### Nuprl Lemma : uniform-partition-refines

`∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀k,m:ℕ+.  uniform-partition([a, b];m * k) refines uniform-partition([a, b];k)`

Proof

Definitions occuring in Statement :  partition-refines: `P refines Q` uniform-partition: `uniform-partition(I;k)` rccint: `[l, u]` rleq: `x ≤ y` real: `ℝ` nat_plus: `ℕ+` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` multiply: `n * m`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` uall: `∀[x:A]. B[x]` sq_stable: `SqStable(P)` squash: `↓T` partition-refines: `P refines Q` frs-refines: `frs-refines(p;q)` l_all: `(∀x∈L.P[x])` uniform-partition: `uniform-partition(I;k)` top: `Top` nat: `ℕ` nat_plus: `ℕ+` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` prop: `ℙ` int_seg: `{i..j-}` i-finite: `i-finite(I)` rccint: `[l, u]` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` rneq: `x ≠ y` rev_implies: `P `` Q` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` real: `ℝ` partition: `partition(I)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` l_exists: `(∃x∈L. P[x])` le: `A ≤ B` uiff: `uiff(P;Q)` subtract: `n - m` less_than': `less_than'(a;b)` less_than: `a < b` cand: `A c∧ B` rless: `x < y` sq_exists: `∃x:{A| B[x]}` rev_uimplies: `rev_uimplies(P;Q)` rdiv: `(x/y)` itermConstant: `"const"` req_int_terms: `t1 ≡ t2`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality setElimination rename hypothesis productElimination independent_functionElimination isectElimination sqequalRule imageMemberEquality baseClosed imageElimination isect_memberEquality voidElimination voidEquality dependent_set_memberEquality natural_numberEquality functionEquality intEquality because_Cache unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality independent_pairFormation computeAll addEquality inrFormation applyEquality multiplyEquality baseApply closedConclusion minusEquality equalityTransitivity equalitySymmetry universeEquality inlFormation

Latex:
\mforall{}a:\mBbbR{}.  \mforall{}b:\{b:\mBbbR{}|  a  \mleq{}  b\}  .  \mforall{}k,m:\mBbbN{}\msupplus{}.    uniform-partition([a,  b];m  *  k)  refines  uniform-partition([a,  b];k\000C)

Date html generated: 2017_10_03-AM-09_46_40
Last ObjectModification: 2017_07_28-AM-07_59_45

Theory : reals

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