Nuprl Lemma : l_contains_cons
∀[T:Type]. ∀L:T List. ∀a:T. ∀as:T List. ([a / as] ⊆ L ⇐⇒ (a ∈ L) ∧ as ⊆ L)
Proof
Definitions occuring in Statement :
l_contains: A ⊆ B,
l_member: (x ∈ l),
cons: [a / b],
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],
member: t ∈ T,
l_contains: A ⊆ B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
list_wf,
and_wf,
l_member_wf,
l_all_wf,
l_all_cons,
cons_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
universeEquality,
independent_pairFormation,
productElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
because_Cache,
addLevel,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}a:T. \mforall{}as:T List. ([a / as] \msubseteq{} L \mLeftarrow{}{}\mRightarrow{} (a \mmember{} L) \mwedge{} as \msubseteq{} L)
Date html generated:
2016_05_14-AM-07_54_39
Last ObjectModification:
2015_12_26-PM-04_48_53
Theory : list_1
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