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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