Nuprl Lemma : rsqrt2-repels-rationals
∀n:ℕ+. ∀m:ℤ. ((r1/r(3 * n * n)) ≤ |rsqrt(r(2)) - (r(m)/r(n))|)
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
nat_plus: ℕ+,
all: ∀x:A. B[x],
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
nat: ℕ,
uiff: uiff(P;Q),
uimplies: b supposing a,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
true: True,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
guard: {T},
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_type: SQType(T),
rneq: x ≠ y,
rdiv: (x/y),
rge: x ≥ y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
int-to-real: r(n),
sq_stable: SqStable(P),
rsqrt: rsqrt(x),
rroot: rroot(i;x),
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
btrue: tt,
rroot-abs: rroot-abs(i;x),
fastexp: i^n,
efficient-exp-ext,
genrec: genrec,
subtract: n - m,
rabs: |x|,
absval: |i|,
iroot: iroot(n;x),
integer-nth-root-ext,
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
genrec-ap: genrec-ap,
divide: n ÷ m,
rmul: a * b,
rinv: rinv(x),
mu-ge: mu-ge(f;n),
lt_int: i <z j,
accelerate: accelerate(k;f),
imax: imax(a;b),
reg-seq-inv: reg-seq-inv(x),
le_int: i ≤z j,
bnot: ¬bb,
bfalse: ff,
reg-seq-mul: reg-seq-mul(x;y),
radd: a + b,
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
nil: [],
it: ⋅
Lemmas referenced :
istype-int,
nat_plus_wf,
rmul_wf,
rabs_wf,
rsub_wf,
rsqrt_wf,
radd_wf,
int-to-real_wf,
rleq-int,
istype-false,
rleq_wf,
subtract_wf,
absval_wf,
rminus_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermAdd_wf,
itermMinus_wf,
itermConstant_wf,
req-int,
squash_wf,
true_wf,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_minus_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
istype-nat,
req_functionality,
req_inversion,
rabs-rmul,
req_weakening,
req_transitivity,
rabs_functionality,
radd_functionality,
rminus_functionality,
rmul-int,
rminus-int,
rabs-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
rmul_functionality,
rsqrt_squared,
radd-int,
decidable__le,
req_wf,
real_wf,
subtype_rel_self,
iff_weakening_equal,
absval-positive,
int_subtype_base,
set_subtype_base,
less_than_wf,
intformand_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
irrational-sqrt-number-lemma,
istype-le,
int_seg_wf,
lelt_wf,
subtype_base_sq,
int_seg_properties,
int_seg_subtype_special,
int_seg_cases,
rleq_functionality,
rmul_preserves_rleq,
rdiv_wf,
rless-int,
mul_bounds_1b,
decidable__lt,
istype-less_than,
mul_nat_plus,
rless_wf,
rinv_wf2,
rneq_functionality,
rneq-int,
int_entire_a,
mul_nzero,
nat_plus_inc_int_nzero,
rinv_functionality2,
rinv-of-rmul,
rmul-rinv3,
rmul-rinv,
rminus-rdiv,
rabs-of-nonneg,
rleq_functionality_wrt_implies,
r-triangle-inequality,
rleq_weakening_equal,
rmul-nonneg-case1,
rsqrt_nonneg,
radd_functionality_wrt_rleq,
rmul_preserves_rleq2,
sq_stable__less_than,
radd-preserves-rleq,
rleq_weakening_rless,
zero-rleq-rabs,
rinv1,
rabs-difference-symmetry,
rmul-identity1,
rinv-as-rdiv,
mul_preserves_le,
nat_plus_subtype_nat,
rleq-int-fractions,
rsub_functionality_wrt_rleq,
rabs-as-rmax,
rmax_ub,
efficient-exp-ext,
integer-nth-root-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
applyEquality,
sqequalRule,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
hypothesisEquality,
minusEquality,
multiplyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
addEquality,
independent_isectElimination,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
promote_hyp,
instantiate,
universeEquality,
equalityIstype,
baseApply,
closedConclusion,
intEquality,
sqequalBase,
productIsType,
cumulativity,
hypothesis_subsumption,
inrFormation_alt,
dependent_set_memberFormation_alt,
computeAll,
inlFormation_alt
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}m:\mBbbZ{}. ((r1/r(3 * n * n)) \mleq{} |rsqrt(r(2)) - (r(m)/r(n))|)
Date html generated:
2019_10_30-AM-08_57_04
Last ObjectModification:
2019_04_03-AM-00_24_17
Theory : reals
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