Nuprl Lemma : Euler-four-square-identity
∀[a1,b1,c1,d1,a,b,c,d:ℤ].
((((a1 * a1) + (b1 * b1) + (c1 * c1) + (d1 * d1)) * ((a * a) + (b * b) + (c * c) + (d * d)))
= ((((a1 * a) + ((-b1) * b) + ((-c1) * c) + ((-d1) * d)) * ((a1 * a) + ((-b1) * b) + ((-c1) * c) + ((-d1) * d)))
+ (((a1 * b) + (b1 * a) + (c1 * d) + ((-d1) * c)) * ((a1 * b) + (b1 * a) + (c1 * d) + ((-d1) * c)))
+ (((a1 * c) + (c1 * a) + (d1 * b) + ((-b1) * d)) * ((a1 * c) + (c1 * a) + (d1 * b) + ((-b1) * d)))
+ (((a1 * d) + (d1 * a) + (b1 * c) + ((-c1) * b)) * ((a1 * d) + (d1 * a) + (b1 * c) + ((-c1) * b))))
∈ ℤ)
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
multiply: n * m,
add: n + m,
minus: -n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_formula_prop_wf,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermMinus_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int
Rules used in proof :
axiomEquality,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
hypothesisEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
isectElimination,
unionElimination,
hypothesis,
because_Cache,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a1,b1,c1,d1,a,b,c,d:\mBbbZ{}].
((((a1 * a1) + (b1 * b1) + (c1 * c1) + (d1 * d1)) * ((a * a) + (b * b) + (c * c) + (d * d)))
= ((((a1 * a) + ((-b1) * b) + ((-c1) * c) + ((-d1) * d))
* ((a1 * a) + ((-b1) * b) + ((-c1) * c) + ((-d1) * d)))
+ (((a1 * b) + (b1 * a) + (c1 * d) + ((-d1) * c))
* ((a1 * b) + (b1 * a) + (c1 * d) + ((-d1) * c)))
+ (((a1 * c) + (c1 * a) + (d1 * b) + ((-b1) * d))
* ((a1 * c) + (c1 * a) + (d1 * b) + ((-b1) * d)))
+ (((a1 * d) + (d1 * a) + (b1 * c) + ((-c1) * b))
* ((a1 * d) + (d1 * a) + (b1 * c) + ((-c1) * b)))))
Date html generated:
2018_05_21-PM-07_23_45
Last ObjectModification:
2017_12_31-PM-07_57_18
Theory : general
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