Nuprl Lemma : member_iff_sublist
∀[T:Type]. ∀x:T. ∀L:T List. ((x ∈ L) ⇐⇒ [x] ⊆ L)
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
l_member: (x ∈ l),
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
sublist: L1 ⊆ L2,
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
top: Top,
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
increasing: increasing(f;k),
subtract: n - m,
sq_type: SQType(T),
select: L[n],
cons: [a / b],
less_than: a < b,
squash: ↓T,
true: True
Lemmas referenced :
length_of_cons_lemma,
length_of_nil_lemma,
exists_wf,
nat_wf,
less_than_wf,
length_wf,
equal_wf,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
int_seg_wf,
cons_wf,
nil_wf,
increasing_wf,
length_wf_nat,
all_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
non_neg_length,
lelt_wf,
list_wf,
false_wf,
le_wf,
length-singleton,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
productElimination,
isectElimination,
lambdaEquality,
productEquality,
setElimination,
rename,
because_Cache,
cumulativity,
hypothesisEquality,
independent_isectElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
functionEquality,
functionExtensionality,
applyEquality,
addEquality,
independent_functionElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
universeEquality,
instantiate,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}L:T List. ((x \mmember{} L) \mLeftarrow{}{}\mRightarrow{} [x] \msubseteq{} L)
Date html generated:
2017_04_14-AM-09_29_39
Last ObjectModification:
2017_02_27-PM-04_02_02
Theory : list_1
Home
Index