Nuprl Lemma : member-s-insert
∀[T:Type]. ∀x:T. ∀L:T List. ∀z:T. ((z ∈ s-insert(x;L)) ⇐⇒ (z = x ∈ T) ∨ (z ∈ L)) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
s-insert: s-insert(x;l),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
guard: {T},
sq_type: SQType(T),
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
s-insert: s-insert(x;l),
implies: P ⇒ Q,
so_apply: x[s],
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
assert_of_le_int,
bnot_of_lt_int,
assert_functionality_wrt_uiff,
assert_of_lt_int,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
le_wf,
le_int_wf,
less_than_wf,
lt_int_wf,
istype-assert,
istype-int,
not_wf,
bnot_wf,
cons_member,
int_subtype_base,
subtype_base_sq,
assert_wf,
bool_wf,
equal-wf-T-base,
eq_int_wf,
istype-universe,
subtype_rel_wf,
list_ind_cons_lemma,
cons_wf,
member_singleton,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
nil_wf,
istype-void,
list_ind_nil_lemma,
list_wf,
equal_wf,
s-insert_wf,
l_member_wf,
iff_wf,
list_induction
Rules used in proof :
equalityElimination,
cumulativity,
Error :inrFormation_alt,
baseClosed,
applyEquality,
universeEquality,
instantiate,
intEquality,
Error :productIsType,
Error :functionIsType,
promote_hyp,
productElimination,
Error :unionIsType,
equalitySymmetry,
equalityTransitivity,
because_Cache,
unionElimination,
Error :inhabitedIsType,
Error :equalityIstype,
Error :inlFormation_alt,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
dependent_functionElimination,
independent_functionElimination,
Error :universeIsType,
unionEquality,
independent_isectElimination,
functionEquality,
Error :lambdaEquality_alt,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
Error :lambdaFormation_alt,
rename,
thin,
hypothesis,
axiomEquality,
sqequalRule,
introduction,
cut,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}L:T List. \mforall{}z:T. ((z \mmember{} s-insert(x;L)) \mLeftarrow{}{}\mRightarrow{} (z = x) \mvee{} (z \mmember{} L)) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2019_06_20-PM-00_42_20
Last ObjectModification:
2019_06_19-AM-10_43_11
Theory : list_0
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