### Nuprl Lemma : l-ordered-filter2

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  ∀L:T List. ∀P:{x:T| (x ∈ L)}  ⟶ 𝔹.  (l-ordered(T;x,y.R[x;y];L) `` l-ordered(T;x,y.R[x;y];filter(P;L)))`

Proof

Definitions occuring in Statement :  l-ordered: `l-ordered(T;x,y.R[x; y];L)` l_member: `(x ∈ l)` filter: `filter(P;l)` list: `T List` bool: `𝔹` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` 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]` prop: `ℙ` implies: `P `` Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_apply: `x[s]` top: `Top` true: `True` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` subtype_rel: `A ⊆r B` guard: `{T}` cand: `A c∧ B` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` false: `False`
Lemmas referenced :  list_induction all_wf l_member_wf bool_wf l-ordered_wf filter_wf5 list_wf filter_nil_lemma true_wf nil_wf l-ordered-nil-true filter_cons_lemma cons_member cons_wf eqtt_to_assert l-ordered-cons subtype_rel_dep_function subtype_rel_sets equal_wf subtype_rel_self set_wf member_filter_2 eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality functionEquality setEquality cumulativity because_Cache hypothesis setElimination rename applyEquality functionExtensionality independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality natural_numberEquality addLevel allFunctionality impliesFunctionality productElimination inlFormation dependent_set_memberEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry independent_isectElimination inrFormation independent_pairFormation dependent_pairFormation promote_hyp instantiate productEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}L:T  List.  \mforall{}P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}.
(l-ordered(T;x,y.R[x;y];L)  {}\mRightarrow{}  l-ordered(T;x,y.R[x;y];filter(P;L)))

Date html generated: 2018_05_21-PM-07_38_27
Last ObjectModification: 2017_07_26-PM-05_12_42

Theory : general

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