Nuprl Lemma : concat-map-map-decide

[T:Type]. ∀[g:Top]. ∀[L:T List]. ∀[f:{x:T| (x ∈ L)}  ⟶ (Top Top)].
  (concat(map(λλy.g[x;y];case f[x] of inl(m) => [m] inr(x) => []);L)) mapfilter(λx.g[x;outl(f[x])];


Definitions occuring in Statement :  mapfilter: mapfilter(f;P;L) l_member: (x ∈ l) concat: concat(ll) map: map(f;as) cons: [a b] nil: [] list: List outl: outl(x) isl: isl(x) uall: [x:A]. B[x] top: Top so_apply: x[s1;s2] so_apply: x[s] set: {x:A| B[x]}  lambda: λx.A[x] function: x:A ⟶ B[x] decide: case of inl(x) => s[x] inr(y) => t[y] union: left right universe: Type sqequal: t
Definitions unfolded in proof :  mapfilter: mapfilter(f;P;L) uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q concat: concat(ll) cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] decidable: Dec(P) nil: [] it: so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) less_than: a < b squash: T less_than': less_than'(a;b) iff: ⇐⇒ Q rev_implies:  Q isl: isl(x) outl: outl(x) assert: b ifthenelse: if then else fi  btrue: tt append: as bs so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] bfalse: ff true: True bool: 𝔹 unit: Unit uiff: uiff(P;Q)
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf l_member_wf top_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases map_nil_lemma filter_nil_lemma reduce_nil_lemma product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int map_cons_lemma filter_cons_lemma list_wf nil_wf subtype_rel_dep_function cons_wf subtype_rel_sets cons_member subtype_rel_self set_wf reduce_cons_lemma list_ind_cons_lemma list_ind_nil_lemma true_wf false_wf not_wf isl_wf bool_wf assert_wf bnot_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination sqequalAxiom functionEquality setEquality cumulativity unionEquality applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination universeEquality inrFormation functionExtensionality inlFormation equalityElimination

\mforall{}[T:Type].  \mforall{}[g:Top].  \mforall{}[L:T  List].  \mforall{}[f:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  (Top  +  Top)].
    (concat(map(\mlambda{}\mlambda{}y.g[x;y];case  f[x]  of  inl(m)  =>  [m]  |  inr(x)  =>  []);L)) 
    \msim{}  mapfilter(\mlambda{}x.g[x;outl(f[x])];\mlambda{}x.isl(f[x]);L))

Date html generated: 2018_05_21-PM-07_35_48
Last ObjectModification: 2017_07_26-PM-05_09_54

Theory : general

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