Nuprl Lemma : divisibility-by-2-rule
∀n:ℕ+. ∀a:ℕn ⟶ ℤ. (2 | Σi<n.a[i]*10^i ⇐⇒ 2 | a[0])
Proof
Definitions occuring in Statement :
power-sum: Σi<n.a[i]*x^i,
divides: b | a,
int_seg: {i..j-},
nat_plus: ℕ+,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
eqmod: a ≡ b mod m,
divides: b | a,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtract: n - m,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
power-sum: Σi<n.a[i]*x^i,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
true: True,
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
int_seg_wf,
istype-int,
nat_plus_wf,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermConstant_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
int_subtype_base,
power-sum_wf,
nat_plus_subtype_nat,
eqmod_wf,
istype-false,
decidable__lt,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
le_wf,
less_than_wf,
eqmod_functionality_wrt_eqmod,
power-sum_functionality_wrt_eqmod,
eqmod_weakening,
subtype_base_sq,
isolate_summand,
exp_wf2,
int_seg_subtype_nat,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
exp0_lemma,
sum_wf,
nat_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
set_subtype_base,
lelt_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
zero_ann_a,
exp-zero,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
itermAdd_wf,
int_term_value_add_lemma,
add_functionality_wrt_eq,
sum_constant,
divides_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_pairFormation_alt,
dependent_functionElimination,
because_Cache,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
equalityIsType4,
inhabitedIsType,
baseClosed,
baseApply,
closedConclusion,
applyEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
int_eqEquality,
productIsType,
productElimination,
promote_hyp,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
imageElimination,
universeEquality,
imageMemberEquality,
equalityElimination,
equalityIsType2,
inrFormation_alt,
addEquality,
minusEquality,
equalityIsType1
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}a:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}. (2 | \mSigma{}i<n.a[i]*10\^{}i \mLeftarrow{}{}\mRightarrow{} 2 | a[0])
Date html generated:
2019_10_15-AM-11_25_58
Last ObjectModification:
2018_10_09-PM-00_15_01
Theory : general
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