Nuprl Lemma : rminimum-split
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[k:ℤ].
(rminimum(n;m;i.x[i]) = rmin(rminimum(n;k;i.x[i]);rminimum(k + 1;m;i.x[i]))) supposing (k < m and (n ≤ k))
Proof
Definitions occuring in Statement :
rminimum: rminimum(n;m;k.x[k]),
rmin: rmin(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,
rminimum: rminimum(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),
top: Top,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mforall{}[k:\mBbbZ{}].
(rminimum(n;m;i.x[i]) = rmin(rminimum(n;k;i.x[i]);rminimum(k + 1;m;i.x[i]))) supposing
(k < m and
(n \mleq{} k))
Date html generated:
2020_05_20-AM-11_14_57
Last ObjectModification:
2019_12_14-PM-00_55_31
Theory : reals
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