Nuprl Lemma : odd-l_sum
∀[T:Type]. ∀[L:T List]. ∀[f:{x:T| (x ∈ L)} ⟶ ℤ]. uiff(↑isOdd(l_sum(map(f;L)));↑isOdd(||filter(λx.isOdd(f x);L)||))
Proof
Definitions occuring in Statement :
isOdd: isOdd(n),
l_sum: l_sum(L),
l_member: (x ∈ l),
length: ||as||,
filter: filter(P;l),
map: map(f;as),
list: T List,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
cons: [a / b],
less_than': less_than'(a;b),
colength: colength(L),
sq_type: SQType(T),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
l_member: (x ∈ l),
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
btrue: tt,
compose: f o g,
bnot: ¬bb,
assert: ↑b,
true: True,
subtract: n - m,
sq_stable: SqStable(P)
Lemmas referenced :
l_member_wf,
istype-int,
list_wf,
istype-universe,
l_sum-sum,
list-subtype,
uiff_wf,
assert_wf,
isOdd_wf,
length_wf,
filter_wf2,
iff_weakening_uiff,
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,
istype-void,
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,
intformless_wf,
int_formula_prop_less_lemma,
select_member,
int_seg_wf,
isOdd-sum,
assert_witness,
istype-assert,
nat_properties,
ge_wf,
istype-less_than,
list-cases,
stuck-spread,
istype-base,
length_of_nil_lemma,
filter_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
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,
length_of_cons_lemma,
filter_cons_lemma,
istype-nat,
nil_wf,
subtype_rel_dep_function,
cons_wf,
subtype_rel_sets_simple,
cons_member,
upto_decomp2,
add_nat_plus,
nat_plus_properties,
add-is-int-iff,
false_wf,
add-subtract-cancel,
eqtt_to_assert,
filter-map,
map-length,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
upto_wf,
filter_wf5,
select-cons-tl,
add-associates,
add-swap,
add-commutes,
zero-add,
sq_stable__l_member,
decidable__equal_int_seg,
subtype_rel_list,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
functionIsType,
setIsType,
universeIsType,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
instantiate,
universeEquality,
setEquality,
functionExtensionality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
because_Cache,
lambdaFormation_alt,
dependent_functionElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
lambdaEquality_alt,
productElimination,
independent_isectElimination,
sqequalRule,
imageElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
promote_hyp,
intWeakElimination,
axiomSqEquality,
functionIsTypeImplies,
inhabitedIsType,
baseClosed,
hypothesis_subsumption,
equalityIstype,
equalityTransitivity,
baseApply,
closedConclusion,
intEquality,
sqequalBase,
inrFormation_alt,
addEquality,
pointwiseFunctionality,
productIsType,
equalityElimination,
cumulativity,
imageMemberEquality,
functionExtensionality_alt
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbZ{}].
uiff(\muparrow{}isOdd(l\_sum(map(f;L)));\muparrow{}isOdd(||filter(\mlambda{}x.isOdd(f x);L)||))
Date html generated:
2020_05_19-PM-10_01_46
Last ObjectModification:
2019_11_13-AM-11_08_47
Theory : num_thy_1
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