Nuprl Lemma : cantor-middle-third-lemma
∀x,a,b:ℝ. ((x ∈ [(2 * a + b)/3, b]) ∨ (x ∈ [a, (a + 2 * b)/3])) supposing ((x ∈ [a, b]) and (a < b))
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
rless: x < y,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
radd: a + b,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
i-member: r ∈ I,
rccint: [l, u],
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
or: P ∨ Q,
cand: A c∧ B,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
sq_stable: SqStable(P)
Lemmas referenced :
rless-cases,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
radd_wf,
int-rmul_wf,
rleq_weakening_rless,
rleq_wf,
i-member_wf,
rccint_wf,
rless_wf,
real_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rmul_wf,
rmul_preserves_rless,
rinv_wf2,
rless_functionality,
int-rdiv-req,
rdiv_functionality,
radd_functionality,
req_weakening,
int-rmul-req,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv3,
rmul_functionality,
radd-preserves-rless,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
rless-implies-rless,
rsub_wf,
sq_stable__rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
productElimination,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
baseClosed,
hypothesisEquality,
because_Cache,
unionElimination,
inlFormation,
independent_pairFormation,
productEquality,
inrFormation,
imageMemberEquality,
computeAll,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
lemma_by_obid,
imageElimination
Latex:
\mforall{}x,a,b:\mBbbR{}.
((x \mmember{} [(2 * a + b)/3, b]) \mvee{} (x \mmember{} [a, (a + 2 * b)/3])) supposing ((x \mmember{} [a, b]) and (a < b))
Date html generated:
2017_10_03-AM-09_47_48
Last ObjectModification:
2017_07_28-AM-08_00_11
Theory : reals
Home
Index