Nuprl Lemma : l_before_filter

`∀[T:Type]. ∀l:T List. ∀P:T ⟶ 𝔹. ∀x,y:T.  (x before y ∈ filter(P;l) `⇐⇒` (↑(P x)) ∧ (↑(P y)) ∧ x before y ∈ l)`

Proof

Definitions occuring in Statement :  l_before: `x before y ∈ l` filter: `filter(P;l)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  l_before: `x before y ∈ l` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` rev_implies: `P `` Q` top: `Top` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False`
Lemmas referenced :  assert_witness sublist_wf cons_wf nil_wf assert_wf l_all_wf_nil l_all_nil select_wf length_of_nil_lemma int_seg_properties decidable__le full-omega-unsat 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 decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf length_wf l_all_cons l_all_wf l_member_wf sublist_filter filter_wf5 subtype_rel_dep_function bool_wf subtype_rel_self set_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  cut independent_pairFormation lambdaFormation introduction sqequalHypSubstitution productElimination thin hypothesis extract_by_obid isectElimination applyEquality hypothesisEquality independent_functionElimination productEquality lambdaEquality voidElimination voidEquality isect_memberEquality dependent_functionElimination functionExtensionality cumulativity because_Cache setElimination rename independent_isectElimination natural_numberEquality unionElimination approximateComputation dependent_pairFormation int_eqEquality intEquality promote_hyp setEquality Error :inhabitedIsType,  Error :universeIsType,  Error :functionIsType,  universeEquality

Latex:
\mforall{}[T:Type]
\mforall{}l:T  List.  \mforall{}P:T  {}\mrightarrow{}  \mBbbB{}.  \mforall{}x,y:T.    (x  before  y  \mmember{}  filter(P;l)  \mLeftarrow{}{}\mRightarrow{}  (\muparrow{}(P  x))  \mwedge{}  (\muparrow{}(P  y))  \mwedge{}  x  before  y  \mmember{}  l)

Date html generated: 2019_06_20-PM-01_25_34
Last ObjectModification: 2018_09_26-PM-05_30_39

Theory : list_1

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