Nuprl Lemma : prime-sum-of-two-squares-iff-one-mod-four
∀p:{p:{2...}| prime(p)} . ((p = 2 ∈ ℤ) ∨ (∃k:ℤ. (p = (1 + (4 * k)) ∈ ℤ)) ⇐⇒ ∃a,b:ℤ. (p = ((a * a) + (b * b)) ∈ ℤ))
Proof
Definitions occuring in Statement :
prime: prime(a),
int_upper: {i...},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
set: {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,
prop: ℙ,
uall: ∀[x:A]. B[x],
int_upper: {i...},
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
rev_implies: P ⇐ Q,
or: P ∨ Q,
decidable: Dec(P),
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
sq_type: SQType(T),
guard: {T},
nat: ℕ,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
le: A ≤ B,
ge: i ≥ j
Lemmas referenced :
or_wf,
equal-wf-T-base,
exists_wf,
int_subtype_base,
set_wf,
int_upper_wf,
prime_wf,
prime-sum-of-two-squares-if-one-mod-four,
int_upper_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
subtype_base_sq,
absval_squared,
div_rem_sum,
absval_wf,
nat_wf,
equal-wf-base,
nequal_wf,
rem_bounds_1,
less_than_wf,
lelt_wf,
int_seg_wf,
divide_wf,
equal_wf,
int_seg_properties,
int_seg_cases,
int_seg_subtype,
false_wf,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
mul-distributes,
mul-distributes-right,
mul-associates,
add-associates,
mul-swap,
mul-commutes,
zero-mul,
zero-add,
add-zero,
add-commutes,
add-swap,
le_wf,
not-prime-mult,
nat_properties,
prime-mult,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
assoced_elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
baseClosed,
sqequalRule,
lambdaEquality,
because_Cache,
baseApply,
closedConclusion,
applyEquality,
natural_numberEquality,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
instantiate,
cumulativity,
dependent_set_memberEquality,
remainderEquality,
imageMemberEquality,
addEquality,
multiplyEquality,
hypothesis_subsumption,
inrFormation,
inlFormation,
imageElimination,
universeEquality,
minusEquality
Latex:
\mforall{}p:\{p:\{2...\}| prime(p)\}
((p = 2) \mvee{} (\mexists{}k:\mBbbZ{}. (p = (1 + (4 * k)))) \mLeftarrow{}{}\mRightarrow{} \mexists{}a,b:\mBbbZ{}. (p = ((a * a) + (b * b))))
Date html generated:
2019_06_20-PM-02_41_46
Last ObjectModification:
2018_09_24-PM-02_53_03
Theory : num_thy_1
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