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
Home
Index