Nuprl Lemma : cantor-common-middle-third-lemma
∀x,y,a,b:ℝ.
(((x ∈ [(2 * a + b)/3, b]) ∧ (y ∈ [(2 * a + b)/3, b]))
∨ ((x ∈ [a, (a + 2 * b)/3]) ∧ (y ∈ [a, (a + 2 * b)/3]))) supposing
(((x ∈ [a, b]) ∧ (y ∈ [a, b]) ∧ (|x - y| ≤ (b - a/r(6)))) and
(a < b))
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
int-rdiv: (a)/k1,
int-rmul: k1 * a,
rsub: x - y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
rdiv: (x/y),
i-member: r ∈ I,
rccint: [l, u],
cand: A c∧ B,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
int-to-real_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rsub_wf,
rless_functionality,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
radd_wf,
rdiv_wf,
rless-int,
rless_wf,
int-rmul_wf,
int-rdiv-req,
i-member_wf,
rccint_wf,
rleq_wf,
rabs_wf,
real_wf,
rmul_wf,
rmul_preserves_rless,
rinv_wf2,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
real_term_value_add_lemma,
real_term_value_mul_lemma,
req_weakening,
rdiv_functionality,
radd_functionality,
int-rmul-req,
rmul-int,
req_transitivity,
rmul_functionality,
rmul-rinv3,
rless-cases,
rless_transitivity2,
rleq_weakening_rless,
sq_stable__rleq,
rabs-bounds,
rleq-implies-rleq,
equal_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rsub_functionality_wrt_rleq,
req-int,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
req_inversion,
rless_transitivity1,
rleq_weakening,
rmul_preserves_rleq,
rminus_wf,
rleq_functionality,
rsub_functionality,
req_functionality,
rinv-of-rmul,
itermMinus_wf,
real_term_value_minus_lemma,
rminus_functionality,
rabs-difference-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
independent_isectElimination,
natural_numberEquality,
hypothesis,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
productElimination,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
baseClosed,
inrFormation,
independent_pairFormation,
imageMemberEquality,
productEquality,
multiplyEquality,
unionElimination,
inlFormation,
imageElimination,
setElimination,
rename,
dependent_pairFormation
Latex:
\mforall{}x,y,a,b:\mBbbR{}.
(((x \mmember{} [(2 * a + b)/3, b]) \mwedge{} (y \mmember{} [(2 * a + b)/3, b]))
\mvee{} ((x \mmember{} [a, (a + 2 * b)/3]) \mwedge{} (y \mmember{} [a, (a + 2 * b)/3]))) supposing
(((x \mmember{} [a, b]) \mwedge{} (y \mmember{} [a, b]) \mwedge{} (|x - y| \mleq{} (b - a/r(6)))) and
(a < b))
Date html generated:
2017_10_03-AM-09_48_17
Last ObjectModification:
2017_07_28-AM-08_00_22
Theory : reals
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