Nuprl Lemma : four-squares
∀n:ℕ+. ∃a,b,c,d:ℤ. (n = ((a * a) + (b * b) + (c * c) + (d * d)) ∈ ℤ)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
false: False,
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
guard: {T},
sq_type: SQType(T),
int_upper: {i...},
Prime: Prime,
and: P ∧ Q,
iff: P ⇐⇒ Q,
true: True,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_seg: {i..j-},
squash: ↓T,
sq_stable: SqStable(P),
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
inject: Inj(A;B;f),
label: ...$L... t,
lelt: i ≤ j < k,
divides: b | a,
eqmod: a ≡ b mod m,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B
Lemmas referenced :
nat_plus_wf,
int_subtype_base,
equal-wf-T-base,
exists_wf,
nat-plus-ind-primes,
equal-wf-base,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermMultiply_wf,
itermAdd_wf,
itermConstant_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
Prime_wf,
subtype_base_sq,
int_upper_properties,
assert-isOdd,
Prime-isOdd,
int_seg_wf,
int-subtype-int_mod,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
less_than_wf,
modulus_wf_int_mod,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
set_wf,
sq_stable__le,
subtype_rel_sets,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformle_wf,
intformand_wf,
decidable__le,
le_wf,
false_wf,
int_upper_subtype_nat,
decidable__prime,
prime_wf,
sq_stable_from_decidable,
pigeon-hole-implies2,
equal_wf,
eqmod_weakening,
subtract_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
int_seg_properties,
mod-eqmod,
eqmod_inversion,
eqmod_transitivity,
eqmod_functionality_wrt_eqmod,
subtract_functionality_wrt_eqmod,
product-eq-0-mod-prime,
lelt_wf,
mul_preserves_le,
int_term_value_minus_lemma,
itermMinus_wf,
sq_stable__equal,
add_functionality_wrt_eqmod,
int_seg_subtype_nat,
mul_bounds_1a,
equal-wf-base-T,
prime-sum-of-four-squares,
iff_weakening_equal,
Euler-four-square-identity,
true_wf,
squash_wf
Rules used in proof :
independent_functionElimination,
because_Cache,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
hypothesis,
hypothesisEquality,
rename,
setElimination,
intEquality,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
cumulativity,
instantiate,
lambdaFormation,
equalitySymmetry,
equalityTransitivity,
productElimination,
minusEquality,
multiplyEquality,
setEquality,
int_eqEquality,
imageElimination,
imageMemberEquality,
independent_pairFormation,
addEquality,
dependent_set_memberEquality,
levelHypothesis,
addLevel,
applyLambdaEquality,
promote_hyp,
equalityUniverse,
universeEquality,
functionEquality
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mexists{}a,b,c,d:\mBbbZ{}. (n = ((a * a) + (b * b) + (c * c) + (d * d)))
Date html generated:
2018_05_21-PM-07_30_42
Last ObjectModification:
2018_01_01-PM-00_36_00
Theory : general
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