Nuprl Lemma : sublist-iff-sub-co-list
∀[T:Type]. ∀L2,L1:T List. (L1 ⊆ L2 ⇐⇒ sub-co-list(T;L1;L2))
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
sub-co-list: sub-co-list(T;s1;s2),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
list: T List,
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
top: Top,
subtype_rel: A ⊆r B,
co-nil: (),
nil: [],
false: False,
or: P ∨ Q
Lemmas referenced :
list_induction,
list_wf,
iff_wf,
sublist_wf,
sub-co-list_wf,
istype-universe,
nil_wf,
co-nil_wf,
nil-sub-co-list,
nil-sublist,
istype-void,
cons_sublist_nil,
cons-sub-co-list-nil,
cons_wf,
cons_sublist_cons,
cons-sub-co-list-cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
functionEquality,
hypothesis,
setElimination,
rename,
Error :universeIsType,
independent_functionElimination,
Error :functionIsType,
because_Cache,
Error :productIsType,
dependent_functionElimination,
instantiate,
universeEquality,
independent_pairFormation,
Error :isect_memberEquality_alt,
voidElimination,
applyEquality,
Error :inhabitedIsType,
productElimination,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
Error :unionIsType,
Error :equalityIstype,
unionElimination,
Error :inlFormation_alt,
Error :inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}L2,L1:T List. (L1 \msubseteq{} L2 \mLeftarrow{}{}\mRightarrow{} sub-co-list(T;L1;L2))
Date html generated:
2019_06_20-PM-01_23_00
Last ObjectModification:
2019_01_02-PM-05_42_20
Theory : list_1
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