### Nuprl Lemma : member_map_filter

[T:Type]
∀P:T ⟶ 𝔹
∀[T':Type]
∀f:{x:T| ↑(P x)}  ⟶ T'. ∀L:T List. ∀x:T'.
((x ∈ mapfilter(f;P;L)) ⇐⇒ ∃y:T. ((y ∈ L) ∧ ((↑(P y)) c∧ (x (f y) ∈ T'))))

Proof

Definitions occuring in Statement :  mapfilter: mapfilter(f;P;L) l_member: (x ∈ l) list: List assert: b bool: 𝔹 uall: [x:A]. B[x] cand: c∧ B all: x:A. B[x] exists: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q set: {x:A| B[x]}  apply: a function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] all: x:A. B[x] mapfilter: mapfilter(f;P;L) member: t ∈ T prop: implies:  Q iff: ⇐⇒ Q and: P ∧ Q exists: x:A. B[x] cand: c∧ B l_member: (x ∈ l) subtype_rel: A ⊆B guard: {T} nat: uimplies: supposing a ge: i ≥  decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) false: False not: ¬A top: Top sq_type: SQType(T) assert: b ifthenelse: if then else fi  btrue: tt true: True so_lambda: λ2x.t[x] so_apply: x[s] rev_implies:  Q
Lemmas referenced :  filter_type list_wf assert_wf l_member_wf equal_wf less_than_wf length_wf subtype_rel_list select_wf nat_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 member_filter assert_elim subtype_base_sq bool_wf bool_subtype_base exists_wf l_member_set2 member-map map_wf iff_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality functionExtensionality applyEquality hypothesis setEquality independent_pairFormation productElimination dependent_pairFormation because_Cache equalityElimination promote_hyp equalitySymmetry hyp_replacement applyLambdaEquality setElimination rename dependent_set_memberEquality equalityTransitivity lambdaEquality sqequalRule productEquality independent_isectElimination dependent_functionElimination natural_numberEquality unionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination addLevel levelHypothesis instantiate impliesFunctionality functionEquality universeEquality

Latex:
\mforall{}[T:Type]
\mforall{}P:T  {}\mrightarrow{}  \mBbbB{}
\mforall{}[T':Type]
\mforall{}f:\{x:T|  \muparrow{}(P  x)\}    {}\mrightarrow{}  T'.  \mforall{}L:T  List.  \mforall{}x:T'.
((x  \mmember{}  mapfilter(f;P;L))  \mLeftarrow{}{}\mRightarrow{}  \mexists{}y:T.  ((y  \mmember{}  L)  \mwedge{}  ((\muparrow{}(P  y))  c\mwedge{}  (x  =  (f  y)))))

Date html generated: 2017_04_17-AM-07_25_40
Last ObjectModification: 2017_02_27-PM-04_04_25

Theory : list_1

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