Nuprl Lemma : l_sum_filter0
∀[L:ℤ List]. (l_sum(L) = l_sum(filter(λx.(¬b(x =z 0));L)) ∈ ℤ)
Proof
Definitions occuring in Statement :
l_sum: l_sum(L),
filter: filter(P;l),
list: T List,
bnot: ¬bb,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
l_sum: l_sum(L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
list_induction,
equal-wf-base,
list_subtype_base,
int_subtype_base,
list_wf,
reduce_nil_lemma,
filter_nil_lemma,
reduce_cons_lemma,
filter_cons_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
lambdaEquality,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
because_Cache,
independent_isectElimination,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
lambdaFormation,
rename,
unionElimination,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[L:\mBbbZ{} List]. (l\_sum(L) = l\_sum(filter(\mlambda{}x.(\mneg{}\msubb{}(x =\msubz{} 0));L)))
Date html generated:
2017_04_17-AM-08_38_13
Last ObjectModification:
2017_02_27-PM-04_58_02
Theory : list_1
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