### Nuprl Lemma : decidable-cantor-to-int

`∀[R:ℤ ⟶ ℤ ⟶ ℙ]. ((∀x,y:ℤ.  Dec(R[x;y])) `` (∀F:(ℕ ⟶ 𝔹) ⟶ ℤ. Dec(∃f,g:ℕ ⟶ 𝔹. R[F f;F g])))`

Proof

Definitions occuring in Statement :  nat: `ℕ` bool: `𝔹` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` nat: `ℕ` sq_type: `SQType(T)` guard: `{T}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` and: `P ∧ Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` bfalse: `ff` so_apply: `x[s1;s2]` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` false: `False` bnot: `¬bb` assert: `↑b` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  cantor-to-int-uniform-continuity nat_wf equal_wf set-value-type le_wf int-value-type subtype_base_sq set_subtype_base int_subtype_base decidable-finite-cantor-to-int bool_wf lt_int_wf eqtt_to_assert assert_of_lt_int int_seg_wf lelt_wf all_wf decidable_wf not_wf exists_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self ifthenelse_wf bfalse_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot less_than_wf int_seg_properties nat_properties satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality hypothesis productElimination rename cutEval dependent_set_memberEquality isectElimination equalityTransitivity equalitySymmetry sqequalRule lambdaEquality independent_isectElimination intEquality natural_numberEquality setElimination promote_hyp instantiate cumulativity independent_functionElimination applyEquality functionExtensionality functionEquality because_Cache unionElimination equalityElimination independent_pairFormation universeEquality inlFormation inrFormation dependent_pairFormation voidElimination addLevel hyp_replacement int_eqEquality isect_memberEquality voidEquality computeAll levelHypothesis

Latex:
\mforall{}[R:\mBbbZ{}  {}\mrightarrow{}  \mBbbZ{}  {}\mrightarrow{}  \mBbbP{}].  ((\mforall{}x,y:\mBbbZ{}.    Dec(R[x;y]))  {}\mRightarrow{}  (\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbZ{}.  Dec(\mexists{}f,g:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  R[F  f;F  g])))

Date html generated: 2017_04_17-AM-09_59_41
Last ObjectModification: 2017_02_27-PM-05_52_36

Theory : continuity

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