Nuprl Lemma : l_sum-triangle-inequality
∀[T:Type]. ∀[L:T List]. ∀[f,g:T ⟶ ℤ].
(|l_sum(map(λa.f[a];L)) - l_sum(map(λa.g[a];L))| ≤ l_sum(map(λa.|f[a] - g[a]|;L)))
Proof
Definitions occuring in Statement :
l_sum: l_sum(L)
,
map: map(f;as)
,
list: T List
,
absval: |i|
,
uall: ∀[x:A]. B[x]
,
so_apply: x[s]
,
le: A ≤ B
,
lambda: λx.A[x]
,
function: x:A ⟶ B[x]
,
subtract: n - m
,
int: ℤ
,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
prop: ℙ
,
squash: ↓T
,
subtype_rel: A ⊆r B
,
uimplies: b supposing a
,
nat: ℕ
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
le: A ≤ B
,
and: P ∧ Q
,
less_than': less_than'(a;b)
,
absval: |i|
,
subtract: n - m
,
false: False
,
not: ¬A
,
implies: P
⇒ Q
,
sq_type: SQType(T)
,
all: ∀x:A. B[x]
,
guard: {T}
,
decidable: Dec(P)
,
or: P ∨ Q
,
uiff: uiff(P;Q)
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
top: Top
,
true: True
Lemmas referenced :
l_sum-triangle-inequality-general,
list_wf,
istype-universe,
le_wf,
squash_wf,
true_wf,
istype-int,
add-zero,
l_sum_wf,
map_wf,
absval_wf,
subtract_wf,
subtype_base_sq,
nat_wf,
set_subtype_base,
int_subtype_base,
istype-false,
minus-zero,
zero-add,
absval_pos,
istype-le,
decidable__equal_int,
add-is-int-iff,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :universeIsType,
instantiate,
universeEquality,
natural_numberEquality,
hyp_replacement,
equalitySymmetry,
sqequalRule,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
equalityTransitivity,
Error :inhabitedIsType,
because_Cache,
cumulativity,
independent_isectElimination,
intEquality,
independent_pairFormation,
Error :lambdaFormation_alt,
setElimination,
rename,
Error :dependent_set_memberEquality_alt,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
pointwiseFunctionality,
promote_hyp,
productElimination,
baseApply,
closedConclusion,
baseClosed,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
Error :functionIsType
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[f,g:T {}\mrightarrow{} \mBbbZ{}].
(|l\_sum(map(\mlambda{}a.f[a];L)) - l\_sum(map(\mlambda{}a.g[a];L))| \mleq{} l\_sum(map(\mlambda{}a.|f[a] - g[a]|;L)))
Date html generated:
2019_06_20-PM-01_44_33
Last ObjectModification:
2018_10_18-PM-00_39_33
Theory : list_1
Home
Index