Nuprl Lemma : insert_property
∀[T:Type]
∀eq:EqDecider(T). ∀a:T. ∀L:T List.
((∀b:T. ((b ∈ insert(a;L)) ⇐⇒ (b = a ∈ T) ∨ (b ∈ L))) ∧ no_repeats(T;insert(a;L)) supposing no_repeats(T;L))
Proof
Definitions occuring in Statement :
insert: insert(a;L),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
uimplies: b supposing a,
cand: A c∧ B,
iff: P ⇐⇒ Q,
guard: {T},
prop: ℙ,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
decidable__l_member,
decidable-equal-deq,
list_wf,
deq_wf,
insert-cases,
equal_wf,
l_member_wf,
and_wf,
or_wf,
no_repeats_witness,
no_repeats_wf,
no_repeats_cons,
cons_wf,
cons_member,
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
cumulativity,
universeEquality,
unionElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
sqequalRule,
inrFormation,
equalitySymmetry,
dependent_set_memberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
hyp_replacement,
Error :applyLambdaEquality,
addLevel,
allFunctionality,
impliesFunctionality,
productEquality,
isectEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}a:T. \mforall{}L:T List.
((\mforall{}b:T. ((b \mmember{} insert(a;L)) \mLeftarrow{}{}\mRightarrow{} (b = a) \mvee{} (b \mmember{} L)))
\mwedge{} no\_repeats(T;insert(a;L)) supposing no\_repeats(T;L))
Date html generated:
2016_10_21-AM-09_51_48
Last ObjectModification:
2016_07_12-AM-05_10_20
Theory : list_0
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