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
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