gcd: `gcd(a;b)` nat: `ℕ` uall: `∀[x:A]. B[x]` le: `A ≤ B` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` nat: `ℕ` prop: `ℙ` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` gcd: `gcd(a;b)` eq_int: `(i =z j)` ifthenelse: `if b then t else f fi ` btrue: `tt` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` less_than': `less_than'(a;b)` nequal: `a ≠ b ∈ T ` int_upper: `{i...}` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` bnot: `¬bb` assert: `↑b` subtype_rel: `A ⊆r B` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` less_than: `a < b` squash: `↓T`
Lemmas referenced :  decidable__equal_int less_than'_wf gcd_wf nat_wf subtype_base_sq int_subtype_base nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf int_upper_subtype_nat false_wf le_wf nequal-le-implies zero-add eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int zero-rem subtype_rel_sets nequal_wf int_upper_properties intformeq_wf int_formula_prop_eq_lemma equal-wf-base gcd-positive decidable__lt intformless_wf int_formula_prop_less_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution dependent_functionElimination thin because_Cache natural_numberEquality hypothesis unionElimination sqequalRule productElimination independent_pairEquality lambdaEquality hypothesisEquality isectElimination setElimination rename axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidElimination instantiate cumulativity intEquality independent_isectElimination independent_functionElimination dependent_pairFormation int_eqEquality voidEquality independent_pairFormation computeAll hypothesis_subsumption dependent_set_memberEquality lambdaFormation equalityElimination promote_hyp applyEquality setEquality applyLambdaEquality baseClosed imageElimination

Latex:
\mforall{}[y,x:\mBbbN{}].    (0  \mleq{}  gcd(x;y))

Date html generated: 2017_04_17-AM-09_45_50
Last ObjectModification: 2017_02_27-PM-05_40_34

Theory : num_thy_1

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