Nuprl Lemma : ab-divides-a^2+b^2+1
∀a,b:ℤ.
((a * b) | (((a * a) + (b * b)) + 1)
⇐⇒ ∃n:ℕ. ((<|a|, |b|> = vexample(n;1;1) ∈ (ℤ × ℤ)) ∨ (<|b|, |a|> = vexample(n;1;1) ∈ (ℤ × ℤ))))
Proof
Definitions occuring in Statement :
vexample: vexample(n;a;b),
divides: b | a,
absval: |i|,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
pair: <a, b>,
product: 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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
squash: ↓T,
guard: {T},
true: True,
divides: b | a,
decidable: Dec(P),
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
uiff: uiff(P;Q),
ge: i ≥ j ,
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
divides_wf,
absval_wf,
istype-nat,
int_subtype_base,
set_subtype_base,
le_wf,
istype-int,
absval-divides,
absval_mul,
iff_weakening_equal,
squash_wf,
true_wf,
add_functionality_wrt_eq,
absval_squared,
subtype_rel_self,
Vieta-jumping-example2-corollary,
decidable__le,
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,
istype-void,
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,
add-is-int-iff,
multiply-is-int-iff,
intformle_wf,
int_formula_prop_le_lemma,
false_wf,
vexample_wf,
nat_properties,
istype-le,
pi2_wf,
pi1_wf,
nat_wf,
equal_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
independent_pairFormation,
Error :universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
multiplyEquality,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
addEquality,
natural_numberEquality,
Error :productIsType,
Error :unionIsType,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
intEquality,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
productElimination,
independent_functionElimination,
promote_hyp,
dependent_functionElimination,
setElimination,
rename,
equalityTransitivity,
imageElimination,
imageMemberEquality,
instantiate,
universeEquality,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :inlFormation_alt,
pointwiseFunctionality,
Error :inrFormation_alt,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality
Latex:
\mforall{}a,b:\mBbbZ{}.
((a * b) | (((a * a) + (b * b)) + 1)
\mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. ((<|a|, |b|> = vexample(n;1;1)) \mvee{} (<|b|, |a|> = vexample(n;1;1))))
Date html generated:
2019_06_20-PM-02_43_55
Last ObjectModification:
2019_03_11-PM-06_33_37
Theory : num_thy_1
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