### Nuprl Lemma : uniform-continuity-pi-pi-prop2

`∀T:Type. ∀F:(ℕ ⟶ 𝔹) ⟶ T.  ((∀x,y:T.  Dec(x = y ∈ T)) `` (∃n:ℕ. ucpB(T;F;n) `⇐⇒` ∃n:ℕ. ucA(T;F;n)))`

Proof

Definitions occuring in Statement :  uniform-continuity-pi-pi: `ucpB(T;F;n)` uniform-continuity-pi: `ucA(T;F;n)` nat: `ℕ` bool: `𝔹` decidable: `Dec(P)` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` exists: `∃x:A. B[x]` uniform-continuity-pi-pi: `ucpB(T;F;n)` uimplies: `b supposing a` nat: `ℕ` sq_type: `SQType(T)` guard: `{T}` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` squash: `↓T` uniform-continuity-pi: `ucA(T;F;n)` sq_stable: `SqStable(P)`
Lemmas referenced :  exists_wf nat_wf uniform-continuity-pi-pi_wf bool_wf uniform-continuity-pi_wf all_wf decidable_wf equal_wf set-value-type le_wf int-value-type subtype_base_sq set_subtype_base int_subtype_base uniform-continuity-pi-dec int_seg_subtype_nat false_wf int_seg_properties nat_properties decidable__lt 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 int_seg_wf uniform-continuity-pi-search_wf subtype_rel_union not_wf top_wf or_wf uniform-continuity-pi-search-prop2 itermAdd_wf itermConstant_wf int_term_value_add_lemma int_term_value_constant_lemma lelt_wf decidable__le intformle_wf int_formula_prop_le_lemma subtype_rel_dep_function subtype_rel_self squash_wf sq_stable__all sq_stable__equal sq_stable__le le_weakening2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis sqequalRule lambdaEquality cumulativity hypothesisEquality functionExtensionality applyEquality functionEquality universeEquality productElimination dependent_pairFormation cutEval dependent_set_memberEquality equalityTransitivity equalitySymmetry independent_isectElimination intEquality natural_numberEquality setElimination rename instantiate dependent_functionElimination independent_functionElimination unionElimination int_eqEquality isect_memberEquality voidElimination voidEquality computeAll because_Cache addEquality applyLambdaEquality imageMemberEquality baseClosed imageElimination axiomEquality

Latex:
\mforall{}T:Type.  \mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  T.    ((\mforall{}x,y:T.    Dec(x  =  y))  {}\mRightarrow{}  (\mexists{}n:\mBbbN{}.  ucpB(T;F;n)  \mLeftarrow{}{}\mRightarrow{}  \mexists{}n:\mBbbN{}.  ucA(T;F;n)))

Date html generated: 2017_04_17-AM-09_59_09
Last ObjectModification: 2017_02_27-PM-05_53_20

Theory : continuity

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