Nuprl Lemma : cantor-to-interval-onto-common
∀a,b:ℝ.
∀x,y:ℝ.
((x ∈ [a, b])
⇒ (y ∈ [a, b])
⇒ (∀n:ℕ
((|x - y| ≤ (2^n * b - a)/6 * 3^n)
⇒ (∃f,g:ℕ ⟶ 𝔹
(((cantor-to-interval(a;b;f) = x) ∧ (cantor-to-interval(a;b;g) = y)) ∧ (f = g ∈ (ℕn ⟶ 𝔹)))))))
supposing a < b
Proof
Definitions occuring in Statement :
cantor-to-interval: cantor-to-interval(a;b;f)
,
rccint: [l, u]
,
i-member: r ∈ I
,
rleq: x ≤ y
,
rless: x < y
,
rabs: |x|
,
int-rdiv: (a)/k1
,
int-rmul: k1 * a
,
rsub: x - y
,
req: x = y
,
real: ℝ
,
exp: i^n
,
int_seg: {i..j-}
,
nat: ℕ
,
bool: 𝔹
,
uimplies: b supposing a
,
all: ∀x:A. B[x]
,
exists: ∃x:A. B[x]
,
implies: P
⇒ Q
,
and: P ∧ Q
,
function: x:A ⟶ B[x]
,
multiply: n * m
,
natural_number: $n
,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
uimplies: b supposing a
,
implies: P
⇒ Q
,
member: t ∈ T
,
uall: ∀[x:A]. B[x]
,
int_nzero: ℤ-o
,
true: True
,
nequal: a ≠ b ∈ T
,
not: ¬A
,
sq_type: SQType(T)
,
guard: {T}
,
false: False
,
nat: ℕ
,
rless: x < y
,
sq_exists: ∃x:A [B[x]]
,
nat_plus: ℕ+
,
ge: i ≥ j
,
decidable: Dec(P)
,
or: P ∨ Q
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
and: P ∧ Q
,
prop: ℙ
,
subtype_rel: A ⊆r B
,
int_seg: {i..j-}
,
lelt: i ≤ j < k
,
le: A ≤ B
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
less_than: a < b
,
squash: ↓T
,
pi1: fst(t)
,
pi2: snd(t)
,
less_than': less_than'(a;b)
,
cand: A c∧ B
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
uiff: uiff(P;Q)
,
subtract: n - m
,
sq_stable: SqStable(P)
,
cantor-interval: cantor-interval(a;b;f;n)
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
bfalse: ff
,
bnot: ¬bb
,
ifthenelse: if b then t else f fi
,
assert: ↑b
,
rneq: x ≠ y
,
rev_uimplies: rev_uimplies(P;Q)
,
rge: x ≥ y
,
top: Top
,
real: ℝ
,
rdiv: (x/y)
,
req_int_terms: t1 ≡ t2
,
rccint: [l, u]
,
i-member: r ∈ I
,
lt_int: i <z j
Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}x,y:\mBbbR{}.
((x \mmember{} [a, b])
{}\mRightarrow{} (y \mmember{} [a, b])
{}\mRightarrow{} (\mforall{}n:\mBbbN{}
((|x - y| \mleq{} (2\^{}n * b - a)/6 * 3\^{}n)
{}\mRightarrow{} (\mexists{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}
(((cantor-to-interval(a;b;f) = x) \mwedge{} (cantor-to-interval(a;b;g) = y)) \mwedge{} (f = g))))))
supposing a < b
Date html generated:
2020_05_20-PM-00_09_15
Last ObjectModification:
2020_01_02-PM-01_56_58
Theory : reals
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