Nuprl Lemma : isEven-sum
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ].  uiff(↑isEven(Σ(f[x] | x < n));↑isEven(||filter(λx.isOdd(f[x]);upto(n))||))
Proof
Definitions occuring in Statement : 
isEven: isEven(n)
, 
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]
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
nat: ℕ
, 
so_apply: x[s]
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
le: A ≤ B
, 
less_than: a < b
, 
squash: ↓T
, 
ge: i ≥ j 
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
prop: ℙ
, 
rev_implies: P 
⇐ Q
, 
so_lambda: λ2x.t[x]
, 
iff: P 
⇐⇒ Q
Lemmas referenced : 
istype-assert, 
isOdd_wf, 
length_wf, 
int_seg_wf, 
filter_wf5, 
upto_wf, 
int_seg_properties, 
nat_properties, 
decidable__le, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
istype-int, 
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, 
istype-le, 
istype-less_than, 
l_member_wf, 
iff_weakening_uiff, 
assert_wf, 
sum_wf, 
isOdd-sum, 
isEven_wf, 
not_wf, 
even-iff-not-odd, 
assert_witness, 
istype-nat
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
cut, 
independent_pairFormation, 
introduction, 
hypothesis, 
sqequalRule, 
sqequalHypSubstitution, 
lambdaEquality_alt, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
voidElimination, 
functionIsTypeImplies, 
inhabitedIsType, 
functionIsType, 
extract_by_obid, 
isectElimination, 
natural_numberEquality, 
setElimination, 
rename, 
because_Cache, 
applyEquality, 
dependent_set_memberEquality_alt, 
productElimination, 
imageElimination, 
unionElimination, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
dependent_pairFormation_alt, 
int_eqEquality, 
isect_memberEquality_alt, 
universeIsType, 
productIsType, 
setIsType, 
lambdaFormation_alt, 
promote_hyp
Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    uiff(\muparrow{}isEven(\mSigma{}(f[x]  |  x  <  n));\muparrow{}isEven(||filter(\mlambda{}x.isOdd(f[x]);upto(n))||))
Date html generated:
2020_05_19-PM-10_01_30
Last ObjectModification:
2019_11_12-PM-02_18_22
Theory : num_thy_1
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