Nuprl Lemma : pos_length
∀[A:Type]. ∀[l:A List]. ||l|| ≥ 1 supposing ¬(l = [] ∈ (A List))
Proof
Definitions occuring in Statement :
length: ||as||,
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
not: ¬A,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
cons: [a / b],
uimplies: b supposing a,
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
top: Top,
guard: {T},
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
list-cases,
product_subtype_list,
less_than'_wf,
length_wf,
not_wf,
equal_wf,
list_wf,
nil_wf,
length_of_cons_lemma,
cons_wf,
non_neg_length,
decidable__le,
false_wf,
not-ge-2,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
sqequalRule,
isect_memberEquality,
independent_pairEquality,
lambdaEquality,
because_Cache,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
universeEquality,
independent_functionElimination,
voidEquality,
addEquality,
independent_pairFormation,
lambdaFormation,
independent_isectElimination,
applyEquality,
intEquality,
minusEquality
Latex:
\mforall{}[A:Type]. \mforall{}[l:A List]. ||l|| \mgeq{} 1 supposing \mneg{}(l = [])
Date html generated:
2016_05_14-AM-06_33_49
Last ObjectModification:
2015_12_26-PM-00_36_55
Theory : list_0
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