### Nuprl Lemma : bar-diverges-iff

`∀[T:Type]. ∀[x:bar-base(T)].  (x↑ `⇐⇒` ∀[a:T]. (¬x↓a))`

Proof

Definitions occuring in Statement :  bar-diverges: `x↑` bar-converges: `x↓a` bar-base: `bar-base(T)` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` not: `¬A` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` not: `¬A` false: `False` prop: `ℙ` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bar-diverges: `x↑` all: `∀x:A. B[x]` uimplies: `b supposing a` bar-converges: `x↓a` exists: `∃x:A. B[x]` isl: `isl(x)` outl: `outl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff`
Lemmas referenced :  bar-converges-not-diverges bar-converges_wf bar-diverges_wf uall_wf not_wf assert_wf isl_wf unit_wf2 bar-val_wf nat_wf bar-base_wf outl_wf equal_wf true_wf false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation thin extract_by_obid sqequalHypSubstitution isectElimination because_Cache hypothesisEquality independent_functionElimination hypothesis voidElimination cumulativity sqequalRule lambdaEquality dependent_functionElimination productElimination independent_pairEquality isect_memberEquality universeEquality independent_isectElimination dependent_pairFormation unionEquality inlEquality unionElimination equalityTransitivity equalitySymmetry

Latex:
\mforall{}[T:Type].  \mforall{}[x:bar-base(T)].    (x\muparrow{}  \mLeftarrow{}{}\mRightarrow{}  \mforall{}[a:T].  (\mneg{}x\mdownarrow{}a))

Date html generated: 2017_04_14-AM-07_46_07
Last ObjectModification: 2017_02_27-PM-03_16_30

Theory : co-recursion

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