Nuprl Lemma : twosquareinv_wf
∀[p:{2...}]. ∀[t:x:ℕ × y:ℕ × {z:ℕ| ((x * x) + (4 * y * z)) = p ∈ ℤ} ].
(twosquareinv(t) ∈ x:ℕ × y:ℕ × {z:ℕ| ((x * x) + (4 * y * z)) = p ∈ ℤ} )
Proof
Definitions occuring in Statement :
twosquareinv: twosquareinv(t),
int_upper: {i...},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
twosquareinv: twosquareinv(t),
spreadn: spread3,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
lt_int_wf,
subtract_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
sq_stable__equal,
nat_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
equal_wf,
nat_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_functionality_wrt_uiff,
assert_wf,
less_than_wf,
int_upper_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
productElimination,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_pairEquality,
dependent_set_memberEquality,
addEquality,
intEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
dependent_functionElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
multiplyEquality,
setEquality,
productEquality,
promote_hyp,
instantiate,
cumulativity,
axiomEquality
Latex:
\mforall{}[p:\{2...\}]. \mforall{}[t:x:\mBbbN{} \mtimes{} y:\mBbbN{} \mtimes{} \{z:\mBbbN{}| ((x * x) + (4 * y * z)) = p\} ].
(twosquareinv(t) \mmember{} x:\mBbbN{} \mtimes{} y:\mBbbN{} \mtimes{} \{z:\mBbbN{}| ((x * x) + (4 * y * z)) = p\} )
Date html generated:
2019_06_20-PM-02_41_02
Last ObjectModification:
2018_09_24-PM-02_52_44
Theory : num_thy_1
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