Nuprl Lemma : olympiad-problem-six
∀k:ℤ. ∀a,b:ℕ. ((((a * a) + (b * b)) = (k * ((a * b) + 1)) ∈ ℤ) ⇒ (∃n:ℤ. (k = (n * n) ∈ ℤ)))
Proof
Definitions occuring in Statement :
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
nat_plus: ℕ+
Lemmas referenced :
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
subtype_rel_self,
nat_properties,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
le_wf,
nat_wf,
equal-wf-base,
lelt_wf,
primrec-wf2,
istype-nat,
mul_bounds_1a,
mul_cancel_in_le,
add_nat_plus,
multiply_nat_wf,
nat_plus_properties,
mul_preserves_le,
square_non_neg,
mul_cancel_in_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
cumulativity,
intEquality,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
Error :inhabitedIsType,
sqequalBase,
Error :functionIsType,
functionEquality,
productEquality,
Error :setIsType,
addEquality,
multiplyEquality
Latex:
\mforall{}k:\mBbbZ{}. \mforall{}a,b:\mBbbN{}. ((((a * a) + (b * b)) = (k * ((a * b) + 1))) {}\mRightarrow{} (\mexists{}n:\mBbbZ{}. (k = (n * n))))
Date html generated:
2019_06_20-PM-02_43_07
Last ObjectModification:
2019_03_10-PM-02_32_31
Theory : num_thy_1
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