Nuprl Lemma : l-union-contained
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs,cs:T List. (as ⋃ bs ⊆ cs ⇐⇒ as ⊆ cs ∧ bs ⊆ cs)
Proof
Definitions occuring in Statement :
l-union: as ⋃ bs,
l_contains: A ⊆ B,
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
l_contains: A ⊆ B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
or: P ∨ Q,
prop: ℙ,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
l_member_wf,
all_wf,
or_wf,
and_wf,
member-union,
l-union_wf,
iff_wf,
l_all_iff,
l_all_wf,
list_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
inlFormation,
lemma_by_obid,
isectElimination,
sqequalRule,
inrFormation,
because_Cache,
lambdaEquality,
functionEquality,
productElimination,
unionElimination,
addLevel,
impliesFunctionality,
allFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
cumulativity,
productEquality,
setElimination,
rename,
setEquality,
andLevelFunctionality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}as,bs,cs:T List. (as \mcup{} bs \msubseteq{} cs \mLeftarrow{}{}\mRightarrow{} as \msubseteq{} cs \mwedge{} bs \msubseteq{} cs)
Date html generated:
2016_05_14-PM-03_24_47
Last ObjectModification:
2015_12_26-PM-06_22_04
Theory : decidable!equality
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