### Nuprl Lemma : frs-non-dec-sorted-by

`∀[p:ℝ List]. (frs-non-dec(p) `⇐⇒` sorted-by(λx,y. (x ≤ y);p))`

Proof

Definitions occuring in Statement :  frs-non-dec: `frs-non-dec(L)` rleq: `x ≤ y` real: `ℝ` sorted-by: `sorted-by(R;L)` list: `T List` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` lambda: `λx.A[x]`
Definitions unfolded in proof :  sorted-by: `sorted-by(R;L)` frs-non-dec: `frs-non-dec(L)` uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` all: `∀x:A. B[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` prop: `ℙ` le: `A ≤ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` sq_type: `SQType(T)` rleq: `x ≤ y` rnonneg: `rnonneg(x)` nat_plus: `ℕ+` subtype_rel: `A ⊆r B`
Lemmas referenced :  rleq_weakening_equal list_wf nat_plus_wf nat_plus_properties rsub_wf less_than'_wf int_formula_prop_eq_lemma intformeq_wf int_subtype_base subtype_base_sq decidable__equal_int int_term_value_constant_lemma itermConstant_wf select_wf rleq_wf le_wf all_wf int_seg_wf int_formula_prop_le_lemma intformle_wf decidable__le lelt_wf int_formula_prop_wf int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt real_wf length_wf int_seg_properties
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut independent_pairFormation lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin setElimination rename dependent_set_memberEquality hypothesisEquality productElimination lemma_by_obid isectElimination natural_numberEquality unionElimination imageElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination because_Cache functionEquality instantiate cumulativity independent_pairEquality applyEquality minusEquality axiomEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}[p:\mBbbR{}  List].  (frs-non-dec(p)  \mLeftarrow{}{}\mRightarrow{}  sorted-by(\mlambda{}x,y.  (x  \mleq{}  y);p))

Date html generated: 2016_05_18-AM-08_53_24
Last ObjectModification: 2016_01_17-AM-02_32_24

Theory : reals

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