Nuprl Lemma : rsum-split-last
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤m - 1} + x[m]) supposing n ≤ m
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
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,
uimplies: b supposing a,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtract: n - m,
uiff: uiff(P;Q),
top: Top,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
itermAdd: left (+) right,
itermVar: vvar,
itermSubtract: left (-) right,
int_term_ind: int_term_ind,
real_term_value: real_term_value(f;t),
req_int_terms: t1 ≡ t2,
itermConstant: "const"
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mSigma{}\{x[i] | n\mleq{}i\mleq{}m\} = (\mSigma{}\{x[i] | n\mleq{}i\mleq{}m - 1\} + x[m]) supposing n \mleq{} m
Date html generated:
2020_05_20-AM-11_11_33
Last ObjectModification:
2019_12_14-PM-00_56_59
Theory : reals
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