Nuprl Lemma : pos_length2
∀[A:Type]. ∀[l:A List]. uiff(¬↑null(l);0 < ||l||)
Proof
Definitions occuring in Statement :
length: ||as||,
null: null(as),
list: T List,
assert: ↑b,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
universe: Type
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,
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,
le: A ≤ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T
Lemmas referenced :
list-cases,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
istype-void,
null_cons_lemma,
length_wf_nat,
decidable__lt,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
istype-int,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
istype-assert,
null_wf,
istype-less_than,
length_wf,
member-less_than,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_functionElimination,
natural_numberEquality,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :isect_memberEquality_alt,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
setElimination,
rename,
addEquality,
independent_isectElimination,
applyEquality,
Error :lambdaEquality_alt,
because_Cache,
minusEquality,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
imageElimination,
Error :functionIsTypeImplies,
independent_pairEquality,
Error :isectIsTypeImplies,
Error :universeIsType,
instantiate,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[l:A List]. uiff(\mneg{}\muparrow{}null(l);0 < ||l||)
Date html generated:
2019_06_20-PM-01_19_55
Last ObjectModification:
2019_01_15-PM-02_20_14
Theory : list_1
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