Nuprl Lemma : disjoint-iff-null-intersection
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a,b:T List]. uiff(l_disjoint(T;a;b);l_intersection(eq;a;b) = [] ∈ (T List))
Proof
Definitions occuring in Statement :
l_intersection: l_intersection(eq;L1;L2),
l_disjoint: l_disjoint(T;l1;l2),
nil: [],
list: T List,
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_disjoint: l_disjoint(T;l1;l2),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
or: P ∨ Q,
cons: [a / b],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
all_wf,
not_wf,
l_member_wf,
equal-wf-T-base,
list_wf,
l_intersection_wf,
deq_wf,
list-cases,
product_subtype_list,
equal_wf,
cons_member,
member-intersection,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
productEquality,
lambdaFormation,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
because_Cache,
baseClosed,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
rename,
inlFormation,
hyp_replacement,
applyLambdaEquality,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[a,b:T List]. uiff(l\_disjoint(T;a;b);l\_intersection(eq;a;b) = [])
Date html generated:
2017_04_17-AM-09_16_20
Last ObjectModification:
2017_02_27-PM-05_21_36
Theory : decidable!equality
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