Nuprl Lemma : list-set-type
∀[T:Type]. ∀[L:T List]. (L ∈ {x:T| (x ∈ L)} List)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b
Lemmas referenced :
list_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le_wf,
equal_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
list-subtype,
nil_wf,
cons_wf,
l_member_wf,
cons_member,
subtype_rel_list_set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
universeEquality,
lambdaFormation,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
intEquality,
instantiate,
setEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. (L \mmember{} \{x:T| (x \mmember{} L)\} List)
Date html generated:
2017_04_14-AM-08_40_24
Last ObjectModification:
2017_02_27-PM-03_31_00
Theory : list_0
Home
Index