Nuprl Lemma : l_exists_reduce
∀[T:Type]. ∀L:T List. ∀P:T ⟶ 𝔹. ((∃x∈L. ↑P[x]) ⇐⇒ ↑reduce(λx,y. (P[x] ∨by);ff;L))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
reduce: reduce(f;k;as),
list: T List,
bor: p ∨bq,
assert: ↑b,
bfalse: ff,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
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,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
lelt: i ≤ j < k,
int_seg: {i..j-},
exists: ∃x:A. B[x],
false: False,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
uimplies: b supposing a,
select: L[n],
l_exists: (∃x∈L. P[x]),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q
Lemmas referenced :
list_induction,
all_wf,
bool_wf,
iff_wf,
l_exists_wf,
l_member_wf,
assert_wf,
reduce_wf,
bor_wf,
bfalse_wf,
reduce_nil_lemma,
istype-void,
reduce_cons_lemma,
list_wf,
assert_of_bor,
assert_witness,
l_exists_cons,
cons_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
less_than_transitivity1,
less_than_irreflexivity,
exists_wf,
int_seg_wf,
false_wf
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,
cumulativity,
hypothesis,
because_Cache,
Error :universeIsType,
setElimination,
rename,
applyEquality,
functionExtensionality,
Error :setIsType,
Error :functionIsType,
independent_functionElimination,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
Error :inhabitedIsType,
Error :productIsType,
universeEquality,
lambdaEquality,
natural_numberEquality,
productElimination,
independent_pairFormation,
voidEquality,
isect_memberEquality,
lambdaFormation,
independent_isectElimination,
baseClosed,
lemma_by_obid,
unionElimination,
Error :inlFormation_alt,
Error :inrFormation_alt,
Error :unionIsType,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. ((\mexists{}x\mmember{}L. \muparrow{}P[x]) \mLeftarrow{}{}\mRightarrow{} \muparrow{}reduce(\mlambda{}x,y. (P[x] \mvee{}\msubb{}y);ff;L))
Date html generated:
2019_06_20-PM-00_41_22
Last ObjectModification:
2018_10_02-PM-06_03_53
Theory : list_0
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