Nuprl Lemma : gcd-property
∀x,y:ℤ. ∃a,b:ℤ. (CoPrime(a,b) ∧ (x = (gcd(x;y) * a) ∈ ℤ) ∧ (y = (gcd(x;y) * b) ∈ ℤ))
Proof
Definitions occuring in Statement :
coprime: CoPrime(a,b),
gcd: gcd(a;b),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
divides: b | a,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
coprime: CoPrime(a,b),
gcd_p: GCD(a;b;y),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
gcd_is_divisor_1,
gcd_is_divisor_2,
decidable__equal_int,
gcd_wf,
subtype_base_sq,
int_subtype_base,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
mul-one,
int_formula_prop_wf,
coprime_wf,
equal-wf-base,
exists_wf,
one_divs_any,
divides_wf,
coprime_bezout_id,
bezout_ident,
gcd_sat_pred,
gcd_unique,
assoced_elim,
gcd-properties,
equal_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
iff_weakening_equal,
mul_cancel_in_eq,
nequal_wf,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
intEquality,
instantiate,
isectElimination,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
productEquality,
applyEquality,
baseClosed,
baseApply,
closedConclusion,
addEquality,
multiplyEquality,
imageElimination,
universeEquality,
imageMemberEquality,
dependent_set_memberEquality
Latex:
\mforall{}x,y:\mBbbZ{}. \mexists{}a,b:\mBbbZ{}. (CoPrime(a,b) \mwedge{} (x = (gcd(x;y) * a)) \mwedge{} (y = (gcd(x;y) * b)))
Date html generated:
2017_04_17-AM-09_45_39
Last ObjectModification:
2017_02_27-PM-05_40_11
Theory : num_thy_1
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