Nuprl Lemma : sorted-by-cons

∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x:T. ∀L:T List.  (sorted-by(R;[x / L]) ⇐⇒ sorted-by(R;L) ∧ (∀z∈L.R x z))


Proof



Definitions occuring in Statement :  sorted-by: sorted-by(R;L),  l_all: (∀x∈L.P[x]),  cons: [a / b],  list: T List,  uall: ∀[x:A]. B[x],  prop: ℙ,  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 :  sorted-by: sorted-by(R;L),  all: ∀x:A. B[x],  member: t ∈ T,  top: Top,  uall: ∀[x:A]. B[x],  iff: P ⇐⇒ Q,  and: P ∧ Q,  implies: P ⇒ Q,  int_seg: {i..j-},  prop: ℙ,  so_lambda: λ2x.t[x],  uimplies: b supposing a,  guard: {T},  lelt: i ≤ j < k,  decidable: Dec(P),  or: P ∨ Q,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  false: False,  not: ¬A,  ge: i ≥ j ,  le: A ≤ B,  so_apply: x[s],  rev_implies: P ⇐ Q,  less_than: a < b,  squash: ↓T,  uiff: uiff(P;Q),  subtract: n - m,  cand: A c∧ B,  sq_type: SQType(T),  subtype_rel: A ⊆r B,  l_all: (∀x∈L.P[x]),  less_than': less_than'(a;b),  select: L[n],  cons: [a / b]
Lemmas referenced :  length_of_cons_lemma,  int_seg_wf,  length_wf,  all_wf,  select_wf,  cons_wf,  int_seg_properties,  decidable__le,  satisfiable-full-omega-tt,  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,  non_neg_length,  decidable__lt,  intformless_wf,  itermAdd_wf,  int_formula_prop_less_lemma,  int_term_value_add_lemma,  l_all_wf,  l_member_wf,  list_wf,  add-member-int_seg2,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  lelt_wf,  add-subtract-cancel,  add-associates,  add-swap,  add-commutes,  zero-add,  squash_wf,  le_wf,  less_than_wf,  and_wf,  equal_wf,  subtype_base_sq,  int_subtype_base,  select_cons_tl,  iff_weakening_equal,  false_wf,  select-cons-tl,  decidable__equal_int,  intformeq_wf,  int_formula_prop_eq_lemma,  add-is-int-iff
Rules used in proof :  sqequalSubstitution,  sqequalRule,  sqequalReflexivity,  sqequalTransitivity,  computationStep,  cut,  introduction,  extract_by_obid,  sqequalHypSubstitution,  dependent_functionElimination,  thin,  isect_memberEquality,  voidElimination,  voidEquality,  hypothesis,  isect_memberFormation,  lambdaFormation,  independent_pairFormation,  isectElimination,  natural_numberEquality,  setElimination,  rename,  hypothesisEquality,  cumulativity,  addEquality,  lambdaEquality,  because_Cache,  applyEquality,  functionExtensionality,  independent_isectElimination,  productElimination,  unionElimination,  dependent_pairFormation,  int_eqEquality,  intEquality,  computeAll,  productEquality,  imageElimination,  setEquality,  functionEquality,  universeEquality,  dependent_set_memberEquality,  equalityTransitivity,  equalitySymmetry,  imageMemberEquality,  baseClosed,  instantiate,  independent_functionElimination,  hyp_replacement,  Error :applyLambdaEquality,  pointwiseFunctionality,  promote_hyp,  baseApply,  closedConclusion
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}x:T.  \mforall{}L:T  List.    (sorted-by(R;[x  /  L])  \mLeftarrow{}{}\mRightarrow{}  sorted-by(R;L)  \mwedge{}  (\mforall{}z\mmember{}L.R  x  z))


Date html generated: 2016_10_21-AM-10_11_02
Last ObjectModification: 2016_07_12-AM-05_30_05

Theory : list_1


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