Nuprl Lemma : rsqrt-irrational
∀n:ℕ. (irrational(rsqrt(r(n))) ∨ (∃m:ℕn + 1. ((m * m) = n ∈ ℤ)))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
irrational: irrational(x),
rsqrt: rsqrt(x),
int-to-real: r(n),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: 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],
member: t ∈ T,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
so_apply: x[s],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
exists: ∃x:A. B[x],
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
irrational: irrational(x),
nat_plus: ℕ+,
rneq: x ≠ y,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
rge: x ≥ y,
less_than: a < b
Lemmas referenced :
decidable__exists_int_seg,
equal_wf,
int_seg_wf,
decidable__equal_int,
irrational_wf,
rsqrt_wf,
rleq-int,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
int-to-real_wf,
rleq_wf,
nat_plus_wf,
exists_wf,
nat_wf,
decidable__lt,
rless_wf,
nat_plus_properties,
rdiv_wf,
rless-int,
intformless_wf,
int_formula_prop_less_lemma,
rmul_wf,
rless_functionality,
req_weakening,
rmul-int,
rmul_functionality,
sq_stable__less_than,
real_wf,
rmul_preserves_rless,
rinv_wf2,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv3,
rinv-as-rdiv,
rmul_preserves_rleq,
rleq_functionality,
rsqrt_squared,
rsqrt_nonneg,
req_wf,
req_inversion,
rless_transitivity1,
rleq_weakening,
rmul-is-positive,
radd_wf,
rsub_wf,
rmul_over_rminus,
rmul-distrib,
rmul_comm,
radd-assoc,
radd-ac,
radd_comm,
radd-rminus-assoc,
radd-rminus-both,
radd_functionality,
radd-zero-both,
rminus_wf,
radd-preserves-rless,
rless-implies-rless,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rless_functionality_wrt_implies,
radd-int,
int_term_value_add_lemma,
itermAdd_wf,
rless_transitivity2,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
intformeq_wf,
irrational-sqrt-number-lemma,
rmul-rdiv-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
intEquality,
multiplyEquality,
because_Cache,
independent_functionElimination,
unionElimination,
inrFormation,
productElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
applyEquality,
inlFormation,
addLevel,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setEquality,
productEquality,
levelHypothesis,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}. (irrational(rsqrt(r(n))) \mvee{} (\mexists{}m:\mBbbN{}n + 1. ((m * m) = n)))
Date html generated:
2017_10_03-AM-11_59_31
Last ObjectModification:
2017_07_28-AM-08_30_14
Theory : reals
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