Nuprl Lemma : involution-with-unique-fixpoint

∀n:ℕ
  ∀[T:Type]
    (T ~ ℕn
    ⇒ (∀f:T ⟶ T
          ((∀x:T. ((f (f x)) = x ∈ T))
          ⇒ (∀x,y:T.  (((f x) = x ∈ T) ⇒ ((f y) = y ∈ T) ⇒ (x = y ∈ T)))
          ⇒ ((n rem 2) = 1 ∈ ℤ ⇐⇒ ∃x:T. ((f x) = x ∈ T)))))


Proof



Definitions occuring in Statement :  equipollent: A ~ B,  int_seg: {i..j-},  nat: ℕ,  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  iff: P ⇐⇒ Q,  implies: P ⇒ Q,  apply: f a,  function: x:A ⟶ B[x],  remainder: n rem m,  natural_number: $n,  int: ℤ,  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  all: ∀x:A. B[x],  uall: ∀[x:A]. B[x],  implies: P ⇒ Q,  iff: P ⇐⇒ Q,  and: P ∧ Q,  member: t ∈ T,  subtype_rel: A ⊆r B,  nat: ℕ,  so_lambda: λ2x.t[x],  so_apply: x[s],  uimplies: b supposing a,  rev_implies: P ⇐ Q,  exists: ∃x:A. B[x],  inject: Inj(A;B;f),  cand: A c∧ B,  squash: ↓T,  prop: ℙ,  true: True,  guard: {T},  l_exists: (∃x∈L. P[x]),  int_seg: {i..j-},  lelt: i ≤ j < k,  less_than: a < b,  le: A ≤ B,  ge: i ≥ j ,  decidable: Dec(P),  or: P ∨ Q,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  false: False,  top: Top,  l_all: (∀x∈L.P[x]),  assert: ↑b,  bnot: ¬bb,  sq_type: SQType(T),  bfalse: ff,  ifthenelse: if b then t else f fi ,  uiff: uiff(P;Q),  btrue: tt,  it: ⋅,  unit: Unit,  bool: 𝔹,  nequal: a ≠ b ∈ T ,  so_apply: x[s1;s2],  so_lambda: λ2x y.t[x; y],  less_than': less_than'(a;b),  no_repeats: no_repeats(T;l),  l_disjoint: l_disjoint(T;l1;l2),  cons: [a / b],  subtract: n - m,  nil: [],  select: L[n],  orbit: orbit(T;f;L),  l_member: (x ∈ l),  nat_plus: ℕ+
Lemmas referenced :  involution-has-fixpoint,  istype-int,  set_subtype_base,  le_wf,  int_subtype_base,  equipollent_wf,  int_seg_wf,  istype-universe,  istype-nat,  count-by-orbits,  equal_wf,  squash_wf,  true_wf,  subtype_rel_self,  iff_weakening_equal,  orbit-of-involution,  l_sum-sum,  list_wf,  length_wf,  l_member_wf,  isolate_summand,  int_seg_properties,  nat_properties,  decidable__le,  full-omega-unsat,  intformand_wf,  intformnot_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  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_less_lemma,  int_formula_prop_wf,  istype-le,  select_wf,  decidable__lt,  neg_assert_of_eq_int,  assert-bnot,  bool_subtype_base,  subtype_base_sq,  bool_cases_sqequal,  eqff_to_assert,  assert_of_eq_int,  eqtt_to_assert,  bool_wf,  eq_int_wf,  select_member,  l_disjoint_wf,  pairwise-implies,  decidable__equal_int,  nat_wf,  int_formula_prop_eq_lemma,  intformeq_wf,  false_wf,  int_seg_subtype_nat,  singleton-orbit,  hd_wf,  set_wf,  equal-wf-T-base,  length_of_cons_lemma,  null_cons_lemma,  product_subtype_list,  length_of_nil_lemma,  null_nil_lemma,  list-cases,  hd_member,  subtype_rel_list,  nil_wf,  orbit_wf,  base_wf,  stuck-spread,  cons_wf,  orbit-transitive,  lelt_wf,  fun_exp_wf,  exists_wf,  l_all_functionality,  int_term_value_add_lemma,  itermAdd_wf,  non_neg_length,  equal-wf-base,  int_term_value_subtract_lemma,  itermSubtract_wf,  subtract_wf,  fun_exp-fixedpoint,  less_than_wf,  ge_wf,  ifthenelse_wf,  length_wf_nat,  sum_functionality,  sum_split1,  int_term_value_mul_lemma,  itermMultiply_wf,  sum_wf,  add_functionality_wrt_eq,  sum_constant,  mul-distributes-right,  add-associates,  add-swap,  add-commutes,  mul-commutes,  mul-distributes,  rem_invariant,  and_wf
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :lambdaFormation_alt,  Error :isect_memberFormation_alt,  independent_pairFormation,  cut,  introduction,  extract_by_obid,  sqequalHypSubstitution,  dependent_functionElimination,  thin,  hypothesisEquality,  isectElimination,  independent_functionElimination,  hypothesis,  Error :equalityIstype,  sqequalRule,  baseApply,  closedConclusion,  baseClosed,  applyEquality,  intEquality,  Error :lambdaEquality_alt,  natural_numberEquality,  independent_isectElimination,  sqequalBase,  equalitySymmetry,  Error :productIsType,  Error :universeIsType,  because_Cache,  Error :functionIsType,  Error :inhabitedIsType,  equalityTransitivity,  setElimination,  rename,  instantiate,  universeEquality,  imageElimination,  imageMemberEquality,  productElimination,  Error :setIsType,  Error :dependent_set_memberEquality_alt,  unionElimination,  approximateComputation,  Error :dependent_pairFormation_alt,  int_eqEquality,  Error :isect_memberEquality_alt,  voidElimination,  lambdaFormation,  voidEquality,  isect_memberEquality,  lambdaEquality,  dependent_pairFormation,  cumulativity,  promote_hyp,  equalityElimination,  dependent_set_memberEquality,  applyLambdaEquality,  functionExtensionality,  setEquality,  hypothesis_subsumption,  addEquality,  axiomEquality,  intWeakElimination,  sqequalIntensionalEquality,  axiomSqEquality,  multiplyEquality,  minusEquality,  hyp_replacement,  remainderEquality
Latex:
\mforall{}n:\mBbbN{}
    \mforall{}[T:Type]
        (T  \msim{}  \mBbbN{}n
        {}\mRightarrow{}  (\mforall{}f:T  {}\mrightarrow{}  T
                    ((\mforall{}x:T.  ((f  (f  x))  =  x))
                    {}\mRightarrow{}  (\mforall{}x,y:T.    (((f  x)  =  x)  {}\mRightarrow{}  ((f  y)  =  y)  {}\mRightarrow{}  (x  =  y)))
                    {}\mRightarrow{}  ((n  rem  2)  =  1  \mLeftarrow{}{}\mRightarrow{}  \mexists{}x:T.  ((f  x)  =  x)))))


Date html generated: 2019_06_20-PM-02_18_38
Last ObjectModification: 2019_03_06-AM-10_53_13

Theory : equipollence!!cardinality!


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