Nuprl Lemma : length_of_not_nil
∀[A:Type]. ∀[as:A List]. uiff(¬(as = [] ∈ (A List));||as|| ≥ 1 )
Proof
Definitions occuring in Statement :
length: ||as||,
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
not: ¬A,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
le: A ≤ B,
prop: ℙ,
less_than': less_than'(a;b),
true: True,
cons: [a / b],
top: Top,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtract: n - m,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
decidable: Dec(P)
Lemmas referenced :
list-cases,
length_of_nil_lemma,
nil_wf,
less_than'_wf,
not_wf,
equal-wf-base,
list_wf,
ge_wf,
product_subtype_list,
length_of_cons_lemma,
length_wf,
equal-wf-T-base,
cons_wf,
cons_neq_nil,
non_neg_length,
length_wf_nat,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
equal_wf,
add-commutes,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
minus-one-mul,
zero-add,
one-mul,
add-mul-special,
add-associates,
two-mul,
mul-distributes-right,
zero-mul,
not-ge-2,
false_wf,
add-swap,
omega-shadow,
less_than_wf,
nat_properties,
decidable__le
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_pairFormation,
isect_memberFormation,
independent_functionElimination,
cumulativity,
voidElimination,
productElimination,
independent_pairEquality,
lambdaEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
because_Cache,
lambdaFormation,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
addEquality,
universeEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
applyEquality,
intEquality,
independent_isectElimination,
multiplyEquality,
minusEquality,
dependent_set_memberEquality,
imageMemberEquality,
setElimination,
rename
Latex:
\mforall{}[A:Type]. \mforall{}[as:A List]. uiff(\mneg{}(as = []);||as|| \mgeq{} 1 )
Date html generated:
2017_04_14-AM-08_36_19
Last ObjectModification:
2017_02_27-PM-03_28_31
Theory : list_0
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