### Nuprl Lemma : strong-continuity2-iff-3

`∀[T:Type]. ∀[F:(ℕ ⟶ T) ⟶ ℕ].  (strong-continuity2(T;F) `⇐⇒` strong-continuity3(T;F))`

Proof

Definitions occuring in Statement :  strong-continuity3: `strong-continuity3(T;F)` strong-continuity2: `strong-continuity2(T;F)` nat: `ℕ` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  guard: `{T}` sq_type: `SQType(T)` not: `¬A` false: `False` less_than': `less_than'(a;b)` le: `A ≤ B` uimplies: `b supposing a` nat: `ℕ` so_apply: `x[s]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` cand: `A c∧ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` strong-continuity2: `strong-continuity2(T;F)` strong-continuity3: `strong-continuity3(T;F)` rev_implies: `P `` Q` prop: `ℙ` member: `t ∈ T` implies: `P `` Q` and: `P ∧ Q` iff: `P `⇐⇒` Q` uall: `∀[x:A]. B[x]`
Lemmas referenced :  and_wf int_subtype_base le_wf set_subtype_base subtype_base_sq strong-continuity3_wf isect_wf exists_wf all_wf isl_wf assert_wf false_wf int_seg_subtype_nat int_seg_wf subtype_rel_dep_function unit_wf2 equal_wf nat_wf strong-continuity2_wf strong-continuity2-implies-3
Rules used in proof :  applyLambdaEquality dependent_set_memberEquality intEquality instantiate universeEquality inlEquality productEquality equalitySymmetry equalityTransitivity independent_isectElimination rename setElimination natural_numberEquality lambdaEquality sqequalRule because_Cache unionEquality dependent_functionElimination dependent_pairFormation productElimination functionEquality applyEquality functionExtensionality cumulativity hypothesis independent_functionElimination hypothesisEquality thin isectElimination sqequalHypSubstitution extract_by_obid introduction cut lambdaFormation independent_pairFormation isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}[T:Type].  \mforall{}[F:(\mBbbN{}  {}\mrightarrow{}  T)  {}\mrightarrow{}  \mBbbN{}].    (strong-continuity2(T;F)  \mLeftarrow{}{}\mRightarrow{}  strong-continuity3(T;F))

Date html generated: 2017_09_29-PM-06_05_14
Last ObjectModification: 2017_09_04-PM-00_14_51

Theory : continuity

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