Nuprl Lemma : isOdd-sum
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. uiff(↑isOdd(Σ(f[x] | x < n));↑isOdd(||filter(λx.isOdd(f[x]);upto(n))||))
Proof
Definitions occuring in Statement :
isOdd: isOdd(n),
upto: upto(n),
sum: Σ(f[x] | x < k),
length: ||as||,
filter: filter(P;l),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
nat_plus: ℕ+,
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
true: True,
bool: 𝔹,
unit: Unit,
btrue: tt,
assert: ↑b,
bnot: ¬bb,
same-parity: same-parity(n;m),
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
isOdd: isOdd(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,
filter_nil_lemma,
length_of_nil_lemma,
isOdd_wf,
istype-assert,
int_seg_wf,
length_wf,
nil_wf,
sum_wf,
istype-le,
subtract-1-ge-0,
filter_wf5,
upto_wf,
int_seg_properties,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
l_member_wf,
istype-nat,
upto_decomp1,
filter_append_sq,
filter_cons_lemma,
sum_split1,
subtype_base_sq,
int_subtype_base,
subtype_rel_function,
subtract_wf,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
itermSubtract_wf,
int_term_value_subtract_lemma,
length-append,
length_wf_nat,
same-parity_wf,
eqtt_to_assert,
same-parity-implies-even-odd,
istype-true,
ifthenelse_wf,
list_wf,
cons_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
not-same-parity-implies-even-odd,
iff_weakening_uiff,
assert_wf,
not_wf,
isOdd-add,
length_of_cons_lemma,
odd-iff-not-even,
isEven_wf,
even-iff-not-odd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
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,
independent_pairEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
because_Cache,
functionIsType,
dependent_set_memberEquality_alt,
applyEquality,
unionElimination,
productIsType,
setIsType,
imageElimination,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
minusEquality,
multiplyEquality,
equalityElimination,
isectIsType,
equalityIstype,
promote_hyp
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. uiff(\muparrow{}isOdd(\mSigma{}(f[x] | x < n));\muparrow{}isOdd(||filter(\mlambda{}x.isOdd(f[x]);upto(n))||))
Date html generated:
2020_05_19-PM-10_01_27
Last ObjectModification:
2019_11_12-PM-01_57_25
Theory : num_thy_1
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