Nuprl Lemma : rminimum-cases
∀n,m:ℤ.
∀x:{n..m + 1-} ⟶ ℝ. (¬¬(∃a:{n..m + 1-}. ((rminimum(n;m;i.x[i]) = x[a]) ∧ (∀j:{n..m + 1-}. (x[a] ≤ x[j])))))
supposing n ≤ m
Proof
Definitions occuring in Statement :
rminimum: rminimum(n;m;k.x[k]),
rleq: x ≤ y,
req: x = y,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
rminimum: rminimum(n;m;k.x[k]),
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
guard: {T},
ge: i ≥ j ,
sq_type: SQType(T),
so_lambda: λ2x.t[x],
squash: ↓T,
less_than: a < b,
so_apply: x[s],
cand: A c∧ B,
top: Top,
lelt: i ≤ j < k,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
real: ℝ,
sq_stable: SqStable(P),
stable: Stable{P},
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y
Latex:
\mforall{}n,m:\mBbbZ{}.
\mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}
(\mneg{}\mneg{}(\mexists{}a:\{n..m + 1\msupminus{}\}. ((rminimum(n;m;i.x[i]) = x[a]) \mwedge{} (\mforall{}j:\{n..m + 1\msupminus{}\}. (x[a] \mleq{} x[j])))))
supposing n \mleq{} m
Date html generated:
2020_05_20-AM-11_15_15
Last ObjectModification:
2020_01_06-PM-01_15_48
Theory : reals
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