Nuprl Lemma : l_contains-append
∀[T:Type]. ∀A,B,C:T List. (A @ B ⊆ C ⇐⇒ A ⊆ C ∧ B ⊆ C)
Proof
Definitions occuring in Statement :
l_contains: A ⊆ B,
append: as @ bs,
list: T List,
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,
member_append,
append_wf,
iff_wf,
l_all_iff,
l_all_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
inlFormation,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
inrFormation,
because_Cache,
lambdaEquality,
functionEquality,
productElimination,
unionElimination,
productEquality,
addLevel,
setElimination,
rename,
setEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}A,B,C:T List. (A @ B \msubseteq{} C \mLeftarrow{}{}\mRightarrow{} A \msubseteq{} C \mwedge{} B \msubseteq{} C)
Date html generated:
2019_06_20-PM-01_26_43
Last ObjectModification:
2018_08_24-PM-11_16_47
Theory : list_1
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