### Nuprl Lemma : filter_type

`∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[l:T List].  (filter(P;l) ∈ {x:T| ↑(P x)}  List)`

Proof

Definitions occuring in Statement :  filter: `filter(P;l)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` subtype_rel: `A ⊆r B` or: `P ∨ Q` top: `Top` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` squash: `↓T` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)` and: `P ∧ Q` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination lambdaEquality dependent_functionElimination axiomEquality equalityTransitivity equalitySymmetry cumulativity applyEquality because_Cache unionElimination isect_memberEquality voidEquality setEquality functionExtensionality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality intEquality instantiate equalityElimination functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[l:T  List].    (filter(P;l)  \mmember{}  \{x:T|  \muparrow{}(P  x)\}    List)

Date html generated: 2017_04_14-AM-08_51_27
Last ObjectModification: 2017_02_27-PM-03_36_36

Theory : list_0

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