Nuprl Lemma : length-filter-lower-bound
∀[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[L:A List]. ∀[T:Type]. ∀[k:ℕ]. ∀[f:{i:ℕ||L||| ¬↑P[L[i]]} ⟶ T].
((||L|| - k) ≤ ||filter(P;L)||) supposing (T ~ ℕk and Inj({i:ℕ||L||| ¬↑P[L[i]]} ;T;f))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
select: L[n],
length: ||as||,
filter: filter(P;l),
list: T List,
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
not: ¬A,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
nat: ℕ,
so_apply: x[s],
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
all: ∀x:A. B[x],
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,
so_lambda: λ2x.t[x],
cand: A c∧ B,
biject: Bij(A;B;f),
equipollent: A ~ B,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
ext-eq: A ≡ B,
istype: istype(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
inject: Inj(A;B;f),
uiff: uiff(P;Q)
Lemmas referenced :
le_witness_for_triv,
equipollent_wf,
int_seg_wf,
inject_wf,
length_wf,
not_wf,
assert_wf,
select_wf,
int_seg_properties,
nat_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,
istype-universe,
nat_wf,
list_wf,
bool_wf,
equipollent-partition,
length_wf_nat,
equipollent_same,
decidable__assert,
equipollent-subtract2,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
subtract_wf,
equipollent_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent-nsub,
equipollent_inversion,
pigeon-hole,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
subtract-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
equipollent_transitivity,
subtype_rel_self,
set_wf,
sq_stable__not,
subtype_rel_sets,
equipollent-filter,
compose_wf,
injection-composition,
satisfiable-full-omega-tt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
Error :universeIsType,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
Error :isect_memberEquality_alt,
because_Cache,
setEquality,
cumulativity,
applyEquality,
functionExtensionality,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
imageElimination,
Error :functionIsType,
Error :setIsType,
Error :inhabitedIsType,
universeEquality,
Error :lambdaFormation_alt,
computeAll,
voidEquality,
isect_memberEquality,
intEquality,
lambdaEquality,
lemma_by_obid,
productEquality,
dependent_pairFormation,
functionEquality,
lambdaFormation,
baseClosed,
imageMemberEquality,
dependent_set_memberEquality,
promote_hyp,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
Error :equalityIsType1,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List]. \mforall{}[T:Type]. \mforall{}[k:\mBbbN{}]. \mforall{}[f:\{i:\mBbbN{}||L||| \mneg{}\muparrow{}P[L[i]]\} {}\mrightarrow{} T].
((||L|| - k) \mleq{} ||filter(P;L)||) supposing (T \msim{} \mBbbN{}k and Inj(\{i:\mBbbN{}||L||| \mneg{}\muparrow{}P[L[i]]\} ;T;f))
Date html generated:
2019_06_20-PM-02_19_48
Last ObjectModification:
2018_10_05-PM-04_11_09
Theory : equipollence!!cardinality!
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