### Nuprl Lemma : concat-map-map-decide

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

Proof

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: `T 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 b of inl(x) => s[x] | inr(y) => t[y]` union: `left + right` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  mapfilter: `mapfilter(f;P;L)` uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` concat: `concat(ll)` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.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: `P `⇐⇒` Q` rev_implies: `P `` Q` isl: `isl(x)` outl: `outl(x)` assert: `↑b` ifthenelse: `if b then t else f 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

Latex:
\mforall{}[T:Type].  \mforall{}[g:Top].  \mforall{}[L:T  List].  \mforall{}[f:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  (Top  +  Top)].
(concat(map(\mlambda{}x.map(\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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