Nuprl Lemma : cons-sub-co-list-nil
∀[T:Type]. ∀x:T. ∀L:colist(T). (sub-co-list(T;[x / L];[]) ⇐⇒ False)
Proof
Definitions occuring in Statement :
sub-co-list: sub-co-list(T;s1;s2),
cons: [a / b],
nil: [],
colist: colist(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
false: False,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sub-co-list: sub-co-list(T;s1;s2),
exists: ∃x:A. B[x],
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uimplies: b supposing a,
nil: [],
list-at: L1@L2,
ifthenelse: if b then t else f fi ,
null: null(as),
bfalse: ff,
cons: [a / b],
top: Top,
not: ¬A,
false: False,
co-cons: [x / L],
prop: ℙ,
list: T List,
rev_implies: P ⇐ Q
Lemmas referenced :
colist_wf,
istype-universe,
colist-ext,
nat_wf,
isaxiom_wf_listunion,
subtype_rel_b-union-left,
unit_wf2,
axiom-listunion,
subtype_rel_b-union-right,
non-axiom-listunion,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
istype-void,
bfalse_wf,
btrue_neq_bfalse,
sub-co-list_wf,
co-cons_wf,
nil_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
universeEquality,
independent_pairFormation,
productElimination,
promote_hyp,
hypothesis_subsumption,
applyEquality,
sqequalRule,
Error :inhabitedIsType,
unionElimination,
equalityElimination,
productEquality,
independent_isectElimination,
rename,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
applyLambdaEquality,
Error :isect_memberEquality_alt,
voidElimination,
because_Cache,
Error :lambdaEquality_alt,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}L:colist(T). (sub-co-list(T;[x / L];[]) \mLeftarrow{}{}\mRightarrow{} False)
Date html generated:
2019_06_20-PM-01_22_07
Last ObjectModification:
2019_01_02-PM-04_52_20
Theory : list_1
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