Nuprl Lemma : l_sum-upper-bound
∀b:ℤ. ∀[L:ℤ List]. ((∀x∈L.x ≤ b) ⇒ (l_sum(L) ≤ (b * ||L||)))
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
l_sum: l_sum(L),
top: Top,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
and: P ∧ Q,
le: A ≤ B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q
Lemmas referenced :
less_than'_wf,
cons_wf,
l_all_cons,
and_wf,
false_wf,
int_term_value_add_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
intformand_wf,
multiply-is-int-iff,
length_of_cons_lemma,
reduce_cons_lemma,
l_all_wf_nil,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
length_of_nil_lemma,
reduce_nil_lemma,
list_wf,
length_wf,
l_sum_wf,
l_member_wf,
le_wf,
l_all_wf,
list_induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesisEquality,
setElimination,
rename,
hypothesis,
setEquality,
multiplyEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
applyEquality,
productElimination,
addEquality,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_pairFormation,
addLevel,
impliesFunctionality,
because_Cache,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}b:\mBbbZ{}. \mforall{}[L:\mBbbZ{} List]. ((\mforall{}x\mmember{}L.x \mleq{} b) {}\mRightarrow{} (l\_sum(L) \mleq{} (b * ||L||)))
Date html generated:
2016_05_14-PM-02_51_30
Last ObjectModification:
2016_01_15-AM-07_31_20
Theory : list_1
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