Nuprl Lemma : list-is-singleton-iff
∀[T:Type]. ∀L:T List. ∀x:T. (L = [x] ∈ (T List) ⇐⇒ no_repeats(T;L) ∧ (∀f:T. ((f ∈ L) ⇐⇒ f = x ∈ T)))
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
l_member: (x ∈ l),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
top: Top,
select: L[n],
cons: [a / b],
cand: A c∧ B,
less_than: a < b,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uiff: uiff(P;Q),
nat_plus: ℕ+,
subtract: n - m
Lemmas referenced :
no_repeats_witness,
l_member_wf,
cons_wf,
nil_wf,
no_repeats_wf,
list_wf,
istype-universe,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
no_repeats_singleton,
member_singleton,
istype-void,
istype-le,
length_of_cons_lemma,
length_of_nil_lemma,
istype-less_than,
length_wf,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
list-cases,
product_subtype_list,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
no_repeats_cons,
cons_member,
add_nat_plus,
add_nat_wf,
length_wf_nat,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
non_neg_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
universeIsType,
equalityIstype,
inhabitedIsType,
productElimination,
sqequalRule,
productIsType,
functionIsType,
because_Cache,
instantiate,
universeEquality,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
dependent_functionElimination,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
setElimination,
rename,
unionElimination,
approximateComputation,
int_eqEquality,
promote_hyp,
hypothesis_subsumption,
inlFormation_alt,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}x:T. (L = [x] \mLeftarrow{}{}\mRightarrow{} no\_repeats(T;L) \mwedge{} (\mforall{}f:T. ((f \mmember{} L) \mLeftarrow{}{}\mRightarrow{} f = x)))
Date html generated:
2020_05_19-PM-09_42_35
Last ObjectModification:
2019_11_13-PM-00_33_02
Theory : list_1
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