Nuprl Lemma : infn-property
∀I:{I:Interval| icompact(I)} . ∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} . ∀e:{e:ℝ| r0 < e} \000C.
∃x:I^n. ((f x) ≤ ((infn(n;I) f) + e))
Proof
Definitions occuring in Statement :
infn: infn(n;I),
interval-vec: I^n,
req-vec: req-vec(n;x;y),
icompact: icompact(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
interval-vec: I^n,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
squash: ↓T,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
infn: infn(n;I),
lelt: i ≤ j < k,
int_seg: {i..j-},
real-vec: ℝ^n,
top: Top,
req_int_terms: t1 ≡ t2,
sq_type: SQType(T),
subtract: n - m,
cand: A c∧ B,
nat_plus: ℕ+,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
req-vec: req-vec(n;x;y),
real-vec-extend: a++z,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bnot: ¬bb,
inf: inf(A) = b,
rneq: x ≠ y,
less_than: a < b,
true: True,
rrange: f[x](x∈I),
rset-member: x ∈ A,
rdiv: (x/y),
rless: x < y,
sq_exists: ∃x:A [B[x]],
rge: x ≥ y
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}n:\mBbbN{}. \mforall{}f:\{f:I\^{}n {}\mrightarrow{} \mBbbR{}| \mforall{}a,b:I\^{}n. (req-vec(n;a;b) {}\mRightarrow{} ((f a) = (f b)))\}\000C .
\mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}x:I\^{}n. ((f x) \mleq{} ((infn(n;I) f) + e))
Date html generated:
2020_05_20-PM-00_40_32
Last ObjectModification:
2020_01_06-PM-11_01_55
Theory : reals
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