### Nuprl Lemma : sorted-by_wf

`∀[T:Type]. ∀[L:T List]. ∀[R:{x:T| (x ∈ L)}  ⟶ {x:T| (x ∈ L)}  ⟶ ℙ].  (sorted-by(R;L) ∈ ℙ)`

Proof

Definitions occuring in Statement :  sorted-by: `sorted-by(R;L)` l_member: `(x ∈ l)` list: `T List` uall: `∀[x:A]. B[x]` prop: `ℙ` member: `t ∈ T` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  sorted-by: `sorted-by(R;L)` uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` prop: `ℙ` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` so_apply: `x[s]`
Lemmas referenced :  list_wf int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le int_seg_properties list-subtype l_member_wf select_wf length_wf int_seg_wf all_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality cumulativity hypothesisEquality hypothesis lambdaEquality because_Cache setElimination rename applyEquality setEquality equalityTransitivity equalitySymmetry independent_isectElimination productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination axiomEquality functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[R:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbP{}].    (sorted-by(R;L)  \mmember{}  \mBbbP{})

Date html generated: 2016_05_14-PM-01_46_41
Last ObjectModification: 2016_01_15-AM-08_20_11

Theory : list_1

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