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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