Nuprl Lemma : prime-sum-of-two-squares-if-one-mod-four
This proof that every prime p that is 1 mod 4 is the sum of two squares
comes from the article by D. Zagier called
"A One-Sentence Proof that Every Prime p == 1 (mod 4) Is a Sum of Two Squares".
It uses the properties
twosquareinv-involution and
twosquareinv-fixpoint
that a certain involution has a unique fixed point to deduce
that the given finite set (see twosquare-type-finite)
has odd cardinality
(see involution-with-unique-fixpoint) ,
so every invoulution has a fixed point (see involution-has-fixpoint),
and this proves the theorem.
⋅
∀p:{p:{2...}| prime(p)} . ((∃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],
implies: 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],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
finite: finite(T),
uall: ∀[x:A]. B[x],
nat: ℕ,
int_upper: {i...},
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
twosquareinv: twosquareinv(t),
spreadn: spread3,
subtract: n - m,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
twosquare-type-finite,
involution-with-unique-fixpoint,
nat_wf,
equal_wf,
twosquareinv_wf,
twosquareinv-involution,
twosquareinv-fixpoint,
subtype_rel_product,
exists_wf,
equal-wf-T-base,
int_subtype_base,
set_wf,
int_upper_wf,
prime_wf,
false_wf,
le_wf,
nat_properties,
sq_stable_from_decidable,
decidable__prime,
upper_subtype_nat,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
decidable__equal_int,
lt_int_wf,
subtract_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_functionality_wrt_uiff,
assert_wf,
less_than_wf,
zero-add,
involution-has-fixpoint,
pi2_wf,
Error :pi1_wf_top,
set_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
productEquality,
hypothesis,
setEquality,
intEquality,
addEquality,
multiplyEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
baseApply,
closedConclusion,
baseClosed,
dependent_pairFormation,
dependent_pairEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
imageElimination,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
applyLambdaEquality,
independent_pairEquality
Latex:
\mforall{}p:\{p:\{2...\}| prime(p)\} . ((\mexists{}k:\mBbbZ{}. (p = (1 + (4 * k)))) {}\mRightarrow{} (\mexists{}a,b:\mBbbZ{}. (p = ((a * a) + (b * b)))))
Date html generated:
2019_06_20-PM-02_41_31
Last ObjectModification:
2018_09_24-PM-02_52_59
Theory : num_thy_1
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