Nuprl Lemma : non_nil_length
∀[T:Type]. ∀[L:T List]. 0 < ||L|| supposing ¬(L = [] ∈ (T List))
Proof
Definitions occuring in Statement :
length: ||as||,
nil: [],
list: T List,
less_than: a < b,
uimplies: b supposing a,
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,
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,
and: P ∧ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True
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,
member-less_than,
length_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
unionElimination,
sqequalRule,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
lambdaFormation,
setElimination,
rename,
natural_numberEquality,
addEquality,
independent_pairFormation,
independent_isectElimination,
applyEquality,
lambdaEquality,
intEquality,
because_Cache,
minusEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. 0 < ||L|| supposing \mneg{}(L = [])
Date html generated:
2019_06_20-PM-00_40_01
Last ObjectModification:
2018_09_14-PM-04_24_24
Theory : list_0
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