### Nuprl Lemma : filter_is_nil

`∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].  filter(P;L) ~ [] supposing (∀x∈L.¬↑(P x))`

Proof

Definitions occuring in Statement :  l_all: `(∀x∈L.P[x])` filter: `filter(P;l)` nil: `[]` list: `T List` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` sqequal: `s ~ t`
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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` 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: `⋅` sq_type: `SQType(T)` less_than: `a < b` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf l_all_wf not_wf assert_wf l_member_wf equal-wf-T-base nat_wf colength_wf_list list-cases filter_nil_lemma l_all_wf_nil product_subtype_list spread_cons_lemma sq_stable__le le_antisymmetry_iff add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel decidable__le false_wf not-le-2 condition-implies-le minus-add minus-one-mul minus-one-mul-top add-commutes le_wf equal_wf subtract_wf not-ge-2 less-iff-le minus-minus add-swap subtype_base_sq set_subtype_base int_subtype_base filter_cons_lemma l_all_cons bool_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot cons_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality sqequalAxiom cumulativity applyEquality functionExtensionality setEquality equalityTransitivity equalitySymmetry because_Cache unionElimination voidEquality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality imageMemberEquality baseClosed imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality intEquality instantiate equalityElimination dependent_pairFormation functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].    filter(P;L)  \msim{}  []  supposing  (\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P  x))

Date html generated: 2017_04_14-AM-08_52_12
Last ObjectModification: 2017_02_27-PM-03_37_15

Theory : list_0

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