Nuprl Lemma : fset-distributive
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a,b,c:fset(T)]. (a ⋂ b ⋃ c = a ⋂ b ⋃ a ⋂ c ∈ fset(T))
Proof
Definitions occuring in Statement :
fset-intersection: a ⋂ b,
fset-union: x ⋃ y,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
member-fset-union,
member-fset-intersection,
iff_weakening_uiff,
iff_transitivity,
uiff_wf,
or_wf,
and_wf,
sq_stable_from_decidable,
decidable__fset-member,
deq_wf,
fset_wf,
fset-member_wf,
fset-member_witness,
fset-union_wf,
fset-intersection_wf,
fset-extensionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
inlFormation,
imageMemberEquality,
baseClosed,
imageElimination,
inrFormation,
voidElimination,
cumulativity,
addLevel,
lambdaFormation,
orFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[a,b,c:fset(T)]. (a \mcap{} b \mcup{} c = a \mcap{} b \mcup{} a \mcap{} c)
Date html generated:
2016_05_14-PM-03_40_28
Last ObjectModification:
2016_01_14-PM-10_41_01
Theory : finite!sets
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