Nuprl Lemma : cons-has-member
∀[S:Type]. ∀a:S. ∀[b:S List]. ∃x:S. (x ∈ [a / b])
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
top: Top,
select: L[n],
cons: [a / b],
cand: A c∧ B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
guard: {T},
uimplies: b supposing a,
sq_stable: SqStable(P),
prop: ℙ
Lemmas referenced :
istype-false,
istype-le,
length_of_cons_lemma,
istype-void,
add_nat_plus,
length_wf_nat,
istype-less_than,
cons_wf,
length_wf,
select_wf,
sq_stable__le,
l_member_wf,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
Error :dependent_pairFormation_alt,
hypothesisEquality,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
hypothesis,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
Error :inhabitedIsType,
setElimination,
rename,
imageElimination,
productElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
Error :productIsType,
independent_isectElimination,
Error :universeIsType,
instantiate,
universeEquality
Latex:
\mforall{}[S:Type]. \mforall{}a:S. \mforall{}[b:S List]. \mexists{}x:S. (x \mmember{} [a / b])
Date html generated:
2019_06_20-PM-00_40_38
Last ObjectModification:
2019_03_06-AM-11_06_29
Theory : list_0
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