Nuprl Lemma : bexists_iff_exists_nth
∀s:DSet. ∀f:|s| ⟶ 𝔹. ∀as:|s| List. (↑∃b x(:|s|) ∈ as. f[x] ⇐⇒ ∃n:ℕ||as||. (↑f[as[n]]))
Proof
Definitions occuring in Statement :
mon_for: For{g} x ∈ as. f[x],
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
bor_mon: <𝔹,∨b>,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bor_mon: <𝔹,∨b>,
grp_id: e,
pi2: snd(t),
pi1: fst(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
grp_op: *,
infix_ap: x f y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m
Lemmas referenced :
list_wf,
set_car_wf,
bool_wf,
dset_wf,
list_induction,
iff_wf,
assert_wf,
mon_for_wf,
bor_mon_wf,
exists_wf,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
mon_for_nil_lemma,
length_of_nil_lemma,
stuck-spread,
istype-base,
mon_for_cons_lemma,
length_of_cons_lemma,
iff_weakening_uiff,
bor_wf,
or_wf,
assert_of_bor,
cons_wf,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
istype-false,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
le_wf,
assert_functionality_wrt_uiff,
select_cons_hd,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-subtract-cancel,
select_cons_tl,
add-associates,
add-swap,
add-commutes,
zero-add,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
functionIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
productElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
imageElimination,
baseClosed,
inhabitedIsType,
promote_hyp,
productIsType,
addEquality,
unionIsType,
dependent_set_memberEquality_alt,
imageMemberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
equalityIsType1,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}s:DSet. \mforall{}f:|s| {}\mrightarrow{} \mBbbB{}. \mforall{}as:|s| List. (\muparrow{}\mexists{}b x(:|s|) \mmember{} as. f[x] \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}||as||. (\muparrow{}f[as[n]]))
Date html generated:
2019_10_16-PM-01_03_22
Last ObjectModification:
2018_10_08-AM-11_16_52
Theory : list_2
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