Nuprl Lemma : l-union_functionality_wrt-l_contains
∀[T:Type]. ∀eq:EqDecider(T). ∀as1,bs1,as2,bs2:T List. (bs1 ⊆ bs2 ⇒ as1 ⊆ as2 ⇒ as1 ⋃ bs1 ⊆ as2 ⋃ bs2)
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],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
l_contains: A ⊆ B,
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
l_all_iff,
l_member_wf,
l-union_wf,
or_wf,
member-union,
all_wf,
l_contains_wf,
list_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
hypothesis,
setEquality,
productElimination,
independent_functionElimination,
unionElimination,
inlFormation,
inrFormation,
because_Cache,
addLevel,
allFunctionality,
impliesFunctionality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}as1,bs1,as2,bs2:T List. (bs1 \msubseteq{} bs2 {}\mRightarrow{} as1 \msubseteq{} as2 {}\mRightarrow{} as1 \mcup{} bs1 \msubseteq{} as2 \mcup{} bs2)
Date html generated:
2016_05_14-PM-03_24_45
Last ObjectModification:
2015_12_26-PM-06_21_57
Theory : decidable!equality
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