Nuprl Lemma : isr-first-success
∀[T:Type]. ∀[A:T ⟶ Type]. ∀[f:x:T ⟶ (A[x]?)]. ∀[L:T List]. (↑isr(first-success(f;L)) ⇐⇒ (∀a∈L.↑isr(f a)))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
first-success: first-success(f;L),
list: T List,
assert: ↑b,
isr: isr(x),
uall: ∀[x:A]. B[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
first-success: first-success(f;L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
btrue: tt,
unit: Unit,
cand: A c∧ B,
true: True,
rev_implies: P ⇐ Q,
list_ind: list_ind,
nil: [],
it: ⋅,
subtype_rel: A ⊆r B,
l_all: (∀x∈L.P[x])
Lemmas referenced :
list_induction,
assert_wf,
isr_wf,
select_wf,
l_all_wf,
l_member_wf,
list_wf,
l_all_nil,
int_seg_wf,
length_wf,
nil_wf,
sq_stable__le,
unit_wf2,
first-success_wf,
list_ind_cons_lemma,
false_wf,
true_wf,
equal_wf,
l_all_cons,
l_all_wf_nil,
cons_wf,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
productEquality,
because_Cache,
applyEquality,
independent_isectElimination,
hypothesis,
cumulativity,
setElimination,
rename,
functionExtensionality,
setEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
unionEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
addLevel,
impliesFunctionality,
independent_pairEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[A:T {}\mrightarrow{} Type]. \mforall{}[f:x:T {}\mrightarrow{} (A[x]?)]. \mforall{}[L:T List].
(\muparrow{}isr(first-success(f;L)) \mLeftarrow{}{}\mRightarrow{} (\mforall{}a\mmember{}L.\muparrow{}isr(f a)))
Date html generated:
2017_04_14-AM-08_41_00
Last ObjectModification:
2017_02_27-PM-03_31_43
Theory : list_0
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