Nuprl Lemma : gcd-reduce
∀p,q:ℤ. ∃g:ℕ. ∃a,b,x,y:ℤ. ((p = (a * g) ∈ ℤ) ∧ (q = (b * g) ∈ ℤ) ∧ (((x * a) + (y * b)) = 1 ∈ ℤ))
Proof
Definitions occuring in Statement :
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
absval: |i|,
sign: sign(x),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
label: ...$L... t,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
cand: A c∧ B,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
decidable: Dec(P),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
squash: ↓T,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
and: P ∧ Q,
exists: ∃x:A. B[x],
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
bnot_of_le_int,
assert_functionality_wrt_uiff,
uiff_transitivity,
assert_wf,
bnot_wf,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
le_int_wf,
assert_of_le_int,
satisfiable-full-omega-tt,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma,
sign_wf,
equal-wf-base-T,
absval_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
mul_assoc,
int_nzero_wf,
mul-commutes,
iff_weakening_equal,
mul_add_distrib,
istype-universe,
true_wf,
squash_wf,
div_rem_sum,
subtract_wf,
divide_wfa,
istype-le,
decidable__le,
istype-less_than,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformless_wf,
intformand_wf,
decidable__lt,
rem_bounds_1,
int-value-type,
equal_wf,
set-value-type,
nequal_wf,
remainder_wfa,
int_term_value_add_lemma,
itermAdd_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_not_lemma,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
nat_properties,
subtype_base_sq,
decidable__equal_int,
subtype_rel_self,
subtype_rel_function,
equal-wf-base,
exists_wf,
nat_wf,
all_wf,
natrec_wf,
istype-nat,
le_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
istype-int,
int_seg_wf
Rules used in proof :
minusEquality,
equalityElimination,
lessCases,
isect_memberFormation,
axiomSqEquality,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
computeAll,
promote_hyp,
dependent_pairEquality,
independent_pairEquality,
axiomEquality,
lambdaEquality,
lambdaFormation,
imageMemberEquality,
universeEquality,
hyp_replacement,
multiplyEquality,
addEquality,
cutEval,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
independent_functionElimination,
cumulativity,
instantiate,
unionElimination,
dependent_functionElimination,
functionEquality,
functionExtensionality,
productEquality,
equalityTransitivity,
equalitySymmetry,
sqequalBase,
baseClosed,
closedConclusion,
baseApply,
independent_isectElimination,
imageElimination,
productElimination,
Error :lambdaEquality_alt,
intEquality,
applyEquality,
Error :equalityIstype,
because_Cache,
Error :productIsType,
hypothesis,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
Error :universeIsType,
Error :functionIsType,
sqequalRule,
hypothesisEquality,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}p,q:\mBbbZ{}. \mexists{}g:\mBbbN{}. \mexists{}a,b,x,y:\mBbbZ{}. ((p = (a * g)) \mwedge{} (q = (b * g)) \mwedge{} (((x * a) + (y * b)) = 1))
Date html generated:
2019_06_20-PM-02_27_10
Last ObjectModification:
2019_06_19-PM-02_31_44
Theory : num_thy_1
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