Nuprl Lemma : bl-exists-as-accum
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹].
((∃x∈L.P[x])_b ~ accumulate (with value p and list item x):
p ∨bP[x]
over list:
L
with starting value:
ff))
Proof
Definitions occuring in Statement :
bl-exists: (∃x∈L.P[x])_b
,
list_accum: list_accum,
list: T List
,
bor: p ∨bq
,
bfalse: ff
,
bool: 𝔹
,
uall: ∀[x:A]. B[x]
,
so_apply: x[s]
,
function: x:A ⟶ B[x]
,
universe: Type
,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
bl-exists: (∃x∈L.P[x])_b
,
squash: ↓T
,
prop: ℙ
,
so_apply: x[s]
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
top: Top
,
true: True
,
subtype_rel: A ⊆r B
,
guard: {T}
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
sq_type: SQType(T)
,
bnot: ¬bb
,
ifthenelse: if b then t else f fi
,
assert: ↑b
,
false: False
,
bor: p ∨bq
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
Lemmas referenced :
subtype_base_sq,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
reduce-as-accum,
bor_wf,
eqtt_to_assert,
testxxx_lemma,
bor_tt_simp,
btrue_wf,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
bfalse_wf,
list_accum_wf,
list_wf,
bor_ff_simp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
functionExtensionality,
sqequalRule,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
universeEquality,
sqequalAxiom,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
((\mexists{}x\mmember{}L.P[x])\_b \msim{} accumulate (with value p and list item x):
p \mvee{}\msubb{}P[x]
over list:
L
with starting value:
ff))
Date html generated:
2017_04_17-AM-08_03_30
Last ObjectModification:
2017_02_27-PM-04_33_38
Theory : list_1
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