Nuprl Lemma : find_property
∀[T:Type]
∀P:T ⟶ 𝔹. ∀as:T List. ∀d:T. (((first a ∈ as s.t. P[a] else d) ∈ as) ∨ ((first a ∈ as s.t. P[a] else d) = d ∈ T))
Proof
Definitions occuring in Statement :
find: (first x ∈ as s.t. P[x] else d),
l_member: (x ∈ l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
so_apply: x[s],
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
uall: ∀[x:A]. B[x],
find: (first x ∈ as s.t. P[x] else d),
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
true: True,
guard: {T},
or: P ∨ Q,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
bool_wf,
list_ind_cons_lemma,
equal_wf,
assert_wf,
bnot_wf,
not_wf,
list_induction,
all_wf,
or_wf,
l_member_wf,
find_wf,
list_wf,
filter_nil_lemma,
list_ind_nil_lemma,
nil_wf,
filter_cons_lemma,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
cons_member,
l_member_subtype,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
applyEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
isectElimination,
because_Cache,
isect_memberFormation,
lambdaFormation,
lambdaEquality,
cumulativity,
functionExtensionality,
independent_functionElimination,
natural_numberEquality,
inrFormation,
rename,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
functionEquality,
universeEquality,
inlFormation
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}as:T List. \mforall{}d:T.
(((first a \mmember{} as s.t. P[a] else d) \mmember{} as) \mvee{} ((first a \mmember{} as s.t. P[a] else d) = d))
Date html generated:
2017_10_01-AM-08_34_16
Last ObjectModification:
2017_07_26-PM-04_25_25
Theory : list!
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