Nuprl Lemma : weak-continuity-implies-strong-cantor

`∀F:(ℕ ⟶ 𝔹) ⟶ ℕ`
`  ∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (ℕ?)`
`   ∀f:ℕ ⟶ 𝔹. ((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (ℕ?) supposing ↑isl(M n f)))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` bool: `𝔹` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` 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 :  all: `∀x:A. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` uall: `∀[x:A]. B[x]` nat: `ℕ` implies: `P `` Q` exposed-it: `exposed-it` 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` bfalse: `ff` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` ext2Cantor: `ext2Cantor(n;f;d)` int_seg: `{i..j-}` lelt: `i ≤ j < k` isl: `isl(x)`
Lemmas referenced :  strong-continuity2-implies-uniform-continuity2-nat nat_wf bool_wf le_int_wf eqtt_to_assert assert_of_le_int ext2Cantor_wf int_seg_wf btrue_wf unit_wf2 eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot le_wf all_wf exists_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self isect_wf assert_wf isl_wf nat_properties satisfiable-full-omega-tt intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf lt_int_wf assert_of_lt_int less_than_wf int_seg_properties intformand_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_less_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality productElimination functionEquality hypothesis dependent_pairFormation lambdaEquality isectElimination setElimination rename because_Cache unionElimination equalityElimination sqequalRule independent_isectElimination inlEquality applyEquality functionExtensionality natural_numberEquality equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity independent_functionElimination voidElimination inrEquality axiomEquality independent_pairFormation productEquality unionEquality int_eqEquality intEquality isect_memberEquality voidEquality computeAll isect_memberFormation

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{}
\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  (\mBbbN{}?)
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}
((\mexists{}n:\mBbbN{}.  ((M  n  f)  =  (inl  (F  f))))  \mwedge{}  (\mforall{}n:\mBbbN{}.  (M  n  f)  =  (inl  (F  f))  supposing  \muparrow{}isl(M  n  f)))

Date html generated: 2017_04_17-AM-10_00_05
Last ObjectModification: 2017_02_27-PM-05_53_14

Theory : continuity

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