Nuprl Lemma : irrational-sqrt-number-lemma
∀a:ℤ. ∀b:ℕ+. ∀n:ℕ. (((a * a) = (n * b * b) ∈ ℤ) ⇒ (∃m:ℕn + 1. ((m * m) = n ∈ ℤ)))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
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],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
nat_plus: ℕ+,
divides: b | a,
exists: ∃x:A. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
true: True,
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
cand: A c∧ B,
sq_stable: SqStable(P),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
equal-wf-base-T,
int_subtype_base,
nat_wf,
nat_plus_wf,
divides_wf,
prime_wf,
prime_divs_prod,
subtype_base_sq,
nat_plus_properties,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
equal-wf-base,
decidable__le,
intformand_wf,
intformle_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
nat_plus_subtype_nat,
int_seg_properties,
int_seg_wf,
subtract_wf,
int_seg_subtype,
false_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
all_wf,
exists_wf,
equal_wf,
decidable__lt,
lelt_wf,
set_wf,
less_than_wf,
primrec-wf2,
itermAdd_wf,
int_term_value_add_lemma,
mul-swap,
mul-commutes,
mul-associates,
one-mul,
absval_wf,
absval-non-neg,
mul_preserves_le,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma,
absval_unfold,
mul_bounds_1a,
mul-distributes,
mul-distributes-right,
add-associates,
add-swap,
add-commutes,
two-mul,
absval_pos,
iff_weakening_equal,
squash_wf,
true_wf,
absval_mul,
least-factor_wf,
subtype_rel_sets,
sq_stable__le,
sq_stable_from_decidable,
decidable__prime,
decidable__divides_ext,
not-lt-2,
add_functionality_wrt_le,
zero-add,
le-add-cancel,
mul_cancel_in_le,
int_upper_properties,
mul_cancel_in_lt,
int_upper_subtype_nat,
mul_cancel_in_eq,
mul_nzero,
nequal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
hypothesis,
multiplyEquality,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
natural_numberEquality,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_pairFormation,
addLevel,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
functionEquality,
addEquality,
minusEquality,
equalityElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
imageMemberEquality,
imageElimination,
equalityUniverse,
universeEquality,
setEquality,
productEquality
Latex:
\mforall{}a:\mBbbZ{}. \mforall{}b:\mBbbN{}\msupplus{}. \mforall{}n:\mBbbN{}. (((a * a) = (n * b * b)) {}\mRightarrow{} (\mexists{}m:\mBbbN{}n + 1. ((m * m) = n)))
Date html generated:
2017_10_03-AM-11_59_04
Last ObjectModification:
2017_07_28-AM-08_29_56
Theory : reals
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