Nuprl Lemma : binomial
∀[r:CRng]. ∀[a,b:|r|]. ∀[n:ℕ].
(((a +r b) ↑r n) = (Σ(r) 0 ≤ i < n + 1. choose(n;i) ⋅r ((a ↑r i) * (b ↑r (n - i)))) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_nat_op: n ⋅r e,
rng_nexp: e ↑r n,
choose: choose(n;i),
rng_sum: rng_sum,
crng: CRng,
rng_times: *,
rng_plus: +r,
rng_car: |r|,
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
subtract: n - m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
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,
crng: CRng,
rng: Rng,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
lelt: i ≤ j < k,
so_apply: x[s],
int_iseg: {i...j},
int_seg: {i..j-},
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
le: A ≤ B,
so_lambda: λ2x.t[x],
infix_ap: x f y,
squash: ↓T,
subtract: n - m,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
ycomb: Y,
choose: choose(n;i),
cand: A c∧ B,
eq_int: (i =z j),
bnot: ¬bb,
sq_type: SQType(T),
assert: ↑b,
bor: p ∨bq,
band: p ∧b q,
nequal: a ≠ b ∈ T ,
nat_plus: 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,
subtract-1-ge-0,
istype-nat,
rng_car_wf,
crng_wf,
iff_weakening_equal,
int_seg_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
itermAdd_wf,
itermSubtract_wf,
subtract_wf,
int_seg_subtype_nat,
rng_nexp_wf,
rng_times_wf,
infix_ap_wf,
int_formula_prop_not_lemma,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
lelt_wf,
subtype_rel_sets,
le_wf,
false_wf,
choose_wf,
rng_nat_op_wf,
rng_sum_unroll_unit,
rng_plus_wf,
rng_nexp_zero,
true_wf,
squash_wf,
equal_wf,
zero-add,
minus-zero,
assert_of_bnot,
assert_of_band,
eqff_to_assert,
bnot_thru_bor,
not_wf,
bnot_wf,
band_wf,
assert_of_eq_int,
assert_of_bor,
eqtt_to_assert,
iff_weakening_uiff,
or_wf,
assert_wf,
equal-wf-base,
iff_transitivity,
bool_wf,
eq_int_wf,
bor_wf,
rng_wf,
nat_wf,
rng_one_wf,
rng_nat_op_one,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
rng_times_one,
istype-universe,
subtract-add-cancel,
istype-le,
rng_sum_unroll_hi,
decidable__lt,
istype-false,
subtype_rel_self,
rng_sum_unroll_lo,
add-subtract-cancel,
add-associates,
minus-add,
minus-minus,
minus-one-mul,
add-zero,
add-swap,
add-commutes,
add-mul-special,
zero-mul,
int_subtype_base,
istype-assert,
bfalse_wf,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
assert-bnot,
neg_assert_of_eq_int,
testxxx_lemma,
btrue_wf,
rng_sum_wf,
set_subtype_base,
intformor_wf,
int_formula_prop_or_lemma,
add_nat_wf,
add-is-int-iff,
rng_nat_op_add,
rng_sum_plus,
rng_nexp_unroll,
rng_times_over_plus,
rng_plus_assoc,
rng_plus_comm,
rng_plus_ac_1,
rng_times_sum_r,
rng_sum_shift,
not-lt-2,
add_functionality_wrt_le,
le-add-cancel,
rng_times_nat_op_r,
rng_times_assoc,
crng_times_ac_1,
crng_times_comm,
itermMultiply_wf,
int_term_value_mul_lemma
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,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
baseClosed,
imageMemberEquality,
computeAll,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
unionElimination,
applyLambdaEquality,
productElimination,
setEquality,
productEquality,
intEquality,
lambdaFormation,
dependent_set_memberEquality,
addEquality,
because_Cache,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
impliesFunctionality,
orFunctionality,
equalityElimination,
instantiate,
dependent_set_memberEquality_alt,
closedConclusion,
setIsType,
productIsType,
multiplyEquality,
baseApply,
inlFormation_alt,
equalityIsType4,
inrFormation_alt,
unionIsType,
promote_hyp,
cumulativity,
equalityIsType1,
functionIsType,
pointwiseFunctionality,
minusEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[a,b:|r|]. \mforall{}[n:\mBbbN{}].
(((a +r b) \muparrow{}r n) = (\mSigma{}(r) 0 \mleq{} i < n + 1. choose(n;i) \mcdot{}r ((a \muparrow{}r i) * (b \muparrow{}r (n - i)))))
Date html generated:
2019_10_15-AM-10_34_09
Last ObjectModification:
2018_10_19-AM-09_35_16
Theory : rings_1
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