Nuprl Lemma : filter-length-less
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].  ||filter(λx.P[x];L)|| < ||L|| supposing ∃x:T. ((x ∈ L) ∧ (¬↑P[x]))
Proof
Definitions occuring in Statement : 
l_member: (x ∈ l)
, 
length: ||as||
, 
filter: filter(P;l)
, 
list: T List
, 
assert: ↑b
, 
bool: 𝔹
, 
less_than: a < b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
and: P ∧ Q
, 
lambda: λx.A[x]
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
, 
all: ∀x:A. B[x]
, 
so_apply: x[s]
, 
prop: ℙ
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
cand: A c∧ B
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
less_than: a < b
, 
squash: ↓T
, 
false: False
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
Lemmas referenced : 
filter-split-length, 
member_filter_2, 
bnot_wf, 
l_member_wf, 
assert_of_bnot, 
l_member_length, 
filter_wf5, 
decidable__lt, 
length_wf, 
add-is-int-iff, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformless_wf, 
itermVar_wf, 
itermConstant_wf, 
intformeq_wf, 
itermAdd_wf, 
istype-int, 
int_formula_prop_and_lemma, 
istype-void, 
int_formula_prop_not_lemma, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_add_lemma, 
int_formula_prop_wf, 
false_wf, 
istype-assert, 
list_wf, 
bool_wf, 
istype-universe
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
productElimination, 
dependent_functionElimination, 
lambdaEquality_alt, 
applyEquality, 
setElimination, 
rename, 
setIsType, 
inhabitedIsType, 
universeIsType, 
independent_functionElimination, 
sqequalRule, 
independent_pairFormation, 
independent_isectElimination, 
equalityTransitivity, 
equalitySymmetry, 
unionElimination, 
imageElimination, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
closedConclusion, 
baseClosed, 
natural_numberEquality, 
approximateComputation, 
dependent_pairFormation_alt, 
int_eqEquality, 
isect_memberEquality_alt, 
voidElimination, 
productIsType, 
functionIsType, 
instantiate, 
universeEquality
Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].
    ||filter(\mlambda{}x.P[x];L)||  <  ||L||  supposing  \mexists{}x:T.  ((x  \mmember{}  L)  \mwedge{}  (\mneg{}\muparrow{}P[x]))
Date html generated:
2020_05_19-PM-09_42_47
Last ObjectModification:
2019_10_29-AM-10_00_34
Theory : list_1
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