Nuprl Lemma : le-l_sum
∀[T:Type]. ∀[f:T ⟶ ℕ]. ∀[L:T List]. ∀[t:T]. ((t ∈ L) ⇒ ((f t) ≤ l_sum(map(f;L))))
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
or: P ∨ Q,
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
le_witness_for_triv,
list-cases,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
l_member_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
cons_member,
map_cons_lemma,
l_sum_cons_lemma,
squash_wf,
true_wf,
l_sum_wf,
map_wf,
subtype_rel_self,
iff_weakening_equal,
cons_wf,
istype-nat,
list_wf,
istype-universe,
l_sum_nonneg,
non_neg_length,
nat_wf,
map_length,
int_seg_properties,
select_wf,
length_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isectIsTypeImplies,
unionElimination,
because_Cache,
promote_hyp,
hypothesis_subsumption,
Error :equalityIstype,
Error :dependent_set_memberEquality_alt,
instantiate,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
addEquality,
functionExtensionality,
imageMemberEquality,
Error :functionIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} \mBbbN{}]. \mforall{}[L:T List]. \mforall{}[t:T]. ((t \mmember{} L) {}\mRightarrow{} ((f t) \mleq{} l\_sum(map(f;L))))
Date html generated:
2019_06_20-PM-01_44_44
Last ObjectModification:
2019_02_23-PM-01_10_57
Theory : list_1
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