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 then else fi n,s. case dec s
                                                                                                  of inl(r) =>
                                                                                                  base r
                                                                                                  inr(x) =>


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: bottom: ifthenelse: if then else fi  uall: [x:A]. B[x] top: Top apply: a lambda: λx.A[x] decide: case of inl(x) => s[x] inr(y) => t[y] add: m natural_number: $n sqequal: 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: implies:  Q false: False ge: i ≥  uimplies: 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: decidable: Dec(P) or: P ∨ Q nat_plus: + subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] has-value: (a)↓ ifthenelse: if then else 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 ≤j lt_int: i <j iff: ⇐⇒ Q rev_implies:  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

\mforall{}[dec,base,ind:Top].  \mforall{}[n:\mBbbN{}].  \mforall{}[s:Top].
    \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)  =>

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

Theory : continuity

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