Nuprl Lemma : rmaximum-split
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[k:ℤ].
(rmaximum(n;m;i.x[i]) = rmax(rmaximum(n;k;i.x[i]);rmaximum(k + 1;m;i.x[i]))) supposing (k < m and (n ≤ k))
Proof
Definitions occuring in Statement :
rmaximum: rmaximum(n;m;k.x[k]),
rmax: rmax(x;y),
req: x = y,
real: ℝ,
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rmaximum: rmaximum(n;m;k.x[k]),
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
guard: {T},
ge: i ≥ j ,
sq_type: SQType(T),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mforall{}[k:\mBbbZ{}].
(rmaximum(n;m;i.x[i]) = rmax(rmaximum(n;k;i.x[i]);rmaximum(k + 1;m;i.x[i]))) supposing
(k < m and
(n \mleq{} k))
Date html generated:
2020_05_20-AM-11_14_11
Last ObjectModification:
2019_12_14-PM-00_56_15
Theory : reals
Home
Index