Nuprl Lemma : before_last
∀[T:Type]. ∀L:T List. ∀x:T. ((x ∈ L) ⇒ x before last(L) ∈ L supposing ¬(x = last(L) ∈ T))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
last: last(L),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
not: ¬A,
false: False,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
top: Top,
or: P ∨ Q,
l_before: x before y ∈ l,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
last: last(L),
select: L[n],
cons: [a / b],
subtract: n - m,
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
not_wf,
equal_wf,
last_wf,
assert_elim,
null_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
assert_wf,
l_before_wf,
list_wf,
nil_member,
nil_wf,
cons_wf,
null_cons_lemma,
istype-void,
bfalse_wf,
cons_member,
cons_sublist_cons,
assert_of_null,
list-cases,
null_nil_lemma,
btrue_wf,
product_subtype_list,
false_wf,
squash_wf,
true_wf,
last_cons,
iff_weakening_equal,
sublist_wf,
subtype_rel_self,
member_iff_sublist,
last_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
cumulativity,
functionEquality,
because_Cache,
hypothesis,
isectEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
Error :universeIsType,
rename,
Error :functionIsType,
Error :inhabitedIsType,
Error :isectIsType,
dependent_functionElimination,
universeEquality,
productElimination,
Error :equalityIsType1,
Error :isect_memberEquality_alt,
Error :unionIsType,
unionElimination,
hyp_replacement,
applyLambdaEquality,
promote_hyp,
hypothesis_subsumption,
natural_numberEquality,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
Error :inlFormation_alt,
instantiate,
independent_pairFormation,
Error :inrFormation_alt,
Error :productIsType
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} x before last(L) \mmember{} L supposing \mneg{}(x = last(L)))
Date html generated:
2019_06_20-PM-01_23_37
Last ObjectModification:
2018_09_29-PM-00_04_11
Theory : list_1
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