Nuprl Lemma : colength-zero
∀[L:Top List]. ((colength(L) = 0 ∈ ℤ) ⇒ (L ~ []))
Proof
Definitions occuring in Statement :
nil: [],
list: T List,
colength: colength(L),
uall: ∀[x:A]. B[x],
top: Top,
implies: P ⇒ Q,
natural_number: $n,
int: ℤ,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
list: T List,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uimplies: b supposing a,
nil: [],
bfalse: ff,
cons: [a / b],
prop: ℙ,
nat: ℕ,
false: False,
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
exists: ∃x:A. B[x],
squash: ↓T,
uiff: uiff(P;Q),
guard: {T},
sq_stable: SqStable(P),
top: Top,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
sq_type: SQType(T),
ge: i ≥ j
Lemmas referenced :
colist-ext,
top_wf,
isaxiom_wf_listunion,
colist_wf,
bool_wf,
subtype_rel_b-union-left,
unit_wf2,
axiom-listunion,
subtype_rel_b-union-right,
non-axiom-listunion,
equal_wf,
equal-wf-T-base,
colength_wf_list,
nat_wf,
list_wf,
colength_wf,
subtype_partial_sqtype_base,
set_subtype_base,
le_wf,
int_subtype_base,
termination,
set-value-type,
int-value-type,
value-type-has-value,
subtype_rel-equal,
base_wf,
le_antisymmetry_iff,
sq_stable__le,
add_functionality_wrt_le,
add-associates,
zero-add,
add-swap,
add-zero,
add-commutes,
le-add-cancel,
nat_properties,
subtract_wf,
minus-zero,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
hypothesis,
promote_hyp,
productElimination,
hypothesis_subsumption,
hypothesisEquality,
applyEquality,
sqequalRule,
unionElimination,
equalityElimination,
productEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
intEquality,
lambdaEquality,
baseClosed,
sqequalAxiom,
because_Cache,
voidElimination,
natural_numberEquality,
callbyvalueAdd,
dependent_pairFormation,
sqequalIntensionalEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
addEquality,
isect_memberEquality,
voidEquality,
minusEquality,
instantiate,
cumulativity
Latex:
\mforall{}[L:Top List]. ((colength(L) = 0) {}\mRightarrow{} (L \msim{} []))
Date html generated:
2017_04_14-AM-07_54_17
Last ObjectModification:
2017_02_27-PM-03_21_23
Theory : list_0
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