Nuprl Lemma : cover-seq-property-ext
∀[A,B:ℝ ⟶ ℙ].
∀d:r:ℝ ⟶ (A[r] + B[r]). ∀a,b:ℝ.
(A[a]
⇒ B[b]
⇒ (∀n:ℕ
(A[fst(cover-seq(d;a;b;n))]
∧ B[snd(cover-seq(d;a;b;n))]
∧ ((cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <a, (a + b/r(2))> ∈ (ℝ × ℝ))
∨ (cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <(a + b/r(2)), b> ∈ (ℝ × ℝ))))))
Proof
Definitions occuring in Statement :
cover-seq: cover-seq(d;a;b;n),
rdiv: (x/y),
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
spread: spread def,
pair: <a, b>,
product: x:A × B[x],
union: left + right,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
cover-seq-property,
subtract: n - m,
spreadn: spread3,
ifthenelse: if b then t else f fi
Lemmas referenced :
cover-seq-property
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A,B:\mBbbR{} {}\mrightarrow{} \mBbbP{}].
\mforall{}d:r:\mBbbR{} {}\mrightarrow{} (A[r] + B[r]). \mforall{}a,b:\mBbbR{}.
(A[a]
{}\mRightarrow{} B[b]
{}\mRightarrow{} (\mforall{}n:\mBbbN{}
(A[fst(cover-seq(d;a;b;n))]
\mwedge{} B[snd(cover-seq(d;a;b;n))]
\mwedge{} ((cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <a, (a + b/r(2))>)
\mvee{} (cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <(a + b/r(2)), b>)))))
Date html generated:
2017_10_03-AM-10_03_33
Last ObjectModification:
2017_07_06-AM-11_19_41
Theory : reals
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