Nuprl Lemma : per-union-elim1
∀[A,B:Type]. ∀x:per-union(A;B). per-or(x ~ inl outl(x);x ~ inr outr(x) )
Proof
Definitions occuring in Statement :
per-union: per-union(A;B),
per-or: per-or(A;B),
outr: outr(x),
outl: outl(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
inr: inr x ,
inl: inl x,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
per-union: per-union(A;B),
uand: uand(A;B),
has-value: (a)↓,
outl: outl(x),
outr: outr(x),
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
top: Top,
guard: {T},
squash: ↓T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
true: True,
false: False
Lemmas referenced :
has-value_wf_base,
is-exception_wf,
member-per-or-left,
uand_wf,
sqle_wf_base,
istype-universe,
istype-top,
istype-void,
if-per-void,
per-void_wf,
member-per-or-right,
equal-wf-base,
member_wf,
squash_wf,
true_wf,
per-or-equal,
subtype_base_sq,
int_subtype_base,
per-union_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
introduction,
pointwiseFunctionality,
sqequalHypSubstitution,
sqequalRule,
pertypeElimination,
cut,
isectElimination,
thin,
baseClosed,
axiomSqleEquality,
divergentSqle,
sqleReflexivity,
extract_by_obid,
hypothesis,
because_Cache,
promote_hyp,
isinlCases,
hypothesisEquality,
independent_isectElimination,
sqequalIntensionalEquality,
baseApply,
closedConclusion,
axiomSqEquality,
Error :isectIsType,
Error :universeIsType,
Error :equalityIsType4,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
voidElimination,
rename,
isaxiomCases,
independent_functionElimination,
isinrCases,
equalityTransitivity,
equalitySymmetry,
isectEquality,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
universeEquality,
independent_pairFormation,
sqequalElimination,
natural_numberEquality,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
imageMemberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}x:per-union(A;B). per-or(x \msim{} inl outl(x);x \msim{} inr outr(x) )
Date html generated:
2019_06_20-AM-11_30_47
Last ObjectModification:
2018_10_06-AM-10_00_54
Theory : per!type
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