Nuprl Lemma : l_disjoint_cons
∀[T:Type]. ∀[a,b:T List]. ∀[x:T]. uiff(l_disjoint(T;a;[x / b]);(¬(x ∈ a)) ∧ l_disjoint(T;a;b))
Proof
Definitions occuring in Statement :
l_disjoint: l_disjoint(T;l1;l2),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
member: t ∈ T,
top: Top,
so_apply: x[s1;s2;s3],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
l_disjoint: l_disjoint(T;l1;l2),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_ind_cons_lemma,
list_ind_nil_lemma,
l_member_wf,
and_wf,
not_wf,
l_disjoint_wf,
l_disjoint_singleton,
cons_wf,
nil_wf,
uiff_wf,
iff_weakening_uiff,
append_wf,
l_disjoint_append,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
independent_pairFormation,
isect_memberFormation,
introduction,
lambdaFormation,
productElimination,
independent_functionElimination,
isectElimination,
hypothesisEquality,
independent_pairEquality,
lambdaEquality,
because_Cache,
addLevel,
independent_isectElimination,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[a,b:T List]. \mforall{}[x:T]. uiff(l\_disjoint(T;a;[x / b]);(\mneg{}(x \mmember{} a)) \mwedge{} l\_disjoint(T;a;b))
Date html generated:
2016_05_14-AM-07_56_09
Last ObjectModification:
2015_12_26-PM-04_50_30
Theory : list_1
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