### Nuprl Lemma : map_select

`∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[as:A List]. ∀[n:ℕ||as||].  (map(f;as)[n] = (f as[n]) ∈ B)`

Proof

Definitions occuring in Statement :  select: `L[n]` length: `||as||` map: `map(f;as)` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` uimplies: `b supposing a` sq_stable: `SqStable(P)` implies: `P `` Q` lelt: `i ≤ j < k` and: `P ∧ Q` squash: `↓T` top: `Top` so_apply: `x[s]` all: `∀x:A. B[x]` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` prop: `ℙ` false: `False` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` not: `¬A` uiff: `uiff(P;Q)` le: `A ≤ B` less_than': `less_than'(a;b)` subtract: `n - m` nat: `ℕ`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality cumulativity hypothesisEquality hypothesis because_Cache functionEquality universeEquality isect_memberFormation sqequalRule isect_memberEquality axiomEquality lambdaEquality functionExtensionality applyEquality setElimination rename independent_isectElimination independent_functionElimination productElimination imageMemberEquality baseClosed imageElimination voidElimination voidEquality dependent_functionElimination lambdaFormation addEquality unionElimination equalityTransitivity equalitySymmetry independent_pairFormation minusEquality intEquality dependent_set_memberEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  B].  \mforall{}[as:A  List].  \mforall{}[n:\mBbbN{}||as||].    (map(f;as)[n]  =  (f  as[n]))

Date html generated: 2017_04_14-AM-08_38_49
Last ObjectModification: 2017_02_27-PM-03_29_52

Theory : list_0

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