### Nuprl Lemma : bag-filter-is-nil

`∀[T:Type]. ∀[p:T ⟶ 𝔹].  ∀[bs:bag(T)]. ([x∈bs|p[x]] ~ []) supposing ∀x:T. (¬↑p[x])`

Proof

Definitions occuring in Statement :  bag-filter: `[x∈b|p[x]]` bag: `bag(T)` nil: `[]` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` not: `¬A` function: `x:A ⟶ B[x]` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bag: `bag(T)` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` bag-filter: `[x∈b|p[x]]` l_all: `(∀x∈L.P[x])` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` less_than: `a < b` squash: `↓T` cons: `[a / b]`
Lemmas referenced :  bag_wf all_wf not_wf assert_wf bool_wf list_wf filter_is_nil nil_wf equal-wf-base permutation_wf select_wf int_seg_properties length_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf list-cases product_subtype_list null_nil_lemma btrue_wf null_cons_lemma bfalse_wf and_wf equal_wf null_wf btrue_neq_bfalse equal-wf-T-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis sqequalAxiom extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality sqequalRule isect_memberEquality because_Cache lambdaEquality applyEquality functionExtensionality equalityTransitivity equalitySymmetry functionEquality universeEquality pointwiseFunctionalityForEquality pertypeElimination productElimination independent_isectElimination productEquality lambdaFormation setElimination rename natural_numberEquality dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination promote_hyp hypothesis_subsumption dependent_set_memberEquality applyLambdaEquality baseClosed

Latex:
\mforall{}[T:Type].  \mforall{}[p:T  {}\mrightarrow{}  \mBbbB{}].    \mforall{}[bs:bag(T)].  ([x\mmember{}bs|p[x]]  \msim{}  [])  supposing  \mforall{}x:T.  (\mneg{}\muparrow{}p[x])

Date html generated: 2017_10_01-AM-08_45_20
Last ObjectModification: 2017_07_26-PM-04_30_40

Theory : bags

Home Index