Nuprl Lemma : member-insert-int
∀[T:Type]. ∀l:T List. ∀x,z:T. ((z ∈ insert-int(x;l)) ⇐⇒ (z = x ∈ T) ∨ (z ∈ l)) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
insert-int: insert-int(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 :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
not: ¬A,
false: False,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
list_induction,
all_wf,
iff_wf,
l_member_wf,
insert-int_wf,
or_wf,
equal_wf,
list_wf,
insert_int_nil_lemma,
member_singleton,
cons_wf,
nil_wf,
insert-int-cons,
subtype_rel_list,
subtype_rel_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
cons_member,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
cumulativity,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
because_Cache,
applyEquality,
intEquality,
universeEquality,
inlFormation,
unionElimination,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
inrFormation,
orFunctionality,
dependent_pairFormation,
promote_hyp,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}l:T List. \mforall{}x,z:T. ((z \mmember{} insert-int(x;l)) \mLeftarrow{}{}\mRightarrow{} (z = x) \mvee{} (z \mmember{} l)) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2017_09_29-PM-05_49_04
Last ObjectModification:
2017_07_26-PM-01_37_26
Theory : list_0
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