Nuprl Lemma : find-hd-filter
ā[T:Type]. ā[P:T ā¶ š¹]. ā[as:T List]. ā[d:Top].
(first a ā as s.t. P[a] else d) = hd(filter(Ī»a.P[a];as)) ā T supposing āa:T. ((a ā as) ā§ (āP[a]))
Proof
Definitions occuring in Statement :
find: (first x ā as s.t. P[x] else d),
l_member: (x ā l),
hd: hd(l),
filter: filter(P;l),
list: T List,
assert: āb,
bool: š¹,
uimplies: b supposing a,
uall: ā[x:A]. B[x],
top: Top,
so_apply: x[s],
exists: āx:A. B[x],
and: P ā§ Q,
lambda: Ī»x.A[x],
function: x:A ā¶ B[x],
universe: Type,
equal: s = t ā T
Definitions unfolded in proof :
uall: ā[x:A]. B[x],
member: t ā T,
all: āx:A. B[x],
nat: ā,
implies: P ā Q,
false: False,
ge: i ā„ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: āx:A. B[x],
not: Ā¬A,
top: Top,
and: P ā§ Q,
prop: ā,
so_lambda: Ī»2x.t[x],
so_apply: x[s],
subtype_rel: A ār B,
guard: {T},
or: P āØ Q,
find: (first x ā as s.t. P[x] else d),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
colength: colength(L),
so_lambda: Ī»2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ā
,
sq_type: SQType(T),
less_than: a < b,
squash: āT,
less_than': less_than'(a;b),
bool: š¹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
cand: A cā§ B,
iff: P āā Q
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,
exists_wf,
l_member_wf,
assert_wf,
top_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
filter_nil_lemma,
list_ind_nil_lemma,
nil_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,
filter_cons_lemma,
cons_wf,
list_wf,
bool_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
list_ind_cons_lemma,
reduce_hd_cons_lemma,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
cons_member,
assert_elim,
not_assert_elim,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
cumulativity,
productEquality,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
functionEquality,
universeEquality,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[as:T List]. \mforall{}[d:Top].
(first a \mmember{} as s.t. P[a] else d) = hd(filter(\mlambda{}a.P[a];as)) supposing \mexists{}a:T. ((a \mmember{} as) \mwedge{} (\muparrow{}P[a]))
Date html generated:
2018_05_21-PM-06_50_59
Last ObjectModification:
2017_07_26-PM-04_57_36
Theory : general
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