Nuprl Lemma : summand-le-l_sum
∀[T:Type]. ∀[L:T List]. ∀[f:{x:T| (x ∈ L)} ⟶ ℤ].
∀x:{x:T| (x ∈ L)} . (f[x] ≤ l_sum(map(f;L))) supposing ∀x:{x:T| (x ∈ L)} . (0 ≤ f[x])
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
sq_stable: SqStable(P),
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
label: ...$L... t,
sq_type: SQType(T)
Lemmas referenced :
le_wf,
squash_wf,
true_wf,
istype-int,
l_sum-sum,
subtype_rel_self,
iff_weakening_equal,
sq_stable__le,
l_member_wf,
sum_wf,
length_wf_nat,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
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,
decidable__lt,
length_wf,
intformless_wf,
int_formula_prop_less_lemma,
select_member,
int_seg_wf,
summand-le-sum,
nat_properties,
istype-le,
istype-less_than,
subtype_base_sq,
int_subtype_base,
equal_wf,
le_witness_for_triv,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
inhabitedIsType,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
productIsType,
cumulativity,
intEquality,
equalityIstype,
setIsType,
functionIsTypeImplies,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbZ{}].
\mforall{}x:\{x:T| (x \mmember{} L)\} . (f[x] \mleq{} l\_sum(map(f;L))) supposing \mforall{}x:\{x:T| (x \mmember{} L)\} . (0 \mleq{} f[x])
Date html generated:
2020_05_19-PM-09_45_56
Last ObjectModification:
2020_01_23-PM-00_48_26
Theory : list_1
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