Nuprl Lemma : bag-filter-trivial

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

Proof

Definitions occuring in Statement :  bag-filter: `[x∈b|p[x]]` bag: `bag(T)` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` bag-filter: `[x∈b|p[x]]` bag: `bag(T)` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_apply: `x[s]` prop: `ℙ` so_lambda: `λ2x.t[x]` squash: `↓T` true: `True` l_all: `(∀x∈L.P[x])` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` less_than: `a < b`
Lemmas referenced :  list_wf quotient-member-eq permutation_wf permutation-equiv filter_wf5 l_member_wf equal_wf equal-wf-base bag_wf all_wf assert_wf bool_wf squash_wf true_wf filter_trivial int_seg_wf length_wf select_wf int_seg_properties 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
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality because_Cache sqequalRule pertypeElimination productElimination thin equalityTransitivity hypothesis equalitySymmetry extract_by_obid isectElimination cumulativity hypothesisEquality lambdaFormation rename lambdaEquality independent_isectElimination dependent_functionElimination applyEquality functionExtensionality setElimination setEquality independent_functionElimination productEquality isect_memberEquality axiomEquality functionEquality universeEquality addLevel hyp_replacement imageElimination natural_numberEquality imageMemberEquality baseClosed levelHypothesis unionElimination dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll

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

Date html generated: 2017_10_01-AM-08_45_23
Last ObjectModification: 2017_07_26-PM-04_30_41

Theory : bags

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