### Nuprl Lemma : strong-continuity-implies3

`∀[F:(ℕ ⟶ ℕ) ⟶ ℕ]`
`  (↓∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)`
`     ∀f:ℕ ⟶ ℕ. (↓∃n:ℕ. (((M n f) = (inl (F f)) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) `` ((M m f) = (inl (F f)) ∈ (ℕ?)))))))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` squash: `↓T` implies: `P `` Q` and: `P ∧ Q` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` squash: `↓T` exists: `∃x:A. B[x]` so_apply: `x[s]` isl: `isl(x)` outl: `outl(x)` guard: `{T}` sq_stable: `SqStable(P)` implies: `P `` Q` not: `¬A` false: `False` less_than': `less_than'(a;b)` le: `A ≤ B` uimplies: `b supposing a` subtype_rel: `A ⊆r B` lelt: `i ≤ j < k` int_seg: `{i..j-}` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` all: `∀x:A. B[x]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` pi1: `fst(t)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` bfalse: `ff` uiff: `uiff(P;Q)` less_than: `a < b` sq_type: `SQType(T)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B`
Lemmas referenced :  strong-continuity-implies2 istype-nat decidable__lt decidable__assert decidable__and2 btrue_neq_bfalse equal_wf and_wf bfalse_wf assert_elim less_than_wf subtype_rel_self le_weakening2 sq_stable__le false_wf int_seg_subtype int_seg_wf subtype_rel_function le_wf unit_wf2 nat_wf isl_wf assert_wf decidable__exists_int_seg nat_properties int_seg_subtype_nat istype-false int_seg_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf true_wf it_wf imax_wf add_nat_wf imax_nat istype-le add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma union_subtype_base set_subtype_base int_subtype_base unit_subtype_base istype-assert decidable_wf btrue_wf pi1_wf subtype_base_sq squash_wf istype-universe iff_weakening_equal bool_wf bool_subtype_base imax_ub intformless_wf int_formula_prop_less_lemma exists_wf all_wf subtype_rel_union
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality imageElimination productElimination sqequalRule imageMemberEquality baseClosed hypothesis functionIsType inhabitedIsType functionEquality isect_memberEquality voidElimination applyLambdaEquality equalityTransitivity equalitySymmetry unionElimination unionEquality independent_functionElimination independent_pairFormation independent_isectElimination because_Cache dependent_set_memberEquality applyEquality productEquality lambdaEquality rename setElimination natural_numberEquality dependent_functionElimination instantiate lambdaFormation dependent_pairFormation_alt lambdaEquality_alt universeIsType functionExtensionality dependent_set_memberEquality_alt lambdaFormation_alt equalityIsType1 inlEquality_alt approximateComputation int_eqEquality isect_memberEquality_alt productIsType inrEquality_alt addEquality pointwiseFunctionality promote_hyp baseApply closedConclusion equalityIstype intEquality sqequalBase dependent_pairEquality_alt cumulativity universeEquality inlFormation_alt inrFormation_alt

Latex:
\mforall{}[F:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}]
(\mdownarrow{}\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  (\mBbbN{}n?)
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}
(\mdownarrow{}\mexists{}n:\mBbbN{}.  (((M  n  f)  =  (inl  (F  f)))  \mwedge{}  (\mforall{}m:\mBbbN{}.  ((\muparrow{}isl(M  m  f))  {}\mRightarrow{}  ((M  m  f)  =  (inl  (F  f))))))))

Date html generated: 2020_05_19-PM-10_04_43
Last ObjectModification: 2019_12_17-PM-06_03_12

Theory : continuity

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