Nuprl Lemma : co-list-cases
∀[T:Type]
∀x:colist(T)
(((↑isaxiom(x)) ∧ (x = Ax ∈ Unit)) ∨ ((↑ispair(x)) ∧ (∃t:T. ∃y:colist(T). (x = <t, y> ∈ (T × colist(T))))))
Proof
Definitions occuring in Statement :
colist: colist(T),
assert: ↑b,
bfalse: ff,
btrue: tt,
uall: ∀[x:A]. B[x],
ispair: if z is a pair then a otherwise b,
isaxiom: if z = Ax then a otherwise b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
unit: Unit,
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
assert: ↑b,
btrue: tt,
bfalse: ff,
or: P ∨ Q,
and: P ∧ Q,
true: True,
prop: ℙ,
false: False,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
co-list-subtype,
b-union_wf,
unit_wf2,
colist_wf,
false_wf,
equal_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
productEquality,
cumulativity,
imageElimination,
productElimination,
unionElimination,
equalityElimination,
inlEquality,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
dependent_functionElimination,
independent_functionElimination,
inrEquality,
natural_numberEquality,
dependent_pairEquality,
because_Cache,
lambdaEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}x:colist(T). (((\muparrow{}isaxiom(x)) \mwedge{} (x = Ax)) \mvee{} ((\muparrow{}ispair(x)) \mwedge{} (\mexists{}t:T. \mexists{}y:colist(T). (x = <t, y>))))
Date html generated:
2018_05_21-PM-10_19_44
Last ObjectModification:
2017_07_26-PM-06_37_09
Theory : eval!all
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