Nuprl Lemma : small-ctrex1-bounds
(r1/r(10))≤r(10^1141) * small-ctrex1()≤(r1/r(4))
Proof
Definitions occuring in Statement :
small-ctrex1: small-ctrex1(),
rdiv: (x/y),
rbetween: x≤y≤z,
rmul: a * b,
int-to-real: r(n),
fastexp: i^n,
natural_number: $n
Definitions unfolded in proof :
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
fastexp: i^n,
efficient-exp-ext,
true: True,
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
rational-approx: (x within 1/n),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
genrec: genrec,
so_lambda: λ2x.t[x],
so_apply: x[s],
rleq: x ≤ y,
rnonneg: rnonneg(x),
label: ...$L... t,
rat_term_to_real: rat_term_to_real(f;t),
rtermMultiply: left "*" right,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
rtermDivide: num "/" denom,
pi2: snd(t),
req_int_terms: t1 ≡ t2,
rbetween: x≤y≤z,
rge: x ≥ y,
rsub: x - y
Lemmas referenced :
fastexp_wf,
false_wf,
le_wf,
ctrex1-calculation,
rabs_wf,
rsub_wf,
small-ctrex1_wf,
rdiv_wf,
int-to-real_wf,
less_than_wf,
rless-int,
rless_wf,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
rational-approx-property,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
req_inversion,
int-rdiv-req,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
rleq_wf,
nat_plus_properties,
decidable__lt,
intformand_wf,
intformless_wf,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
decidable__equal_int,
rneq-int,
int_entire_a,
rmul_wf,
square-nonzero,
rneq_functionality,
rmul-int,
rmul_functionality,
rdiv_functionality,
rmul-neq-zero,
rabs-difference-bound-rleq,
radd_wf,
set_subtype_base,
radd_functionality,
rmul-rdiv2,
rmul_preserves_rleq2,
rleq-int,
le_witness_for_triv,
rmul-distrib2,
rmul_comm,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermVar_wf,
rtermDivide_wf,
rtermConstant_wf,
rleq-implies-rleq,
itermSubtract_wf,
itermAdd_wf,
req-iff-rsub-is-0,
uiff_transitivity,
req_transitivity,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
iff_weakening_uiff,
rmul-rsub-distrib,
rmul-one-both,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd-preserves-rleq,
rminus_wf,
radd_comm,
radd-rminus-assoc,
rleq-int-fractions,
istype-false,
radd-int-fractions,
efficient-exp-ext
Rules used in proof :
cut,
sqequalSubstitution,
sqequalRule,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
natural_numberEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
lambdaFormation,
hypothesis,
hypothesisEquality,
imageMemberEquality,
baseClosed,
applyEquality,
because_Cache,
multiplyEquality,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
addLevel,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
universeIsType,
inhabitedIsType,
lambdaFormation_alt,
closedConclusion,
setElimination,
rename,
inrFormation_alt,
applyLambdaEquality,
int_eqEquality,
equalityIstype,
sqequalBase,
baseApply,
isect_memberFormation_alt,
functionIsTypeImplies,
addEquality
Latex:
(r1/r(10))\mleq{}r(10\^{}1141) * small-ctrex1()\mleq{}(r1/r(4))
Date html generated:
2019_10_31-AM-06_23_20
Last ObjectModification:
2019_04_02-PM-10_38_53
Theory : reals_2
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