### Nuprl Lemma : sorted-by-reverse

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀L:T List. (sorted-by(R;L) `⇐⇒` sorted-by(λx,y. (R y x);rev(L)))`

Proof

Definitions occuring in Statement :  sorted-by: `sorted-by(R;L)` reverse: `rev(as)` list: `T List` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` sorted-by: `sorted-by(R;L)` member: `t ∈ T` int_seg: `{i..j-}` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` rev_implies: `P `` Q` top: `Top` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` squash: `↓T` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` true: `True` subtract: `n - m` cand: `A c∧ B` sq_type: `SQType(T)`
Lemmas referenced :  int_seg_wf length_wf reverse_wf sorted-by_wf subtype_rel_dep_function l_member_wf subtype_rel_self set_wf list_wf length-reverse lelt_wf int_seg_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_wf subtract_wf decidable__le intformle_wf itermConstant_wf itermSubtract_wf int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma select-reverse iff_weakening_equal all_wf squash_wf true_wf add-associates minus-add minus-minus minus-one-mul add-swap add-mul-special add-commutes zero-add zero-mul add-zero select_wf le_wf less_than_wf and_wf equal_wf subtype_base_sq int_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation sqequalHypSubstitution sqequalRule cut introduction extract_by_obid isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis cumulativity applyEquality instantiate lambdaEquality functionEquality universeEquality setEquality independent_isectElimination because_Cache functionExtensionality dependent_functionElimination isect_memberEquality voidElimination voidEquality dependent_set_memberEquality productElimination unionElimination imageElimination dependent_pairFormation int_eqEquality intEquality computeAll equalityTransitivity equalitySymmetry independent_functionElimination imageMemberEquality baseClosed multiplyEquality addEquality productEquality hyp_replacement Error :applyLambdaEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    \mforall{}L:T  List.  (sorted-by(R;L)  \mLeftarrow{}{}\mRightarrow{}  sorted-by(\mlambda{}x,y.  (R  y  x);rev(L)))

Date html generated: 2016_10_21-AM-10_11_57
Last ObjectModification: 2016_07_12-AM-05_29_50

Theory : list_1

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