Nuprl Lemma : rsum_unroll
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ].
(Σ{x[k] | n≤k≤m} = if m <z n then r0 if (m =z n) then x[n] else Σ{x[k] | n≤k≤m - 1} + x[m] fi )
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rsum: Σ{x[k] | n≤k≤m},
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
nequal: a ≠ b ∈ T ,
has-value: (a)↓,
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
top: Top,
from-upto: [n, m),
subtype_rel: A ⊆r B,
istype: istype(T),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
true: True,
req_int_terms: t1 ≡ t2
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
(\mSigma{}\{x[k] | n\mleq{}k\mleq{}m\} = if m <z n then r0 if (m =\msubz{} n) then x[n] else \mSigma{}\{x[k] | n\mleq{}k\mleq{}m - 1\} + x[m] fi )
Date html generated:
2020_05_20-AM-11_10_37
Last ObjectModification:
2019_12_14-PM-00_55_13
Theory : reals
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