Nuprl Lemma : list-powerset_wf

∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List].  (list-powerset(eq;L) ∈ {p:fset(fset(T))| ∀x:fset(T). (x ∈ p ⇐⇒ x ⊆ L)} )


Proof



Definitions occuring in Statement :  list-powerset: list-powerset(eq;L),  deq-fset: deq-fset(eq),  f-subset: xs ⊆ ys,  fset-member: a ∈ s,  fset: fset(T),  list: T List,  deq: EqDecider(T),  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  member: t ∈ T,  set: {x:A| B[x]} ,  universe: Type
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  all: ∀x:A. B[x],  nat: ℕ,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  uimplies: b supposing a,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  top: Top,  and: P ∧ Q,  prop: ℙ,  or: P ∨ Q,  list-powerset: list-powerset(eq;L),  cons: [a / b],  le: A ≤ B,  less_than': less_than'(a;b),  colength: colength(L),  nil: [],  it: ⋅,  so_lambda: λ2x.t[x],  so_apply: x[s],  sq_type: SQType(T),  guard: {T},  less_than: a < b,  squash: ↓T,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  decidable: Dec(P),  subtype_rel: A ⊆r B,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  empty-fset: {},  f-subset: xs ⊆ ys,  fset-member: a ∈ s,  eqof: eqof(d),  sq_stable: SqStable(P),  deq: EqDecider(T),  cand: A c∧ B,  uiff: uiff(P;Q),  bool: 𝔹,  unit: Unit,  btrue: tt,  bfalse: ff,  bnot: ¬bb,  ifthenelse: if b then t else f fi ,  assert: ↑b,  true: True
Lemmas referenced :  nat_properties,  full-omega-unsat,  intformand_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  istype-int,  int_formula_prop_and_lemma,  istype-void,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_less_lemma,  int_formula_prop_wf,  ge_wf,  less_than_wf,  list-cases,  reduce_nil_lemma,  product_subtype_list,  colength-cons-not-zero,  colength_wf_list,  istype-false,  le_wf,  subtract-1-ge-0,  subtype_base_sq,  nat_wf,  set_subtype_base,  int_subtype_base,  spread_cons_lemma,  decidable__equal_int,  subtract_wf,  intformnot_wf,  intformeq_wf,  itermSubtract_wf,  itermAdd_wf,  int_formula_prop_not_lemma,  int_formula_prop_eq_lemma,  int_term_value_subtract_lemma,  int_term_value_add_lemma,  decidable__le,  reduce_cons_lemma,  list_wf,  deq_wf,  fset-singleton_wf,  fset_wf,  empty-fset_wf,  iff_weakening_uiff,  fset-member_wf,  deq-fset_wf,  equal-wf-T-base,  member-fset-singleton,  f-subset_wf,  f-subset-empty,  fset-union_wf,  fset-image_wf,  fset-add_wf,  member-fset-union,  cons_wf,  list_subtype_fset,  squash_wf,  exists_wf,  equal_wf,  member-fset-image-iff,  fset-member_witness,  deq_member_cons_lemma,  assert-deq-member,  assert_wf,  bor_wf,  eqof_wf,  deq-member_wf,  or_wf,  l_member_wf,  iff_transitivity,  assert_of_bor,  safe-assert-deq,  sq_stable_from_decidable,  decidable__f-subset,  member-fset-add,  member_wf,  decidable__fset-member,  fset-filter_wf,  bnot_wf,  member-fset-filter,  not_wf,  assert_of_bnot,  fset-extensionality,  eqtt_to_assert,  eqff_to_assert,  bool_cases_sqequal,  bool_wf,  bool_subtype_base,  assert-bnot,  true_wf,  istype-universe,  subtype_rel_self,  iff_weakening_equal
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :isect_memberFormation_alt,  introduction,  cut,  thin,  Error :lambdaFormation_alt,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  hypothesisEquality,  hypothesis,  setElimination,  rename,  sqequalRule,  intWeakElimination,  natural_numberEquality,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality,  dependent_functionElimination,  Error :isect_memberEquality_alt,  voidElimination,  independent_pairFormation,  Error :universeIsType,  axiomEquality,  equalityTransitivity,  equalitySymmetry,  Error :functionIsTypeImplies,  Error :inhabitedIsType,  unionElimination,  promote_hyp,  hypothesis_subsumption,  productElimination,  Error :equalityIsType1,  because_Cache,  Error :dependent_set_memberEquality_alt,  instantiate,  cumulativity,  intEquality,  imageElimination,  applyLambdaEquality,  Error :equalityIsType4,  addEquality,  applyEquality,  universeEquality,  baseClosed,  lambdaFormation,  Error :functionIsType,  Error :productIsType,  Error :inlFormation_alt,  productEquality,  Error :inrFormation_alt,  imageMemberEquality,  Error :unionIsType,  hyp_replacement,  independent_pairEquality,  equalityElimination
Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[L:T  List].
    (list-powerset(eq;L)  \mmember{}  \{p:fset(fset(T))|  \mforall{}x:fset(T).  (x  \mmember{}  p  \mLeftarrow{}{}\mRightarrow{}  x  \msubseteq{}  L)\}  )


Date html generated: 2019_06_20-PM-02_00_41
Last ObjectModification: 2018_10_05-PM-04_22_23

Theory : finite!sets


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