Nuprl Lemma : nil-contains
∀[T:Type]. ∀L:T List. (L ⊆ [] ⇐⇒ ↑null(L))
Proof
Definitions occuring in Statement :
l_contains: A ⊆ B,
null: null(as),
nil: [],
list: T List,
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
top: Top,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
true: True,
prop: ℙ,
rev_implies: P ⇐ Q,
false: False,
l_contains: A ⊆ B,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
guard: {T},
decidable: Dec(P),
uiff: uiff(P;Q),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
ge: i ≥ j
Lemmas referenced :
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
list_wf,
l_contains_wf,
nil_wf,
l_contains_nil,
true_wf,
length_of_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
lelt_wf,
length_wf,
btrue_wf,
member-implies-null-eq-bfalse,
select_wf,
cons_wf,
non_neg_length,
intformle_wf,
int_formula_prop_le_lemma,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
cumulativity,
universeEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
pointwiseFunctionality,
baseApply,
closedConclusion,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
independent_functionElimination,
addEquality,
because_Cache
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. (L \msubseteq{} [] \mLeftarrow{}{}\mRightarrow{} \muparrow{}null(L))
Date html generated:
2017_04_17-AM-07_28_33
Last ObjectModification:
2017_02_27-PM-04_06_19
Theory : list_1
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