Nuprl Lemma : l_find_wf
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹].
(l_find(L;P) ∈ (∃x:T [(∃i:ℕ||L||. ((x = L[i] ∈ T) ∧ (↑(P x)) ∧ (∀j:ℕi. (¬↑(P L[j])))))]) ∨ (↓∀i:ℕ||L||. (¬↑(P L[i]))))
Proof
Definitions occuring in Statement :
l_find: l_find(L;P),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
l_find: l_find(L;P),
int_seg: {i..j-},
lelt: i ≤ j < k,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
sq_type: SQType(T),
less_than: a < b,
less_than': less_than'(a;b),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
nat_plus: ℕ+,
true: True,
cand: A c∧ B,
l_member: (x ∈ l),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
l_member_wf,
bool_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
length_of_nil_lemma,
stuck-spread,
base_wf,
reduce_nil_lemma,
int_seg_properties,
assert_wf,
nil_wf,
int_seg_wf,
sq_exists_wf,
exists_wf,
all_wf,
not_wf,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
length_of_cons_lemma,
reduce_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
cons_wf,
list_wf,
false_wf,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
lelt_wf,
length_wf,
select-cons-hd,
select_wf,
int_seg_subtype_nat,
non_neg_length,
list-subtype,
squash_wf,
subtype_rel_dep_function,
subtype_rel_sets,
cons_member,
subtype_rel_self,
set_wf,
add-member-int_seg2,
select-cons-tl,
add-subtract-cancel,
select-cons,
le_int_wf,
assert_of_le_int,
or_wf,
subtype_rel_list_set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
setEquality,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
baseClosed,
inrEquality,
productElimination,
functionExtensionality,
imageMemberEquality,
productEquality,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
instantiate,
imageElimination,
equalityElimination,
universeEquality,
inlEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}].
(l\_find(L;P) \mmember{} (\mexists{}x:T [(\mexists{}i:\mBbbN{}||L||. ((x = L[i]) \mwedge{} (\muparrow{}(P x)) \mwedge{} (\mforall{}j:\mBbbN{}i. (\mneg{}\muparrow{}(P L[j])))))])
\mvee{} (\mdownarrow{}\mforall{}i:\mBbbN{}||L||. (\mneg{}\muparrow{}(P L[i]))))
Date html generated:
2018_05_21-PM-06_36_00
Last ObjectModification:
2017_07_26-PM-04_52_43
Theory : general
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