Nuprl Lemma : insert-not-nil
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a:T]. ∀[L:T List]. (¬(insert(a;L) = [] ∈ (T List)))
Proof
Definitions occuring in Statement :
insert: insert(a;L),
nil: [],
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
insert: insert(a;L),
prop: ℙ,
has-value: (a)↓,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
top: Top,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
equal-wf-T-base,
list_wf,
insert_wf,
deq_wf,
value-type-has-value,
list-value-type,
eval_list_wf,
deq-member_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
and_wf,
equal_wf,
null_wf,
btrue_neq_bfalse,
assert_wf,
bnot_wf,
not_wf,
l_member_wf,
null_cons_lemma,
bfalse_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
because_Cache,
hypothesis,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
baseClosed,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
universeEquality,
callbyvalueReduce,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
setElimination,
rename,
productElimination,
voidEquality,
unionElimination,
instantiate,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[a:T]. \mforall{}[L:T List]. (\mneg{}(insert(a;L) = []))
Date html generated:
2017_04_14-AM-08_53_56
Last ObjectModification:
2017_02_27-PM-03_38_42
Theory : list_0
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