### Nuprl Lemma : simple-decidable-finite-cantor

`∀[T:Type]. ∀[R:T ⟶ ℙ].  ((∀x:T. Dec(R[x])) `` (∀n:ℕ. ∀F:(ℕn ⟶ 𝔹) ⟶ T.  Dec(∃f:ℕn ⟶ 𝔹. R[F f])))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` nat: `ℕ` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` and: `P ∧ Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` guard: `{T}` subtype_rel: `A ⊆r B` sq_exists: `∃x:A [B[x]]` cand: `A c∧ B` sq_stable: `SqStable(P)` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` le: `A ≤ B` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` select: `L[n]` cons: `[a / b]` nil: `[]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]`
Lemmas referenced :  int_seg_wf bool_wf nat_wf all_wf decidable_wf equal_wf length_wf subtract_wf list_wf isect_wf sq_exists_wf nat_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermVar_wf itermSubtract_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_wf lelt_wf select_wf int_seg_properties decidable__le intformle_wf int_formula_prop_le_lemma intformeq_wf int_formula_prop_eq_lemma subtype_rel_self set_wf less_than_wf primrec-wf2 not_wf sq_stable__and sq_stable__all sq_stable__equal squash_wf sq_stable_from_decidable true_wf iff_weakening_equal subtype_base_sq int_subtype_base decidable__equal_int itermAdd_wf int_term_value_add_lemma append_wf cons_wf btrue_wf nil_wf length-append length-singleton add_functionality_wrt_eq select_append_front bfalse_wf select-append subtype_rel_list top_wf int_seg_subtype_nat false_wf lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot not_functionality_wrt_uiff assert_wf bool_cases non_neg_length length_of_cons_lemma length_of_nil_lemma exists_wf stuck-spread base_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut functionEquality introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis sqequalRule lambdaEquality applyEquality cumulativity universeEquality axiomEquality intEquality because_Cache productEquality functionExtensionality dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality equalityTransitivity equalitySymmetry instantiate inlFormation inrFormation dependent_set_memberFormation imageMemberEquality baseClosed imageElimination hyp_replacement addEquality promote_hyp equalityElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  \mBbbP{}].    ((\mforall{}x:T.  Dec(R[x]))  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}F:(\mBbbN{}n  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  T.    Dec(\mexists{}f:\mBbbN{}n  {}\mrightarrow{}  \mBbbB{}.  R[F  f])))

Date html generated: 2019_06_20-PM-02_49_49
Last ObjectModification: 2018_09_26-AM-09_54_22

Theory : continuity

Home Index