Nuprl Lemma : infn-rleq
∀[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)))} . ∀x:I^n. ((infn(n;I) f) ≤ (f x))
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,
req: x = y,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
less_than': less_than'(a;b),
interval-vec: I^n,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
infn: infn(n;I),
lelt: i ≤ j < k,
int_seg: {i..j-},
real-vec: ℝ^n,
top: Top,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
req-vec: req-vec(n;x;y),
guard: {T},
sq_type: SQType(T),
subtract: n - m,
cand: A c∧ B,
nat_plus: ℕ+,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bnot: ¬bb,
so_lambda: λ2x.t[x],
so_apply: x[s],
real-vec-extend: a++z,
inf: inf(A) = b,
lower-bound: lower-bound(A;b),
rrange: f[x](x∈I),
rset-member: x ∈ A,
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)))\} . \mforall{}x:I\^{}n.
((infn(n;I) f) \mleq{} (f x))
Date html generated:
2020_05_20-PM-00_40_02
Last ObjectModification:
2020_01_07-AM-00_51_30
Theory : reals
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