Nuprl Lemma : list-closed-test_wf
∀[T:Type]. ∀L:T List. ∀f:T ⟶ (T List). ∀d:EqDecider(T). (list-closed-test(f;d;L) ∈ {b:𝔹| ↑b ⇐⇒ list-closed(T;L;f)} )
Proof
Definitions occuring in Statement :
list-closed-test: list-closed-test(f;d;L),
list-closed: list-closed(T;L;f),
list: T List,
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
false: False,
not: ¬A,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
isl: isl(x),
or: P ∨ Q,
decidable: Dec(P),
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
list-closed-test: list-closed-test(f;d;L),
decidable__list-closed2-ext,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
btrue_neq_bfalse,
assert_elim,
bfalse_wf,
assert_of_tt,
istype-assert,
btrue_wf,
istype-universe,
list-closed_wf,
decidable_wf,
deq_wf,
list_wf,
decidable__list-closed2-ext
Rules used in proof :
voidElimination,
independent_isectElimination,
Error :productIsType,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
unionElimination,
universeEquality,
Error :functionIsTypeImplies,
axiomEquality,
independent_functionElimination,
dependent_functionElimination,
Error :equalityIstype,
instantiate,
because_Cache,
thin,
sqequalHypSubstitution,
extract_by_obid,
Error :universeIsType,
Error :functionIsType,
Error :inhabitedIsType,
Error :isectIsType,
hypothesis,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
isectElimination,
Error :lambdaEquality_alt,
applyEquality,
sqequalRule,
Error :lambdaFormation_alt,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}f:T {}\mrightarrow{} (T List). \mforall{}d:EqDecider(T).
(list-closed-test(f;d;L) \mmember{} \{b:\mBbbB{}| \muparrow{}b \mLeftarrow{}{}\mRightarrow{} list-closed(T;L;f)\} )
Date html generated:
2019_06_20-PM-01_51_45
Last ObjectModification:
2019_06_19-PM-04_35_16
Theory : list_1
Home
Index