Nuprl Lemma : twosquare-type-finite
∀p:{p:{2...}| prime(p)} . finite(x:ℕ × y:ℕ × {z:ℕ| ((x * x) + (4 * y * z)) = p ∈ ℤ} )
Proof
Definitions occuring in Statement :
prime: prime(a),
finite: finite(T),
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,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
int_upper: {i...},
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
so_apply: x[s],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
spreadn: spread3,
so_lambda: λ2x.t[x],
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
nat: ℕ,
ext-eq: A ≡ B,
ge: i ≥ j ,
guard: {T},
top: Top,
sq_type: SQType(T),
iff: P ⇐⇒ Q
Lemmas referenced :
prime_wf,
istype-int_upper,
le_wf,
int_upper_wf,
int_subtype_base,
istype-int,
lelt_wf,
set_subtype_base,
equal-wf-base,
int_seg_wf,
finite-decidable-subset,
false_wf,
upper_subtype_nat,
nat_wf,
subtype_rel_set,
nsub_finite,
finite-product,
decidable__equal_int,
equal_wf,
decidable__squash,
istype-nat,
istype-false,
int_seg_subtype_nat,
istype-le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_upper_properties,
int_seg_properties,
int_term_value_mul_lemma,
itermMultiply_wf,
mul_preserves_le,
nat_properties,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
istype-less_than,
decidable__lt,
intformless_wf,
istype-void,
int_formula_prop_less_lemma,
subtype_base_sq,
not-prime-square,
finite_functionality_wrt_ext-eq
Rules used in proof :
hypothesisEquality,
rename,
setElimination,
universeIsType,
hypothesis,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
setIsType,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
productIsType,
independent_isectElimination,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
intEquality,
productElimination,
lambdaEquality_alt,
sqequalRule,
because_Cache,
productEquality,
dependent_functionElimination,
lambdaFormation,
independent_pairFormation,
lambdaEquality,
multiplyEquality,
addEquality,
sqequalBase,
inhabitedIsType,
equalityIstype,
voidElimination,
Error :memTop,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
independent_pairEquality,
isect_memberEquality_alt,
cumulativity,
instantiate,
setEquality
Latex:
\mforall{}p:\{p:\{2...\}| prime(p)\} . finite(x:\mBbbN{} \mtimes{} y:\mBbbN{} \mtimes{} \{z:\mBbbN{}| ((x * x) + (4 * y * z)) = p\} )
Date html generated:
2020_05_19-PM-10_04_09
Last ObjectModification:
2019_12_26-AM-11_44_54
Theory : num_thy_1
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