Nuprl Lemma : length_one
∀[T:Type]. ∀L:T List. uiff(||L|| = 1 ∈ ℤ;∃x:T. (L = [x] ∈ (T List)))
Proof
Definitions occuring in Statement :
length: ||as||,
cons: [a / b],
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
true: True,
false: False,
cons: [a / b],
top: Top,
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
squash: ↓T,
subtract: n - m,
sq_stable: SqStable(P),
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j
Lemmas referenced :
list-cases,
length_of_nil_lemma,
subtype_base_sq,
int_subtype_base,
product_subtype_list,
length_of_cons_lemma,
cons_wf,
nil_wf,
equal_wf,
list_wf,
length_wf_nat,
nat_wf,
subtype_rel-equal,
base_wf,
le_antisymmetry_iff,
length_wf,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
sq_stable__le,
add_functionality_wrt_le,
zero-add,
le-add-cancel2,
equal-wf-T-base,
exists_wf,
subtract_wf,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_functionElimination,
unionElimination,
sqequalRule,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
because_Cache,
sqequalIntensionalEquality,
applyEquality,
applyLambdaEquality,
setElimination,
imageMemberEquality,
baseClosed,
addEquality,
lambdaEquality,
minusEquality,
imageElimination,
dependent_set_memberEquality,
hyp_replacement,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. uiff(||L|| = 1;\mexists{}x:T. (L = [x]))
Date html generated:
2017_04_14-AM-08_35_38
Last ObjectModification:
2017_02_27-PM-03_28_05
Theory : list_0
Home
Index