Nuprl Lemma : insert-cases2
∀[T:Type]. ∀eq:EqDecider(T). ∀a:T. ∀L:T List. (((a ∈ L) ∧ (insert(a;L) ~ L)) ∨ ((¬(a ∈ L)) ∧ (insert(a;L) ~ [a / L])))
Proof
Definitions occuring in Statement :
insert: insert(a;L),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
insert: insert(a;L),
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
list_wf,
deq_wf,
subtype_rel_list,
top_wf,
value-type-has-value,
list-value-type,
deq-member_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
l_member_wf,
not_wf,
bnot_wf,
eval_list_sq,
iff_transitivity,
iff_weakening_uiff,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
universeEquality,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
baseClosed,
inlFormation,
independent_pairFormation,
productEquality,
sqequalIntensionalEquality,
independent_functionElimination,
baseApply,
closedConclusion,
inrFormation,
callbyvalueReduce,
unionElimination,
equalityElimination,
dependent_functionElimination,
productElimination,
impliesFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}a:T. \mforall{}L:T List.
(((a \mmember{} L) \mwedge{} (insert(a;L) \msim{} L)) \mvee{} ((\mneg{}(a \mmember{} L)) \mwedge{} (insert(a;L) \msim{} [a / L])))
Date html generated:
2017_04_14-AM-08_53_48
Last ObjectModification:
2017_02_27-PM-03_37_44
Theory : list_0
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