Nuprl Lemma : prime-sum-of-four-squares
∀p:Prime. ∀r:ℕ. ∀a,b,c,d:ℤ.
(((((a * a) + (b * b) + (c * c) + (d * d)) = (p * r) ∈ ℤ) ∧ 0 < r ∧ r < p)
⇒ (∃a,b,c,d:ℤ. (p = ((a * a) + (b * b) + (c * c) + (d * d)) ∈ ℤ)))
Proof
Definitions occuring in Statement :
Prime: Prime,
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
nat_plus: ℕ+,
cand: A c∧ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
ge: i ≥ j ,
sq_type: SQType(T),
squash: ↓T,
sq_stable: SqStable(P),
nat: ℕ,
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
and: P ∧ Q,
lelt: i ≤ j < k,
int_upper: {i...},
false: False,
Prime: Prime,
int_seg: {i..j-},
guard: {T},
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
divides: b | a,
eqmod: a ≡ b mod m,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
true: True,
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
less_than: a < b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹
Lemmas referenced :
Prime_wf,
nat_wf,
primrec-wf2,
set_wf,
lelt_wf,
decidable__lt,
all_wf,
equal-wf-base-T,
multiply_functionality_wrt_eqmod,
less_than_wf,
small-eqmod,
exists_wf,
equal-wf-T-base,
int_term_value_mul_lemma,
int_term_value_add_lemma,
itermMultiply_wf,
itermAdd_wf,
nat_properties,
int_subtype_base,
subtype_base_sq,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
intformeq_wf,
itermSubtract_wf,
intformnot_wf,
decidable__le,
le_wf,
int_upper_subtype_nat,
decidable__prime,
prime_wf,
sq_stable_from_decidable,
false_wf,
int_seg_subtype,
subtract_wf,
decidable__equal_int,
int_seg_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
int_upper_properties,
int_seg_properties,
add_functionality_wrt_eqmod,
eqmod_functionality_wrt_eqmod,
eqmod_weakening,
multiply-is-int-iff,
mul_bounds_1a,
iff_weakening_equal,
square_non_neg,
absval_pos,
equal_wf,
absval_wf,
squash_wf,
true_wf,
absval_mul,
le_functionality,
le_weakening,
multiply_functionality_wrt_le,
mul_preserves_lt,
mul_preserves_le,
add_functionality_wrt_le,
int_entire,
eqmod_wf,
int_term_value_minus_lemma,
itermMinus_wf,
nequal_wf,
equal-wf-base,
mul_cancel_in_eq,
assoced_wf,
or_wf,
assoced_elim,
divides-prime,
Euler-four-square-identity,
add_functionality_wrt_eq,
eqmod_transitivity,
minus_functionality_wrt_eqmod,
mul_nzero,
absval_nat_plus,
mul_nat_plus,
mul_bounds_1b,
int_formula_prop_or_lemma,
intformor_wf,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
top_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
absval_unfold,
eqmod_inversion,
eqmod-zero
Rules used in proof :
addEquality,
functionEquality,
multiplyEquality,
productEquality,
closedConclusion,
baseApply,
cumulativity,
instantiate,
hypothesis_subsumption,
levelHypothesis,
imageElimination,
baseClosed,
imageMemberEquality,
dependent_set_memberEquality,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
applyEquality,
addLevel,
unionElimination,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
productElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
because_Cache,
natural_numberEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
promote_hyp,
pointwiseFunctionality,
equalityUniverse,
universeEquality,
minusEquality,
orFunctionality,
sqequalAxiom,
isect_memberFormation,
lessCases,
equalityElimination
Latex:
\mforall{}p:Prime. \mforall{}r:\mBbbN{}. \mforall{}a,b,c,d:\mBbbZ{}.
(((((a * a) + (b * b) + (c * c) + (d * d)) = (p * r)) \mwedge{} 0 < r \mwedge{} r < p)
{}\mRightarrow{} (\mexists{}a,b,c,d:\mBbbZ{}. (p = ((a * a) + (b * b) + (c * c) + (d * d)))))
Date html generated:
2018_05_21-PM-07_28_03
Last ObjectModification:
2018_01_01-AM-11_30_57
Theory : general
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