Nuprl Lemma : apply-cycle-member

∀[n:ℕ]. ∀[L:ℕn List].
  ∀[i:ℕ||L||]. ((cycle(L) L[i]) = if (i =z ||L|| - 1) then L[0] else L[i + 1] fi  ∈ ℕn) supposing no_repeats(ℕn;L)


Proof



Definitions occuring in Statement :  cycle: cycle(L),  no_repeats: no_repeats(T;l),  select: L[n],  length: ||as||,  list: T List,  int_seg: {i..j-},  nat: ℕ,  ifthenelse: if b then t else f fi ,  eq_int: (i =z j),  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  apply: f a,  subtract: n - m,  add: n + m,  natural_number: $n,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  uimplies: b supposing a,  cycle: cycle(L),  let: let,  member: t ∈ T,  nat: ℕ,  prop: ℙ,  subtype_rel: A ⊆r B,  int_seg: {i..j-},  so_lambda: λ2x.t[x],  so_apply: x[s],  not: ¬A,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  lelt: i ≤ j < k,  and: P ∧ Q,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  all: ∀x:A. B[x],  top: Top,  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  ifthenelse: if b then t else f fi ,  bfalse: ff,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  cons: [a / b],  decidable: Dec(P),  nequal: a ≠ b ∈ T ,  assert: ↑b,  bnot: ¬bb,  sq_type: SQType(T),  guard: {T},  subtract: n - m,  so_apply: x[s1;s2],  so_lambda: λ2x y.t[x; y],  nil: [],  select: L[n],  or: P ∨ Q,  less_than: a < b,  squash: ↓T,  cand: A c∧ B,  so_lambda: so_lambda(x,y,z.t[x; y; z]),  so_apply: x[s1;s2;s3],  le: A ≤ B,  less_than': less_than'(a;b),  colength: colength(L),  true: True,  l_member: (x ∈ l)
Lemmas referenced :  int_seg_wf,  length_wf,  no_repeats_wf,  list_wf,  istype-nat,  null_wf,  equal-wf-base,  bool_wf,  list_subtype_base,  set_subtype_base,  lelt_wf,  istype-int,  int_subtype_base,  assert_wf,  bnot_wf,  not_wf,  istype-assert,  istype-void,  length_of_nil_lemma,  int_seg_properties,  nat_properties,  full-omega-unsat,  intformand_wf,  intformeq_wf,  itermVar_wf,  itermConstant_wf,  intformless_wf,  intformle_wf,  int_formula_prop_and_lemma,  int_formula_prop_eq_lemma,  int_term_value_var_lemma,  int_term_value_constant_lemma,  int_formula_prop_less_lemma,  int_formula_prop_le_lemma,  int_formula_prop_wf,  uiff_transitivity,  eqtt_to_assert,  assert_of_null,  iff_transitivity,  iff_weakening_uiff,  eqff_to_assert,  assert_of_bnot,  btrue_neq_bfalse,  btrue_wf,  null_nil_lemma,  and_wf,  bfalse_wf,  null_cons_lemma,  length_of_cons_lemma,  product_subtype_list,  neg_assert_of_eq_int,  assert-bnot,  bool_subtype_base,  subtype_base_sq,  bool_cases_sqequal,  equal_wf,  assert_of_eq_int,  eq_int_wf,  satisfiable-full-omega-tt,  decidable__equal_int,  base_wf,  stuck-spread,  list-cases,  hd_wf,  decidable__le,  intformnot_wf,  int_formula_prop_not_lemma,  reduce_hd_cons_lemma,  istype-base,  select_wf,  decidable__lt,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  itermAdd_wf,  int_term_value_add_lemma,  ge_wf,  istype-less_than,  list_ind_nil_lemma,  colength-cons-not-zero,  colength_wf_list,  istype-false,  istype-le,  subtract-1-ge-0,  spread_cons_lemma,  le_wf,  list_ind_cons_lemma,  nil_wf,  cons_wf,  squash_wf,  true_wf,  istype-universe,  eq_int_eq_true,  subtype_rel_self,  iff_weakening_equal,  intformimplies_wf,  int_formual_prop_imp_lemma,  non_neg_length,  select_cons_tl,  add-is-int-iff,  false_wf,  no_repeats_cons,  subtract-is-int-iff,  add-associates,  add-swap,  add-commutes,  zero-add,  le_weakening2,  add-subtract-cancel,  subtract-add-cancel
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :isect_memberFormation_alt,  sqequalRule,  Error :universeIsType,  cut,  introduction,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  natural_numberEquality,  setElimination,  rename,  because_Cache,  hypothesis,  hypothesisEquality,  baseApply,  closedConclusion,  baseClosed,  applyEquality,  independent_isectElimination,  intEquality,  Error :lambdaEquality_alt,  Error :equalityIstype,  sqequalBase,  equalitySymmetry,  Error :functionIsType,  applyLambdaEquality,  equalityTransitivity,  productElimination,  approximateComputation,  independent_functionElimination,  Error :dependent_pairFormation_alt,  int_eqEquality,  dependent_functionElimination,  Error :isect_memberEquality_alt,  voidElimination,  independent_pairFormation,  Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination,  equalityElimination,  hypothesis_subsumption,  dependent_set_memberEquality,  cumulativity,  instantiate,  promote_hyp,  minusEquality,  computeAll,  lambdaEquality,  dependent_pairFormation,  voidEquality,  isect_memberEquality,  lambdaFormation,  imageElimination,  addEquality,  Error :productIsType,  intWeakElimination,  axiomEquality,  Error :functionIsTypeImplies,  Error :dependent_set_memberEquality_alt,  universeEquality,  imageMemberEquality,  pointwiseFunctionality
Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[L:\mBbbN{}n  List].
    \mforall{}[i:\mBbbN{}||L||].  ((cycle(L)  L[i])  =  if  (i  =\msubz{}  ||L||  -  1)  then  L[0]  else  L[i  +  1]  fi  ) 
    supposing  no\_repeats(\mBbbN{}n;L)


Date html generated: 2019_06_20-PM-01_39_56
Last ObjectModification: 2018_11_28-PM-03_20_47

Theory : list_1


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