### Nuprl Lemma : reduce-mapfilter

`∀[f1,x:Top]. ∀[T,A:Type]. ∀[as:T List]. ∀[P:{a:T| (a ∈ as)}  ⟶ 𝔹]. ∀[f2:{a:T| (a ∈ as) ∧ (↑(P a))}  ⟶ A].`
`  (reduce(f1;x;mapfilter(f2;P;as)) ~ reduce(λu,z. if P u then f1 (f2 u) z else z fi ;x;as))`

Proof

Definitions occuring in Statement :  mapfilter: `mapfilter(f;P;L)` l_member: `(x ∈ l)` reduce: `reduce(f;k;as)` list: `T List` assert: `↑b` ifthenelse: `if b then t else f fi ` bool: `𝔹` uall: `∀[x:A]. B[x]` top: `Top` and: `P ∧ Q` set: `{x:A| B[x]} ` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` 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` mapfilter: `mapfilter(f;P;L)` 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` cand: `A c∧ B` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b`
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 assert_wf bool_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases reduce_nil_lemma filter_nil_lemma map_nil_lemma nil_wf 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 reduce_cons_lemma filter_cons_lemma subtype_rel_dep_function cons_wf subtype_rel_sets cons_member subtype_rel_self set_wf eqtt_to_assert map_cons_lemma eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_wf top_wf
Rules used in proof :  cut thin sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation introduction 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 sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom functionEquality setEquality cumulativity productEquality applyEquality functionExtensionality dependent_set_memberEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality addEquality baseClosed instantiate imageElimination inrFormation inlFormation equalityElimination universeEquality isect_memberFormation

Latex:
\mforall{}[f1,x:Top].  \mforall{}[T,A:Type].  \mforall{}[as:T  List].  \mforall{}[P:\{a:T|  (a  \mmember{}  as)\}    {}\mrightarrow{}  \mBbbB{}].
\mforall{}[f2:\{a:T|  (a  \mmember{}  as)  \mwedge{}  (\muparrow{}(P  a))\}    {}\mrightarrow{}  A].
(reduce(f1;x;mapfilter(f2;P;as))  \msim{}  reduce(\mlambda{}u,z.  if  P  u  then  f1  (f2  u)  z  else  z  fi  ;x;as))

Date html generated: 2017_04_17-AM-07_30_26
Last ObjectModification: 2017_02_27-PM-04_08_28

Theory : list_1

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