### Nuprl Lemma : kripke2b-baire-seq_wf

`∀[a:ℕ ⟶ ℕ]. ∀[x:ℕ]. ∀[F:∀b:{b:ℕ ⟶ ℕ| a = b ∈ (ℕx ⟶ ℕ)} . ∃n:ℕ. ((b n) ≥ ((a x) + 1) )].`
`  (kripke2b-baire-seq(a;x;F) ∈ (ℕ ⟶ 𝔹) ⟶ ℕ)`

Proof

Definitions occuring in Statement :  kripke2b-baire-seq: `kripke2b-baire-seq(a;x;F)` int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` uall: `∀[x:A]. B[x]` ge: `i ≥ j ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` kripke2b-baire-seq: `kripke2b-baire-seq(a;x;F)` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` nat: `ℕ` ge: `i ≥ j ` prop: `ℙ` so_apply: `x[s]` exists: `∃x:A. B[x]` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  eq-finite-seqs_wf bool_wf eqtt_to_assert min-inc-seq_wf pi1_wf ge_wf exists_wf nat_wf cantor2baire_wf add_nat_wf false_wf le_wf nat_properties decidable__le add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf equal_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot all_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat subtype_rel_self eq-finite-seqs-implies-eq-upto
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule lambdaEquality extract_by_obid sqequalHypSubstitution isectElimination thin functionExtensionality applyEquality hypothesisEquality because_Cache hypothesis lambdaFormation unionElimination equalityElimination productElimination independent_isectElimination setElimination rename addEquality natural_numberEquality dependent_set_memberEquality independent_pairFormation equalityTransitivity equalitySymmetry applyLambdaEquality dependent_functionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination instantiate cumulativity functionEquality axiomEquality setEquality

Latex:
\mforall{}[a:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}].  \mforall{}[x:\mBbbN{}].  \mforall{}[F:\mforall{}b:\{b:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}|  a  =  b\}  .  \mexists{}n:\mBbbN{}.  ((b  n)  \mgeq{}  ((a  x)  +  1)  )].
(kripke2b-baire-seq(a;x;F)  \mmember{}  (\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{})

Date html generated: 2017_09_29-PM-06_09_09
Last ObjectModification: 2017_04_22-PM-05_37_55

Theory : continuity

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