Nuprl Lemma : l_sum-upper-bound-map
∀[b:ℤ]. ∀[T:Type]. ∀[f:T ⟶ {...b}]. ∀[L:T List]. (l_sum(map(f;L)) ≤ (b * ||L||))
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
length: ||as||,
map: map(f;as),
list: T List,
int_lower: {...i},
uall: ∀[x:A]. B[x],
le: A ≤ B,
function: x:A ⟶ B[x],
multiply: n * m,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
int_lower: {...i},
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_lower_properties,
l_member_wf,
le_wf,
l_all_iff,
list_wf,
int_lower_wf,
subtype_rel_dep_function,
l_sum_wf,
length_wf,
less_than'_wf,
map_wf,
l_sum-upper-bound,
map-length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
intEquality,
applyEquality,
because_Cache,
sqequalRule,
independent_functionElimination,
productElimination,
independent_pairEquality,
lambdaEquality,
multiplyEquality,
independent_isectElimination,
lambdaFormation,
setElimination,
rename,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
setEquality,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[b:\mBbbZ{}]. \mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} \{...b\}]. \mforall{}[L:T List]. (l\_sum(map(f;L)) \mleq{} (b * ||L||))
Date html generated:
2016_05_14-PM-02_51_40
Last ObjectModification:
2016_01_15-AM-07_32_31
Theory : list_1
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