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),
natural_number: $n,
fastexp: i^n
Definitions unfolded in proof :
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
le: A ≤ B,
nat: ℕ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
true: True,
efficient-exp-ext,
fastexp: i^n,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rational-approx: (x within 1/n),
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
label: ...$L... t,
real: ℝ,
rnonneg: rnonneg(x),
rleq: x ≤ y,
rbetween: x≤y≤z,
rge: x ≥ y,
rsub: x - y
Lemmas referenced :
efficient-exp-ext,
radd-int-fractions,
rleq-int-fractions,
radd-preserves-rleq,
rminus_wf,
radd_comm,
radd-rminus-assoc,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rmul-rsub-distrib,
rmul-one-both,
rmul-assoc,
req_functionality,
rmul_functionality,
req_transitivity,
uiff_transitivity,
rmul_preserves_rleq2,
rleq-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
less_than'_wf,
rmul-distrib2,
rmul_comm,
rmul-rdiv-cancel,
req-iff-bdd-diff,
trivial-bdd-diff,
rmul-rdiv2,
radd_functionality,
radd_wf,
rabs-difference-bound-rleq,
rmul-neq-zero,
rmul-int,
rdiv_functionality,
rmul_wf,
equal_wf,
int_entire_a,
rneq-int,
decidable__equal_int,
nat_plus_wf,
real_wf,
rleq_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
int-rdiv-req,
req_inversion,
req_weakening,
rsub_functionality,
rabs_functionality,
rleq_functionality,
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,
fastexp_wf,
false_wf,
le_wf
Rules used in proof :
cut,
baseClosed,
imageMemberEquality,
introduction,
hypothesisEquality,
hypothesis,
lambdaFormation,
dependent_set_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
natural_numberEquality,
independent_pairFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalRule,
sqequalSubstitution,
voidElimination,
equalitySymmetry,
equalityTransitivity,
intEquality,
cumulativity,
instantiate,
addLevel,
independent_functionElimination,
productElimination,
dependent_functionElimination,
inrFormation,
independent_isectElimination,
multiplyEquality,
because_Cache,
applyEquality,
equalityEquality,
computeAll,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
setEquality,
lambdaEquality,
rename,
setElimination,
promote_hyp,
levelHypothesis,
axiomEquality,
minusEquality,
independent_pairEquality,
isect_memberFormation,
functionEquality,
impliesFunctionality,
addEquality
Latex:
(r1/r(10))\mleq{}r(10\^{}1141) * small-ctrex1()\mleq{}(r1/r(4))
Date html generated:
2016_05_18-AM-10_48_58
Last ObjectModification:
2016_01_17-AM-00_37_10
Theory : reals
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