Nuprl Lemma : gcd-reduce-coprime
∀p,q:ℤ. ∃x,y:ℤ. (((x * p) + (y * q)) = 1 ∈ ℤ) supposing CoPrime(p,q)
Proof
Definitions occuring in Statement :
coprime: CoPrime(a,b),
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
coprime: CoPrime(a,b),
gcd_p: GCD(a;b;y),
cand: A c∧ B,
divides: b | a,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
gcd-reduce-ext,
coprime_wf,
subtype_base_sq,
int_subtype_base,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
equal-wf-base,
exists_wf,
equal-wf-base-T,
divisor_bound,
less_than_wf,
intformle_wf,
int_formula_prop_le_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation,
productElimination,
isectElimination,
intEquality,
promote_hyp,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
dependent_pairFormation,
setElimination,
rename,
unionElimination,
natural_numberEquality,
approximateComputation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
multiplyEquality,
dependent_set_memberEquality,
imageMemberEquality
Latex:
\mforall{}p,q:\mBbbZ{}. \mexists{}x,y:\mBbbZ{}. (((x * p) + (y * q)) = 1) supposing CoPrime(p,q)
Date html generated:
2018_05_21-PM-00_59_22
Last ObjectModification:
2018_05_19-AM-06_35_23
Theory : num_thy_1
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