Nuprl Lemma : no-value-bottom
∀[T:Type]. ∀[x:partial(T)]. x ~ ⊥ supposing ¬(x)↓ supposing value-type(T)
Proof
Definitions occuring in Statement :
partial: partial(T),
value-type: value-type(T),
has-value: (a)↓,
bottom: ⊥,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
prop: ℙ,
top: Top,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
partial: partial(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T
Lemmas referenced :
value-type_wf,
partial_wf,
has-value_wf-partial,
not_wf,
istype-void,
bottom-sqle,
partial-not-exception,
has-value_wf_base,
is-exception_wf,
base-partial_wf,
per-partial_wf,
istype-sqle
Rules used in proof :
universeEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
sqequalRule,
independent_isectElimination,
hypothesisEquality,
Error :universeIsType,
axiomSqEquality,
hypothesis,
voidElimination,
Error :isect_memberEquality_alt,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalSqle,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
pointwiseFunctionality,
divergentSqle,
independent_functionElimination,
pertypeElimination,
promote_hyp,
productElimination,
Error :productIsType,
Error :equalityIstype,
sqequalBase,
sqleExtensionalEquality,
independent_pairFormation,
Error :lambdaFormation_alt,
baseClosed,
imageMemberEquality,
hyp_replacement,
Error :dependent_set_memberEquality_alt,
Error :inhabitedIsType,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}[T:Type]. \mforall{}[x:partial(T)]. x \msim{} \mbot{} supposing \mneg{}(x)\mdownarrow{} supposing value-type(T)
Date html generated:
2019_06_20-PM-00_34_42
Last ObjectModification:
2018_11_28-PM-00_40_13
Theory : partial_1
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