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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