Nuprl Lemma : hd-wf-not-null
∀[A:Type]. ∀[l:A List]. hd(l) ∈ A supposing ¬↑null(l)
Proof
Definitions occuring in Statement :
hd: hd(l),
null: null(as),
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
implies: P ⇒ Q,
true: True,
false: False,
cons: [a / b],
top: Top,
bfalse: ff,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
hd_wf,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
length_wf_nat,
nat_wf,
decidable__le,
false_wf,
not-ge-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
equal_wf,
not_wf,
assert_wf,
null_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_functionElimination,
natural_numberEquality,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
lambdaFormation,
setElimination,
rename,
addEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
lambdaEquality,
intEquality,
because_Cache,
minusEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[l:A List]. hd(l) \mmember{} A supposing \mneg{}\muparrow{}null(l)
Date html generated:
2017_04_14-AM-09_26_30
Last ObjectModification:
2017_02_27-PM-04_00_41
Theory : list_1
Home
Index