Nuprl Lemma : all-even-implies-sum-even
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ↑isEven(Σ(f[x] | x < n)) supposing ∀x:ℕn. (↑isEven(f[x]))
Proof
Definitions occuring in Statement :
isEven: isEven(n),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
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,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: 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,
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
btrue: tt,
true: True
Lemmas referenced :
isEven-sum,
int_seg_wf,
filter_is_nil,
isOdd_wf,
upto_wf,
l_all_iff,
not_wf,
assert_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,
even-iff-not-odd,
istype-assert,
assert_witness,
isEven_wf,
sum_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
universeIsType,
natural_numberEquality,
setElimination,
rename,
hypothesis,
productElimination,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality_alt,
imageElimination,
independent_pairFormation,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
setIsType,
lambdaFormation_alt,
functionIsType,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \muparrow{}isEven(\mSigma{}(f[x] | x < n)) supposing \mforall{}x:\mBbbN{}n. (\muparrow{}isEven(f[x]))
Date html generated:
2020_05_19-PM-10_01_50
Last ObjectModification:
2019_11_12-PM-02_22_12
Theory : num_thy_1
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