Nuprl Lemma : pos-length
∀[A:Type]. ∀[l:A List]. uiff(¬(l = [] ∈ (A List));0 < ||l||)
Proof
Definitions occuring in Statement :
length: ||as||,
nil: [],
list: T List,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
less_than: a < b,
squash: ↓T
Lemmas referenced :
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal_wf,
not_wf,
equal-wf-T-base,
list_wf,
cons_neq_nil,
less_than_wf,
length_wf,
member-less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_functionElimination,
cumulativity,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
lambdaFormation,
setElimination,
rename,
natural_numberEquality,
addEquality,
independent_isectElimination,
applyEquality,
lambdaEquality,
intEquality,
because_Cache,
minusEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
imageElimination,
independent_pairEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[l:A List]. uiff(\mneg{}(l = []);0 < ||l||)
Date html generated:
2017_04_14-AM-08_54_45
Last ObjectModification:
2017_02_27-PM-03_38_52
Theory : list_0
Home
Index