Nuprl Lemma : odd-lsum-of-odd
∀[T:Type]. ∀[L:T List].
∀[f:{x:T| (x ∈ L)} ⟶ ℤ]. ↑isOdd(Σ(f[x] | x ∈ L)) supposing (∀x∈L.↑isOdd(f[x])) supposing ↑isOdd(||L||)
Proof
Definitions occuring in Statement :
isOdd: isOdd(n),
lsum: Σ(f[x] | x ∈ L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
length: ||as||,
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
int: ℤ,
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,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
cons: [a / b],
uiff: uiff(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
btrue: tt,
true: True,
l_all: (∀x∈L.P[x]),
select: L[n],
l_member: (x ∈ l),
less_than': less_than'(a;b),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
same-parity: same-parity(n;m),
bfalse: ff,
rev_uimplies: rev_uimplies(P;Q),
nat_plus: ℕ+
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,
assert_witness,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
non_neg_length,
length_wf,
list-cases,
product_subtype_list,
isOdd_wf,
lsum_wf,
l_member_wf,
l_all_wf,
assert_wf,
istype-assert,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
length_wf_nat,
list_wf,
istype-universe,
odd-iff-not-even,
nil_wf,
length_of_nil_lemma,
length_of_cons_lemma,
lsum_cons_lemma,
lsum_nil_lemma,
add-zero,
cons_wf,
select_wf,
subtype_rel_dep_function,
subtype_rel_sets_simple,
cons_member,
add-member-int_seg2,
select_cons_tl_sq2,
int_seg_subtype_nat,
istype-false,
isOdd-add,
isEven_wf,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
add_nat_plus,
add_nat_wf,
nat_plus_properties,
add-is-int-iff,
false_wf,
odd-plus-even,
odd-plus-odd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
promote_hyp,
hypothesis_subsumption,
imageElimination,
setIsType,
functionIsType,
addEquality,
universeEquality,
voidEquality,
imageMemberEquality,
baseClosed,
equalityIstype,
setEquality,
intEquality,
inrFormation_alt,
closedConclusion,
cumulativity,
pointwiseFunctionality,
baseApply
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List].
\mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbZ{}]. \muparrow{}isOdd(\mSigma{}(f[x] | x \mmember{} L)) supposing (\mforall{}x\mmember{}L.\muparrow{}isOdd(f[x]))
supposing \muparrow{}isOdd(||L||)
Date html generated:
2020_05_19-PM-10_01_40
Last ObjectModification:
2019_11_13-AM-10_43_00
Theory : num_thy_1
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