Nuprl Lemma : decidable-bar-rec-equal-spector

∀[dec,base,ind:Top]. ∀[n:ℕ]. ∀[s:Top].
  (decidable-bar-rec(dec;base;ind;n;s) ~ spector-bar-rec(λn,s. if dec n s then 0 else n + 1 fi ;λn,s. case dec n s
                                                                                                  of inl(r) =>
                                                                                                  base n s r
                                                                                                  | inr(x) =>
                                                                                                  ⊥;ind;n;s))


Proof



Definitions occuring in Statement :  decidable-bar-rec: decidable-bar-rec(dec;base;ind;n;s),  spector-bar-rec: spector-bar-rec(Y;G;H;n;s),  nat: ℕ,  bottom: ⊥,  ifthenelse: if b then t else f fi ,  uall: ∀[x:A]. B[x],  top: Top,  apply: f a,  lambda: λx.A[x],  decide: case b of inl(x) => s[x] | inr(y) => t[y],  add: n + m,  natural_number: $n,  sqequal: s ~ t
Definitions unfolded in proof :  member: t ∈ T,  uall: ∀[x:A]. B[x],  spector-bar-rec: spector-bar-rec(Y;G;H;n;s),  decidable-bar-rec: decidable-bar-rec(dec;base;ind;n;s),  nat: ℕ,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  uimplies: b supposing a,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  not: ¬A,  all: ∀x:A. B[x],  top: Top,  and: P ∧ Q,  prop: ℙ,  decidable: Dec(P),  or: P ∨ Q,  nat_plus: ℕ+,  subtype_rel: A ⊆r B,  so_lambda: λ2x.t[x],  so_apply: x[s],  has-value: (a)↓,  ifthenelse: if b then t else f fi ,  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  bfalse: ff,  sq_type: SQType(T),  guard: {T},  bnot: ¬bb,  assert: ↑b,  le_int: i ≤z j,  lt_int: i <z j,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  squash: ↓T,  true: True,  less_than': less_than'(a;b),  less_than: a < b
Lemmas referenced :  istype-top,  nat_wf,  nat_properties,  satisfiable-full-omega-tt,  intformand_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  int_formula_prop_and_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,  ge_wf,  less_than_wf,  base_wf,  fun_exp0_lemma,  strictness-apply,  bottom-sqle,  decidable__le,  subtract_wf,  intformnot_wf,  itermSubtract_wf,  int_formula_prop_not_lemma,  int_term_value_subtract_lemma,  fun_exp_unroll_1,  has-value_wf_base,  int_subtype_base,  set_subtype_base,  le_wf,  is-exception_wf,  top_wf,  equal_wf,  le_int_wf,  bool_wf,  eqtt_to_assert,  assert_of_le_int,  eqff_to_assert,  bool_cases_sqequal,  subtype_base_sq,  bool_subtype_base,  assert-bnot,  itermAdd_wf,  int_term_value_add_lemma,  subtract-1-ge-0,  istype-base,  istype-less_than,  istype-void,  istype-int,  full-omega-unsat,  int-value-type,  union-value-type,  value-type-has-value,  assert_wf,  iff_weakening_uiff,  assert_of_lt_int,  lt_int_wf,  not-exception-has-value,  set-value-type
Rules used in proof :  extract_by_obid,  Error :universeIsType,  Error :isectIsTypeImplies,  isectElimination,  Error :isect_memberEquality_alt,  hypothesisEquality,  Error :inhabitedIsType,  axiomSqEquality,  hypothesis,  sqequalHypSubstitution,  sqequalSqle,  thin,  sqequalRule,  cut,  introduction,  Error :isect_memberFormation_alt,  sqequalReflexivity,  computationStep,  sqequalTransitivity,  sqequalSubstitution,  fixpointLeast,  setElimination,  rename,  intWeakElimination,  lambdaFormation,  natural_numberEquality,  independent_isectElimination,  dependent_pairFormation,  lambdaEquality,  int_eqEquality,  intEquality,  dependent_functionElimination,  isect_memberEquality,  voidElimination,  voidEquality,  independent_pairFormation,  computeAll,  independent_functionElimination,  axiomSqleEquality,  unionElimination,  dependent_set_memberEquality,  because_Cache,  divergentSqle,  baseApply,  closedConclusion,  baseClosed,  applyEquality,  callbyvalueDecide,  equalityTransitivity,  equalitySymmetry,  unionEquality,  equalityElimination,  productElimination,  sqleReflexivity,  promote_hyp,  instantiate,  cumulativity,  addEquality,  sqleRule,  decideExceptionCases,  Error :lambdaEquality_alt,  exceptionLess,  exceptionSqequal,  Error :dependent_set_memberEquality_alt,  Error :functionIsTypeImplies,  Error :dependent_pairFormation_alt,  approximateComputation,  Error :lambdaFormation_alt,  callbyvalueLess,  Error :equalityIsType1,  Error :equalityIsType2,  imageElimination,  imageMemberEquality,  lessCases,  lessExceptionCases
Latex:
\mforall{}[dec,base,ind:Top].  \mforall{}[n:\mBbbN{}].  \mforall{}[s:Top].
    (decidable-bar-rec(dec;base;ind;n;s) 
    \msim{}  spector-bar-rec(\mlambda{}n,s.  if  dec  n  s  then  0  else  n  +  1  fi  ;\mlambda{}n,s.  case  dec  n  s
                                                                                                                          of  inl(r)  =>
                                                                                                                          base  n  s  r
                                                                                                                          |  inr(x)  =>
                                                                                                                          \mbot{};ind;n;s))


Date html generated: 2019_06_20-PM-03_06_04
Last ObjectModification: 2019_03_27-PM-03_15_35

Theory : continuity


Home Index