Nuprl Lemma : twosquareinv-fixpoint
∀p:{p:{2...}| prime(p)} . ∀t:x:ℕ × y:ℕ × {z:ℕ| ((x * x) + (4 * y * z)) = p ∈ ℤ} .
  ((twosquareinv(t) = t ∈ (ℕ × ℕ × ℕ)) 
⇒ (t ~ <1, 1, (p - 1) ÷ 4>))
Proof
Definitions occuring in Statement : 
twosquareinv: twosquareinv(t)
, 
prime: prime(a)
, 
int_upper: {i...}
, 
nat: ℕ
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
set: {x:A| B[x]} 
, 
pair: <a, b>
, 
product: x:A × B[x]
, 
divide: n ÷ m
, 
multiply: n * m
, 
subtract: n - m
, 
add: n + m
, 
natural_number: $n
, 
int: ℤ
, 
sqequal: s ~ t
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
assert: ↑b
, 
bnot: ¬bb
, 
bfalse: ff
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
ifthenelse: if b then t else f fi 
, 
uiff: uiff(P;Q)
, 
btrue: tt
, 
it: ⋅
, 
unit: Unit
, 
bool: 𝔹
, 
spreadn: spread3, 
twosquareinv: twosquareinv(t)
, 
and: P ∧ Q
, 
exists: ∃x:A. B[x]
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
or: P ∨ Q
, 
decidable: Dec(P)
, 
ge: i ≥ j 
, 
guard: {T}
, 
sq_type: SQType(T)
, 
false: False
, 
not: ¬A
, 
prop: ℙ
, 
int_upper: {i...}
, 
uimplies: b supposing a
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
nat: ℕ
, 
subtype_rel: A ⊆r B
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
less_than: a < b
, 
top: Top
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
nat_plus: ℕ+
, 
squash: ↓T
, 
sq_stable: SqStable(P)
Lemmas referenced : 
istype-le, 
decidable__le, 
equal_wf, 
int_formula_prop_le_lemma, 
intformle_wf, 
int_term_value_subtract_lemma, 
itermSubtract_wf, 
istype-less_than, 
less_than_wf, 
assert_wf, 
iff_weakening_uiff, 
assert-bnot, 
bool_subtype_base, 
bool_wf, 
bool_cases_sqequal, 
eqff_to_assert, 
pi1_wf, 
pi2_wf, 
assert_of_lt_int, 
eqtt_to_assert, 
subtract_wf, 
lt_int_wf, 
int_formula_prop_wf, 
int_term_value_constant_lemma, 
int_term_value_add_lemma, 
int_term_value_var_lemma, 
int_term_value_mul_lemma, 
int_formula_prop_eq_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
itermConstant_wf, 
itermAdd_wf, 
itermVar_wf, 
itermMultiply_wf, 
intformeq_wf, 
intformnot_wf, 
intformand_wf, 
full-omega-unsat, 
decidable__equal_int, 
int_upper_properties, 
nat_properties, 
not-prime-square, 
istype-nat, 
product_subtype_base, 
istype-int_upper, 
prime_wf, 
int_upper_wf, 
int_subtype_base, 
istype-int, 
le_wf, 
set_subtype_base, 
equal-wf-base, 
nat_wf, 
subtype_base_sq, 
not-prime-mult, 
add-subtract-cancel, 
istype-void, 
assoced_nelim, 
prime-mul, 
int_formula_prop_less_lemma, 
intformless_wf, 
decidable__lt, 
div_mul_cancel, 
mul-commutes, 
istype-false, 
upper_subtype_nat, 
decidable__prime, 
sq_stable_from_decidable
Rules used in proof : 
dependent_set_memberEquality_alt, 
setIsType, 
productIsType, 
addEquality, 
promote_hyp, 
applyLambdaEquality, 
equalityElimination, 
voidElimination, 
universeIsType, 
independent_pairFormation, 
Error :memTop, 
int_eqEquality, 
dependent_pairFormation_alt, 
approximateComputation, 
unionElimination, 
multiplyEquality, 
dependent_functionElimination, 
independent_functionElimination, 
sqequalBase, 
equalitySymmetry, 
equalityTransitivity, 
equalityIstype, 
productElimination, 
inhabitedIsType, 
because_Cache, 
rename, 
setElimination, 
independent_isectElimination, 
natural_numberEquality, 
lambdaEquality_alt, 
applyEquality, 
hypothesisEquality, 
baseClosed, 
closedConclusion, 
baseApply, 
sqequalRule, 
intEquality, 
setEquality, 
hypothesis, 
productEquality, 
cumulativity, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
instantiate, 
thin, 
cut, 
lambdaFormation_alt, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
isect_memberEquality_alt, 
dependent_pairEquality_alt, 
imageElimination, 
imageMemberEquality
Latex:
\mforall{}p:\{p:\{2...\}|  prime(p)\}  .  \mforall{}t:x:\mBbbN{}  \mtimes{}  y:\mBbbN{}  \mtimes{}  \{z:\mBbbN{}|  ((x  *  x)  +  (4  *  y  *  z))  =  p\}  .
    ((twosquareinv(t)  =  t)  {}\mRightarrow{}  (t  \msim{}  ə,  1,  (p  -  1)  \mdiv{}  4>))
Date html generated:
2020_05_19-PM-10_04_01
Last ObjectModification:
2019_12_26-AM-11_45_09
Theory : num_thy_1
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