Nuprl Lemma : frs-non-dec-sum
∀p:ℝ List. (1 < ||p|| ⇒ frs-non-dec(p) ⇒ (Σ{p[i + 1] - p[i] | 0≤i≤||p|| - 2} = (last(p) - p[0])))
Proof
Definitions occuring in Statement :
frs-non-dec: frs-non-dec(L),
rsum: Σ{x[k] | n≤k≤m},
rsub: x - y,
req: x = y,
real: ℝ,
last: last(L),
select: L[n],
length: ||as||,
list: T List,
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtract: n - m,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
cons: [a / b],
le: A ≤ B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
so_apply: x[s],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
colength: colength(L),
sq_type: SQType(T),
subtype_rel: A ⊆r B,
length: ||as||,
list_ind: list_ind,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
frs-non-dec: frs-non-dec(L),
true: True,
cand: A c∧ B,
iff: P ⇐⇒ Q,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rev_implies: P ⇐ Q,
req_int_terms: t1 ≡ t2,
last: last(L),
bool: 𝔹,
unit: Unit,
bnot: ¬bb,
eq_int: (i =z j),
lt_int: i <z j
Latex:
\mforall{}p:\mBbbR{} List. (1 < ||p|| {}\mRightarrow{} frs-non-dec(p) {}\mRightarrow{} (\mSigma{}\{p[i + 1] - p[i] | 0\mleq{}i\mleq{}||p|| - 2\} = (last(p) - p[0])))
Date html generated:
2020_05_20-AM-11_36_19
Last ObjectModification:
2020_01_02-PM-01_59_58
Theory : reals
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