### Nuprl Lemma : Riemann-sum-alt-req

`∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ].`
`  ((∀x,y:ℝ.  ((x ∈ [a, b]) `` (x = y) `` ((f y) = (f x))))`
`  `` (∀[k:ℕ+]. (Riemann-sum-alt(f;a;b;k) = Riemann-sum(f;a;b;k))))`

Proof

Definitions occuring in Statement :  Riemann-sum-alt: `Riemann-sum-alt(f;a;b;k)` Riemann-sum: `Riemann-sum(f;a;b;k)` rfun: `I ⟶ℝ` rccint: `[l, u]` i-member: `r ∈ I` rleq: `x ≤ y` req: `x = y` real: `ℝ` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a`
Definitions unfolded in proof :  cand: `A c∧ B` rccint: `[l, u]` i-member: `r ∈ I` top: `Top` not: `¬A` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` decidable: `Dec(P)` rev_implies: `P `` Q` or: `P ∨ Q` guard: `{T}` rneq: `x ≠ y` squash: `↓T` exists: `∃x:A. B[x]` real: `ℝ` so_apply: `x[s]` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` uimplies: `b supposing a` has-value: `(a)↓` Riemann-sum-alt: `Riemann-sum-alt(f;a;b;k)` and: `P ∧ Q` iff: `P `⇐⇒` Q` all: `∀x:A. B[x]` implies: `P `` Q` sq_stable: `SqStable(P)` prop: `ℙ` member: `t ∈ T` uall: `∀[x:A]. B[x]` rfun: `I ⟶ℝ` rev_uimplies: `rev_uimplies(P;Q)` uiff: `uiff(P;Q)` lelt: `i ≤ j < k` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` le: `A ≤ B` rnonneg: `rnonneg(x)` rleq: `x ≤ y` rge: `x ≥ y` subtract: `n - m` less_than: `a < b` partition-sum: `partition-sum(f;x;p)` default-partition-choice: `default-partition-choice(p)` has-valueall: `has-valueall(a)` callbyvalueall: callbyvalueall Riemann-sum: `Riemann-sum(f;a;b;k)` sq_type: `SQType(T)` nat: `ℕ` full-partition: `full-partition(I;p)` uniform-partition: `uniform-partition(I;k)` pointwise-req: `x[k] = y[k] for k ∈ [n,m]` cons: `[a / b]` select: `L[n]` true: `True` assert: `↑b` bnot: `¬bb` bfalse: `ff` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` rsub: `x - y`
Rules used in proof :  imageElimination computeAll voidEquality voidElimination isect_memberEquality int_eqEquality dependent_pairFormation unionElimination inrFormation baseClosed imageMemberEquality independent_pairFormation functionEquality natural_numberEquality lambdaEquality intEquality independent_isectElimination callbyvalueReduce sqequalRule productElimination dependent_functionElimination introduction independent_functionElimination because_Cache hypothesis dependent_set_memberEquality hypothesisEquality isectElimination sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution sqequalHypSubstitution lemma_by_obid cut rename thin setElimination isect_memberFormation lambdaFormation applyEquality multiplyEquality addEquality equalitySymmetry equalityTransitivity axiomEquality minusEquality independent_pairEquality instantiate setEquality cumulativity universeEquality equalityEquality promote_hyp equalityElimination

Latex:
\mforall{}[a:\mBbbR{}].  \mforall{}[b:\{b:\mBbbR{}|  a  \mleq{}  b\}  ].  \mforall{}[f:[a,  b]  {}\mrightarrow{}\mBbbR{}].
((\mforall{}x,y:\mBbbR{}.    ((x  \mmember{}  [a,  b])  {}\mRightarrow{}  (x  =  y)  {}\mRightarrow{}  ((f  y)  =  (f  x))))
{}\mRightarrow{}  (\mforall{}[k:\mBbbN{}\msupplus{}].  (Riemann-sum-alt(f;a;b;k)  =  Riemann-sum(f;a;b;k))))

Date html generated: 2016_05_18-AM-10_45_38
Last ObjectModification: 2016_01_17-AM-00_42_13

Theory : reals

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