Nuprl Lemma : equipollent-filter
∀[A:Type]. ∀P:A ⟶ 𝔹. ∀L:A List. {x:ℕ||L||| ↑P[L[x]]} ~ ℕ||filter(P;L)||
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
select: L[n],
length: ||as||,
filter: filter(P;l),
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
ext-eq: A ≡ B,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
sq_type: SQType(T),
true: True
Lemmas referenced :
last_induction,
equipollent_wf,
int_seg_wf,
length_wf,
assert_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
list_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
filter_nil_lemma,
equipollent-zero,
filter_append,
cons_wf,
nil_wf,
length-append,
length_of_cons_lemma,
filter_cons_lemma,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
equipollent-add,
length_wf_nat,
false_wf,
le_wf,
append_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
equipollent_functionality_wrt_equipollent2,
equipollent_inversion,
union_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent-split,
sq_stable_from_decidable,
decidable__assert,
less_than_wf,
decidable__squash,
equipollent_functionality_wrt_equipollent,
lelt_wf,
assert_functionality_wrt_uiff,
select_append_front,
subtype_rel_sets,
subtype_rel_set,
int_seg_subtype,
equal-wf-base-T,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
length-singleton,
select_append_back,
select-cons-hd,
ext-eq_wf,
equipollent_transitivity,
equipollent-one,
add-zero,
add-commutes,
subtype_base_sq,
set_subtype_base,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
setEquality,
natural_numberEquality,
cumulativity,
hypothesis,
applyEquality,
functionExtensionality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
baseClosed,
functionEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
dependent_set_memberEquality,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
unionEquality,
imageMemberEquality,
instantiate,
addLevel,
hyp_replacement,
levelHypothesis
Latex:
\mforall{}[A:Type]. \mforall{}P:A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List. \{x:\mBbbN{}||L||| \muparrow{}P[L[x]]\} \msim{} \mBbbN{}||filter(P;L)||
Date html generated:
2017_04_17-AM-09_35_10
Last ObjectModification:
2017_02_27-PM-05_35_25
Theory : equipollence!!cardinality!
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