Nuprl Lemma : implies-sum-of-two-squares
∀n:ℕ. ((∃x:ℤ-o. ∃w,y:ℤ. ((n * x * x) = ((w * w) + (y * y)) ∈ ℤ)) ⇒ (∃a,b:ℤ. (n = ((a * a) + (b * b)) ∈ ℤ)))
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
nat_plus: ℕ+,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
divides: b | a,
Prime: Prime,
int_upper: {i...},
sq_stable: SqStable(P),
less_than': less_than'(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
gt: i > j,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eqmod: a ≡ b mod m,
prime: prime(a),
true: True,
sq_exists: ∃x:A [B[x]],
mul-list: Π(ns) ,
reduce: reduce(f;k;as),
list_ind: list_ind,
cons: [a / b],
l_member: (x ∈ l),
select: L[n],
uiff: uiff(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹
Lemmas referenced :
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
subtype_rel_self,
nat_properties,
lelt_wf,
decidable__exists-divisor,
le_wf,
divides_wf,
decidable__cand,
decidable__divides_ext,
less_than_wf,
equal-wf-base,
primrec-wf2,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
istype-nat,
itermMultiply_wf,
int_term_value_mul_lemma,
mul_preserves_le,
multiply_nat_wf,
mul_bounds_1b,
subtype_rel_sets,
prime_wf,
sq_stable__le,
sq_stable_from_decidable,
decidable__prime,
upper_subtype_nat,
istype-false,
int_upper_properties,
Prime_wf,
mul_cancel_in_eq,
mul_nzero,
nequal_wf,
pos_mul_arg_bounds,
mul_preserves_lt,
prime-sum-of-two-squares,
eqmod_wf,
int_upper_wf,
istype-int_upper,
prime_divs_prod,
equal_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal,
mul-associates,
prime-factors,
sq_stable__equal,
l_member_wf,
list_induction,
mul-list_wf,
subtype_rel_list,
list_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
cons_wf,
cons_member,
list_subtype_base,
mul_list_nil_lemma,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
false_wf,
length_wf,
equal-wf-T-base,
int_nzero_wf,
exists_wf,
absval_wf,
int_term_value_minus_lemma,
itermMinus_wf,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
int_nzero_properties,
top_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
absval_unfold,
absval_square,
absval-non-neg,
absval_mul
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
cumulativity,
intEquality,
inhabitedIsType,
productEquality,
multiplyEquality,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
functionIsType,
functionEquality,
addEquality,
imageElimination,
setIsType,
promote_hyp,
setEquality,
imageMemberEquality,
universeEquality,
inrFormation_alt,
pointwiseFunctionality,
lambdaEquality,
lambdaFormation,
dependent_pairFormation,
voidEquality,
isect_memberEquality,
axiomSqEquality,
isect_memberFormation,
lessCases,
equalityElimination,
minusEquality,
dependent_set_memberEquality,
sqequalIntensionalEquality
Latex:
\mforall{}n:\mBbbN{}. ((\mexists{}x:\mBbbZ{}\msupminus{}\msupzero{}. \mexists{}w,y:\mBbbZ{}. ((n * x * x) = ((w * w) + (y * y)))) {}\mRightarrow{} (\mexists{}a,b:\mBbbZ{}. (n = ((a * a) + (b * b)))))
Date html generated:
2019_10_15-AM-11_12_43
Last ObjectModification:
2019_06_26-PM-04_24_32
Theory : general
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