Nuprl Lemma : l_subset_cons
∀[T:Type]. ∀x:T. ∀L1:T List. ∀[L2:T List]. (l_subset(T;[x / L1];L2) ⇐⇒ (x ∈ L2) ∧ l_subset(T;L1;L2))
Proof
Definitions occuring in Statement :
l_subset: l_subset(T;as;bs),
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
l_subset: l_subset(T;as;bs),
member: t ∈ T,
rev_implies: P ⇐ Q,
or: P ∨ Q,
prop: ℙ,
guard: {T}
Lemmas referenced :
cons_member,
l_member_wf,
equal_wf,
l_subset_wf,
cons_wf,
and_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
productElimination,
inlFormation,
cumulativity,
sqequalRule,
inrFormation,
unionElimination,
equalitySymmetry,
dependent_set_memberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
hyp_replacement,
Error :applyLambdaEquality,
productEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}x:T. \mforall{}L1:T List. \mforall{}[L2:T List]. (l\_subset(T;[x / L1];L2) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L2) \mwedge{} l\_subset(T;L1;L2))
Date html generated:
2016_10_21-AM-10_05_13
Last ObjectModification:
2016_07_12-AM-05_25_20
Theory : list_1
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