Nuprl Lemma : rleq-infn
∀[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)))} . ∀c:ℝ.
c ≤ (infn(n;I) f) supposing ∀a:I^n. (c ≤ (f a))
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,
uimplies: b supposing a,
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,
infn: infn(n;I),
lelt: i ≤ j < k,
int_seg: {i..j-},
real-vec: ℝ^n,
subtype_rel: A ⊆r B,
top: Top,
guard: {T},
sq_type: SQType(T),
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B,
nat_plus: ℕ+,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
req-vec: req-vec(n;x;y),
real-vec-extend: a++z
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{}c:\mBbbR{}.
c \mleq{} (infn(n;I) f) supposing \mforall{}a:I\^{}n. (c \mleq{} (f a))
Date html generated:
2020_05_20-PM-00_39_29
Last ObjectModification:
2020_01_06-PM-10_03_35
Theory : reals
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