Nuprl Lemma : iseg-mapfilter

`∀[T:Type]`
`  ∀P:T ⟶ 𝔹. ∀[T':Type]. ∀f:{x:T| ↑(P x)}  ⟶ T'. ∀L1,L2:T List.  (L1 ≤ L2 `` mapfilter(f;P;L1) ≤ mapfilter(f;P;L2))`

Proof

Definitions occuring in Statement :  iseg: `l1 ≤ l2` mapfilter: `mapfilter(f;P;L)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` implies: `P `` Q` prop: `ℙ` so_apply: `x[s]` mapfilter: `mapfilter(f;P;L)` top: `Top` or: `P ∨ Q` cons: `[a / b]` uimplies: `b supposing a` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` ge: `i ≥ j ` le: `A ≤ B` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B` squash: `↓T`
Lemmas referenced :  list_induction all_wf list_wf iseg_wf mapfilter_wf assert_wf filter_nil_lemma map_nil_lemma filter_cons_lemma bool_wf nil_iseg nil_wf cons_wf list-cases product_subtype_list iseg_length length_of_cons_lemma length_of_nil_lemma eqtt_to_assert mapfilter_nil_lemma map_cons_lemma non_neg_length satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot cons_iseg equal-wf-T-base bnot_wf not_wf assert_elim not_assert_elim and_wf btrue_neq_bfalse uiff_transitivity assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity hypothesis functionEquality functionExtensionality applyEquality setEquality independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality rename because_Cache universeEquality unionElimination promote_hyp hypothesis_subsumption productElimination independent_isectElimination equalityElimination equalityTransitivity equalitySymmetry natural_numberEquality dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll instantiate baseClosed dependent_set_memberEquality applyLambdaEquality setElimination imageMemberEquality imageElimination

Latex:
\mforall{}[T:Type]
\mforall{}P:T  {}\mrightarrow{}  \mBbbB{}
\mforall{}[T':Type]
\mforall{}f:\{x:T|  \muparrow{}(P  x)\}    {}\mrightarrow{}  T'.  \mforall{}L1,L2:T  List.    (L1  \mleq{}  L2  {}\mRightarrow{}  mapfilter(f;P;L1)  \mleq{}  mapfilter(f;P;L2))

Date html generated: 2017_04_17-AM-08_50_07
Last ObjectModification: 2017_02_27-PM-05_07_04

Theory : list_1

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