Nuprl Lemma : prime-sum-of-two-squares
∀p:Prime
((∃a,b:ℤ. ((¬((a ≡ 0 mod p) ∧ (b ≡ 0 mod p))) ∧ (((a * a) + (b * b)) ≡ 0 mod p)))
⇒ (∃a,b:ℤ. (p = ((a * a) + (b * b)) ∈ ℤ)))
Proof
Definitions occuring in Statement :
Prime: Prime,
eqmod: a ≡ b mod m,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
not: ¬A,
member: t ∈ T,
uall: ∀[x:A]. B[x],
Prime: Prime,
int_upper: {i...},
prop: ℙ,
false: False,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
eqmod: a ≡ b mod m,
divides: b | a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
sq_stable: SqStable(P),
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j
Lemmas referenced :
istype-int,
eqmod_wf,
istype-void,
Prime_wf,
small-eqmod,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
eqmod_functionality_wrt_eqmod,
add_functionality_wrt_eqmod,
multiply_functionality_wrt_eqmod,
eqmod_inversion,
eqmod_weakening,
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
decidable__equal_int,
intformeq_wf,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
int_subtype_base,
set_subtype_base,
int_upper_wf,
prime_wf,
istype-int_upper,
le_wf,
absval_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
add_functionality_wrt_eq,
absval_pos,
square_non_neg,
istype-le,
subtype_rel_self,
iff_weakening_equal,
absval_mul,
itermAdd_wf,
int_term_value_add_lemma,
Prime-isOdd,
assert-isOdd,
sq_stable_from_decidable,
decidable__prime,
upper_subtype_nat,
istype-false,
decidable__le,
multiply-is-int-iff,
false_wf,
product-eq-0-mod-prime,
nat_properties,
int_seg_wf,
int_seg_properties,
mul_preserves_le,
int_seg_subtype_nat,
subtype_rel_sets,
sq_stable__le,
prime-sum-of-two-squares-lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
productIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
functionIsType,
universeIsType,
isectElimination,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
addEquality,
multiplyEquality,
dependent_functionElimination,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
promote_hyp,
minusEquality,
inhabitedIsType,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype,
instantiate,
cumulativity,
baseApply,
closedConclusion,
applyEquality,
intEquality,
sqequalBase,
universeEquality,
sqequalIntensionalEquality,
pointwiseFunctionality,
applyLambdaEquality,
setEquality,
setIsType
Latex:
\mforall{}p:Prime
((\mexists{}a,b:\mBbbZ{}. ((\mneg{}((a \mequiv{} 0 mod p) \mwedge{} (b \mequiv{} 0 mod p))) \mwedge{} (((a * a) + (b * b)) \mequiv{} 0 mod p)))
{}\mRightarrow{} (\mexists{}a,b:\mBbbZ{}. (p = ((a * a) + (b * b)))))
Date html generated:
2020_05_20-AM-08_08_27
Last ObjectModification:
2019_11_27-PM-02_22_09
Theory : general
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