Nuprl Lemma : filter-for-max
∀[A:Type]. ∀[l:A List]. ∀[m:ℤ]. ∀[g:A ⟶ ℤ].
(||filter(λe.(g[e] =z m);l)|| ≥ 1 ) supposing ((imax-list(map(λe.g[e];l)) = m ∈ ℤ) and 0 < ||l||)
Proof
Definitions occuring in Statement :
imax-list: imax-list(L),
length: ||as||,
filter: filter(P;l),
map: map(f;as),
list: T List,
eq_int: (i =z j),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
ge: i ≥ j ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
top: Top,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
sq_type: SQType(T),
guard: {T},
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
nat: ℕ,
true: True,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
less_than'_wf,
length_wf,
filter_wf5,
eq_int_wf,
l_member_wf,
equal-wf-T-base,
imax-list_wf,
map-length,
int_subtype_base,
less_than_wf,
pos_length,
imax-list-member,
map_wf,
subtype_base_sq,
list_wf,
squash_wf,
true_wf,
map_length,
iff_weakening_equal,
equal_wf,
map_select,
lelt_wf,
member_filter,
select_wf,
nat_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,
select_member,
assert_of_eq_int,
eta_conv,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
cumulativity,
applyEquality,
functionExtensionality,
setElimination,
rename,
hypothesis,
setEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
intEquality,
lambdaFormation,
instantiate,
independent_functionElimination,
baseClosed,
imageElimination,
imageMemberEquality,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[l:A List]. \mforall{}[m:\mBbbZ{}]. \mforall{}[g:A {}\mrightarrow{} \mBbbZ{}].
(||filter(\mlambda{}e.(g[e] =\msubz{} m);l)|| \mgeq{} 1 ) supposing ((imax-list(map(\mlambda{}e.g[e];l)) = m) and 0 < ||l||)
Date html generated:
2017_04_17-AM-07_50_39
Last ObjectModification:
2017_02_27-PM-04_24_30
Theory : list_1
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