Nuprl Lemma : insert-cases
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a:T]. ∀[L:T List].
(insert(a;L) ~ L supposing (a ∈ L) ∧ insert(a;L) ~ [a / L] supposing ¬(a ∈ L))
Proof
Definitions occuring in Statement :
insert: insert(a;L),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
insert: insert(a;L),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
has-value: (a)↓,
all: ∀x:A. B[x],
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,
l_member_wf,
not_wf,
subtype_rel_list,
top_wf,
value-type-has-value,
list-value-type,
deq-member_wf,
bool_wf,
equal-wf-T-base,
assert_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,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
sqequalAxiom,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
cumulativity,
universeEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
voidElimination,
voidEquality,
baseClosed,
independent_pairFormation,
independent_functionElimination,
callbyvalueReduce,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_functionElimination,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[a:T]. \mforall{}[L:T List].
(insert(a;L) \msim{} L supposing (a \mmember{} L) \mwedge{} insert(a;L) \msim{} [a / L] supposing \mneg{}(a \mmember{} L))
Date html generated:
2017_04_14-AM-08_53_39
Last ObjectModification:
2017_02_27-PM-03_37_49
Theory : list_0
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