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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