Nuprl Lemma : last-insert
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. ∀[x:T]. (last(insert(x;L)) = if null(L) then x else last(L) fi ∈ T)
Proof
Definitions occuring in Statement :
insert: insert(a;L),
last: last(L),
null: null(as),
list: T List,
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
top: Top,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
bfalse: ff,
last: last(L),
select: L[n],
length: ||as||,
list_ind: list_ind,
insert: insert(a;L),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
insert_nil_lemma,
null_nil_lemma,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le_wf,
equal_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
null_cons_lemma,
list_wf,
deq_wf,
eval_list_sq,
cons_wf,
subtype_rel_list,
top_wf,
deq_member_cons_lemma,
bor_wf,
deq-member_wf,
bool_wf,
eqtt_to_assert,
assert-deq-member,
last_wf,
assert_elim,
null_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
assert_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
l_member_wf,
squash_wf,
true_wf,
last_cons,
bfalse_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
because_Cache,
cumulativity,
applyEquality,
unionElimination,
voidEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
instantiate,
universeEquality,
equalityElimination,
addLevel,
levelHypothesis,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List]. \mforall{}[x:T].
(last(insert(x;L)) = if null(L) then x else last(L) fi )
Date html generated:
2017_04_14-AM-08_54_05
Last ObjectModification:
2017_02_27-PM-03_39_10
Theory : list_0
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