### Nuprl Lemma : kripke2b-seq_wf

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

Proof

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

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-seq(a;x;F)  \mmember{}  (\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{})

Date html generated: 2017_04_20-AM-07_37_24
Last ObjectModification: 2017_04_19-PM-03_34_36

Theory : continuity

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