Nuprl Lemma : last_cons2
∀[L:Top List]. ∀[x:Top]. (last([x / L]) ~ if null(L) then x else last(L) fi )
Proof
Definitions occuring in Statement :
last: last(L),
null: null(as),
cons: [a / b],
list: T List,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
sqequal: s ~ t
Definitions unfolded in proof :
last: last(L),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uall: ∀[x:A]. B[x],
select: L[n],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
nil: [],
cons: [a / b],
subtype_rel: A ⊆r B,
ge: i ≥ j ,
le: A ≤ B,
length: ||as||,
list_ind: list_ind,
has-value: (a)↓,
nat_plus: ℕ+,
nat: ℕ
Lemmas referenced :
length_of_cons_lemma,
spread_cons_lemma,
lt_int_wf,
subtract_wf,
length_wf,
top_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
less_than_wf,
null_wf,
assert_of_null,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
list_wf,
list-cases,
length_of_nil_lemma,
stuck-spread,
base_wf,
nil_wf,
product_subtype_list,
non_neg_length,
length_wf_nat,
nat_wf,
subtype_rel-equal,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
minus-minus,
add-commutes,
add_functionality_wrt_le,
zero-add,
le-add-cancel2,
not-lt-2,
null_nil_lemma,
btrue_wf,
and_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
value-type-has-value,
int-value-type,
int_subtype_base,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
false_wf,
omega-shadow,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isect_memberFormation,
isectElimination,
addEquality,
hypothesisEquality,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
lessCases,
sqequalAxiom,
because_Cache,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
hypothesis_subsumption,
sqequalIntensionalEquality,
applyEquality,
lambdaEquality,
intEquality,
minusEquality,
cumulativity,
callbyvalueReduce,
sqleReflexivity,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
multiplyEquality
Latex:
\mforall{}[L:Top List]. \mforall{}[x:Top]. (last([x / L]) \msim{} if null(L) then x else last(L) fi )
Date html generated:
2017_04_14-AM-08_37_51
Last ObjectModification:
2017_02_27-PM-03_29_32
Theory : list_0
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