Nuprl Lemma : l-union-left-contains
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs,cs:T List. (as ⊆ bs ⇒ as ⊆ 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],
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,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
or: P ∨ Q,
guard: {T},
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
l_all: (∀x∈L.P[x]),
cand: A c∧ B
Lemmas referenced :
l_all_iff,
l_member_wf,
l-union_wf,
member-union,
all_wf,
or_wf,
l_contains_wf,
list_wf,
deq_wf,
lelt_wf,
length_wf,
and_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
cumulativity,
setElimination,
rename,
hypothesis,
setEquality,
productElimination,
independent_functionElimination,
inlFormation,
because_Cache,
addLevel,
allFunctionality,
impliesFunctionality,
functionEquality,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation,
hyp_replacement,
equalitySymmetry,
levelHypothesis,
equalityTransitivity,
applyEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}as,bs,cs:T List. (as \msubseteq{} bs {}\mRightarrow{} as \msubseteq{} bs \mcup{} cs)
Date html generated:
2016_10_21-AM-10_38_27
Last ObjectModification:
2016_07_12-AM-05_49_06
Theory : decidable!equality
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