Nuprl Lemma : twosquareinv-involution
∀p:{p:{2...}| prime(p)} . ∀t:x:ℕ × y:ℕ × {z:ℕ| ((x * x) + (4 * y * z)) = p ∈ ℤ} . (twosquareinv(twosquareinv(t)) ~ t)
Proof
Definitions occuring in Statement :
twosquareinv: twosquareinv(t),
prime: prime(a),
int_upper: {i...},
nat: ℕ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
int_upper: {i...},
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
guard: {T},
sq_type: SQType(T),
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
not: ¬A,
twosquareinv: twosquareinv(t),
spreadn: spread3,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
subtract: n - m
Lemmas referenced :
istype-int_upper,
prime_wf,
int_upper_wf,
int_subtype_base,
le_wf,
set_subtype_base,
istype-int,
istype-nat,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformand_wf,
decidable__equal_int,
subtype_base_sq,
istype-le,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
int_upper_properties,
nat_properties,
not-prime-mult,
not-prime-square,
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,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
not_functionality_wrt_uiff,
assert_wf,
less_than_wf,
nat_wf,
product_subtype_base,
add-associates,
minus-add,
minus-one-mul,
minus-one-mul-top,
two-mul,
add-swap,
mul-distributes-right,
add-commutes,
zero-mul,
add-zero,
add-mul-special,
zero-add,
minus-minus
Rules used in proof :
universeIsType,
equalitySymmetry,
sqequalBase,
independent_isectElimination,
natural_numberEquality,
lambdaEquality_alt,
intEquality,
isectElimination,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
baseClosed,
closedConclusion,
baseApply,
sqequalRule,
equalityIstype,
setIsType,
because_Cache,
extract_by_obid,
introduction,
productIsType,
hypothesis,
cut,
rename,
setElimination,
thin,
productElimination,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
int_eqEquality,
cumulativity,
instantiate,
multiplyEquality,
voidElimination,
Error :memTop,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
equalityTransitivity,
inhabitedIsType,
lambdaFormation,
equalityElimination,
addEquality,
dependent_pairFormation,
lambdaEquality,
isect_memberEquality,
voidEquality,
imageMemberEquality,
imageElimination,
promote_hyp,
productEquality,
independent_pairEquality,
minusEquality
Latex:
\mforall{}p:\{p:\{2...\}| prime(p)\} . \mforall{}t:x:\mBbbN{} \mtimes{} y:\mBbbN{} \mtimes{} \{z:\mBbbN{}| ((x * x) + (4 * y * z)) = p\} .
(twosquareinv(twosquareinv(t)) \msim{} t)
Date html generated:
2020_05_19-PM-10_03_54
Last ObjectModification:
2019_12_26-AM-11_44_29
Theory : num_thy_1
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