Nuprl Lemma : last_member
∀[T:Type]. ∀L:T List. (last(L) ∈ L) supposing ¬↑null(L)
Proof
Definitions occuring in Statement :
last: last(L),
l_member: (x ∈ l),
null: null(as),
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
universe: Type
Definitions unfolded in proof :
last: last(L),
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
or: P ∨ Q,
gt: i > j,
cons: [a / b],
top: Top,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
exists: ∃x:A. B[x],
nat: ℕ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
cand: A c∧ B,
sq_stable: SqStable(P)
Lemmas referenced :
assert_wf,
null_wf,
assert_of_null,
equal-wf-T-base,
list_wf,
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
equal_wf,
subtract_wf,
length_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
select_wf,
sq_stable__le,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesis,
rename,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
promote_hyp,
unionElimination,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
setElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
addEquality,
applyEquality,
intEquality,
minusEquality,
because_Cache,
productEquality,
imageElimination,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. (last(L) \mmember{} L) supposing \mneg{}\muparrow{}null(L)
Date html generated:
2017_04_14-AM-08_37_45
Last ObjectModification:
2017_02_27-PM-03_29_27
Theory : list_0
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