Nuprl Lemma : Vieta-jumping-example2-corollary
∀k:ℤ. ∀a,b:ℕ. (((((a * a) + (b * b)) + 1) = (k * a * b) ∈ ℤ) ⇒ (a ≤ b) ⇒ (∃n:ℕ. (<a, b> = vexample(n;1;1) ∈ (ℤ × ℤ)))\000C)
Proof
Definitions occuring in Statement :
vexample: vexample(n;a;b),
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
pair: <a, b>,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
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],
nat: ℕ,
ge: i ≥ j ,
subtract: n - m,
vexample: vexample(n;a;b),
nat_plus: ℕ+,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
nequal: a ≠ b ∈ T ,
has-value: (a)↓,
fun_exp: f^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
compose: f o g
Lemmas referenced :
Vieta-jumping-example2,
subtype_base_sq,
int_subtype_base,
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,
set_subtype_base,
lelt_wf,
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,
le_wf,
nat_wf,
equal-wf-base,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
itermMultiply_wf,
int_term_value_mul_lemma,
int_entire,
exp_preserves_lt,
less_than_wf,
squash_wf,
true_wf,
exp2,
iff_weakening_equal,
mul_bounds_1a,
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
itermMinus_wf,
int_term_value_minus_lemma,
exp_preserves_le,
absval_wf,
absval_squared,
le_functionality,
multiply_functionality_wrt_le,
le_weakening,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
add-subtract-cancel,
value-type-has-value,
set-value-type,
int-value-type,
product_subtype_base,
ge_wf,
subtract-1-ge-0,
fun_exp_add_apply,
base_wf,
subtract-add-cancel,
fun_exp_com
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
productElimination,
natural_numberEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
unionElimination,
applyEquality,
Error :inhabitedIsType,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
because_Cache,
Error :productIsType,
hypothesis_subsumption,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
Error :functionIsType,
functionEquality,
productEquality,
Error :setIsType,
addEquality,
independent_pairEquality,
multiplyEquality,
imageElimination,
imageMemberEquality,
universeEquality,
minusEquality,
equalityElimination,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
promote_hyp,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
callbyvalueReduce,
sqleReflexivity,
intWeakElimination,
Error :functionIsTypeImplies,
sqequalIntensionalEquality,
functionExtensionality,
spreadEquality
Latex:
\mforall{}k:\mBbbZ{}. \mforall{}a,b:\mBbbN{}.
(((((a * a) + (b * b)) + 1) = (k * a * b)) {}\mRightarrow{} (a \mleq{} b) {}\mRightarrow{} (\mexists{}n:\mBbbN{}. (<a, b> = vexample(n;1;1))))
Date html generated:
2019_06_20-PM-02_43_47
Last ObjectModification:
2019_03_11-PM-06_29_12
Theory : num_thy_1
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