Nuprl Lemma : gcd-non-zero
∀a,b:ℤ. ((a ≠ 0 ∨ b ≠ 0) ⇒ gcd(a;b) ≠ 0)
Proof
Definitions occuring in Statement :
gcd: gcd(a;b),
all: ∀x:A. B[x],
nequal: a ≠ b ∈ T ,
implies: P ⇒ Q,
or: P ∨ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ
Lemmas referenced :
nequal_wf,
or_wf,
gcd_wf,
equal_wf,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_formula_prop_or_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
intformor_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_subtype_base,
subtype_base_sq,
gcd-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
productElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache
Latex:
\mforall{}a,b:\mBbbZ{}. ((a \mneq{} 0 \mvee{} b \mneq{} 0) {}\mRightarrow{} gcd(a;b) \mneq{} 0)
Date html generated:
2016_05_14-PM-09_24_31
Last ObjectModification:
2016_01_14-PM-11_32_58
Theory : num_thy_1
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