Nuprl Lemma : orbit-cover

∀[T:Type]
  ((∀x,y:T.  Dec(x = y ∈ T))
  ⇒ finite-type(T)
  ⇒ (∀f:T ⟶ T
        ∃orbits:T List List
         ((∀orbit∈orbits.0 < ||orbit||
                 ∧ no_repeats(T;orbit)
                 ∧ (∀i:ℕ||orbit||. (orbit[i] = (f^i orbit[0]) ∈ T))
                 ∧ (∀b:T. ((b ∈ orbit) ⇐⇒ ∃n:ℕ. (b = (f^n orbit[0]) ∈ T))))
         ∧ (∀a:T. (∃orbit∈orbits. (a ∈ orbit)))
         ∧ (∀o1∈orbits.(∀o2∈orbits.(o1[0] ∈ o2) ⇒ o1 ⊆ o2)))))


Proof



Definitions occuring in Statement :  finite-type: finite-type(T),  l_contains: A ⊆ B,  l_exists: (∃x∈L. P[x]),  l_all: (∀x∈L.P[x]),  no_repeats: no_repeats(T;l),  l_member: (x ∈ l),  select: L[n],  length: ||as||,  list: T List,  fun_exp: f^n,  int_seg: {i..j-},  nat: ℕ,  less_than: a < b,  decidable: Dec(P),  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  iff: P ⇐⇒ Q,  implies: P ⇒ Q,  and: P ∧ Q,  apply: f a,  function: x:A ⟶ B[x],  natural_number: $n,  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  implies: P ⇒ Q,  all: ∀x:A. B[x],  member: t ∈ T,  exists: ∃x:A. B[x],  finite-type: finite-type(T),  prop: ℙ,  and: P ∧ Q,  int_seg: {i..j-},  uimplies: b supposing a,  guard: {T},  lelt: i ≤ j < k,  decidable: Dec(P),  or: P ∨ Q,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  false: False,  top: Top,  less_than: a < b,  squash: ↓T,  subtype_rel: A ⊆r B,  le: A ≤ B,  less_than': less_than'(a;b),  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  pi1: fst(t),  nat: ℕ,  cand: A c∧ B,  so_lambda: λ2x.t[x],  ge: i ≥ j ,  so_apply: x[s],  compose: f o g,  cons: [a / b],  uiff: uiff(P;Q),  subtract: n - m,  true: True,  surject: Surj(A;B;f),  l_all: (∀x∈L.P[x]),  l_contains: A ⊆ B,  l_member: (x ∈ l)
Lemmas referenced :  orbit-exists,  finite-type_wf,  decidable_wf,  equal_wf,  no_repeats_wf,  int_seg_wf,  length_wf,  select_wf,  int_seg_properties,  decidable__le,  full-omega-unsat,  intformand_wf,  intformnot_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  istype-int,  int_formula_prop_and_lemma,  istype-void,  int_formula_prop_not_lemma,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_wf,  decidable__lt,  intformless_wf,  int_formula_prop_less_lemma,  fun_exp_wf,  int_seg_subtype_nat,  l_member_wf,  nat_wf,  map_wf,  list_wf,  compose_wf,  upto_wf,  l_all_wf,  less_than_wf,  all_wf,  nat_properties,  istype-false,  iff_wf,  exists_wf,  l_exists_wf,  l_all_iff,  member_map,  le_wf,  fun_exp0_lemma,  list-cases,  length_of_nil_lemma,  nil_member,  product_subtype_list,  length_of_cons_lemma,  length_wf_nat,  not-lt-2,  condition-implies-le,  minus-add,  minus-one-mul,  zero-add,  minus-one-mul-top,  add-commutes,  add_functionality_wrt_le,  add-associates,  add-zero,  le-add-cancel,  l_exists_iff,  squash_wf,  true_wf,  subtype_rel_self,  iff_weakening_equal,  member_upto2,  itermAdd_wf,  int_term_value_add_lemma,  fun_exp_add_sq,  l_contains_wf
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  cut,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  independent_functionElimination,  hypothesis,  dependent_functionElimination,  promote_hyp,  productElimination,  Error :functionIsType,  Error :universeIsType,  Error :inhabitedIsType,  sqequalRule,  universeEquality,  applyEquality,  functionExtensionality,  Error :productIsType,  natural_numberEquality,  Error :equalityIsType1,  cumulativity,  because_Cache,  setElimination,  rename,  independent_isectElimination,  unionElimination,  approximateComputation,  Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality,  Error :isect_memberEquality_alt,  voidElimination,  independent_pairFormation,  imageElimination,  equalityTransitivity,  equalitySymmetry,  productEquality,  Error :setIsType,  hyp_replacement,  applyLambdaEquality,  Error :dependent_set_memberEquality_alt,  hypothesis_subsumption,  addEquality,  minusEquality,  imageMemberEquality,  baseClosed,  instantiate,  functionEquality
Latex:
\mforall{}[T:Type]
    ((\mforall{}x,y:T.    Dec(x  =  y))
    {}\mRightarrow{}  finite-type(T)
    {}\mRightarrow{}  (\mforall{}f:T  {}\mrightarrow{}  T
                \mexists{}orbits:T  List  List
                  ((\mforall{}orbit\mmember{}orbits.0  <  ||orbit||
                                  \mwedge{}  no\_repeats(T;orbit)
                                  \mwedge{}  (\mforall{}i:\mBbbN{}||orbit||.  (orbit[i]  =  (f\^{}i  orbit[0])))
                                  \mwedge{}  (\mforall{}b:T.  ((b  \mmember{}  orbit)  \mLeftarrow{}{}\mRightarrow{}  \mexists{}n:\mBbbN{}.  (b  =  (f\^{}n  orbit[0])))))
                  \mwedge{}  (\mforall{}a:T.  (\mexists{}orbit\mmember{}orbits.  (a  \mmember{}  orbit)))
                  \mwedge{}  (\mforall{}o1\mmember{}orbits.(\mforall{}o2\mmember{}orbits.(o1[0]  \mmember{}  o2)  {}\mRightarrow{}  o1  \msubseteq{}  o2)))))


Date html generated: 2019_06_20-PM-01_39_03
Last ObjectModification: 2018_10_04-PM-02_56_28

Theory : list_1


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