Nuprl Lemma : Long-theorem
∀[x,y:Atom].
∀[a,b:ℤ]. ∀[n,k:ℕ+].
(Moessner(ℤ-rng;x;y;((a - b)*atom(x)+(b)*atom(y));λi.if (i =z 0) then 1
if (i =z 1) then n - 1
else 0
fi ;k)[bag-rep(n;x)]
= ((a + ((k - 1) * b)) * k^(n - 1))
∈ ℤ)
supposing ¬(x = y ∈ Atom)
Proof
Definitions occuring in Statement :
Moessner: Moessner(r;x;y;h;d;k),
fps-scalar-mul: (c)*f,
fps-add: (f+g),
fps-atom: atom(x),
fps-coeff: f[b],
bag-rep: bag-rep(n;x),
exp: i^n,
atom-deq: AtomDeq,
nat_plus: ℕ+,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
lambda: λx.A[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
atom: Atom,
equal: s = t ∈ T,
int_ring: ℤ-rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
integ_dom: IntegDom{i},
uimplies: b supposing a,
rng_car: |r|,
pi1: fst(t),
int_ring: ℤ-rng,
crng: CRng,
rng: Rng,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
int_upper: {i...},
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
true: True,
eq_int: (i =z j),
rng_one: 1,
pi2: snd(t),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
istype: istype(T),
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
nequal: a ≠ b ∈ T ,
so_apply: x[s],
power-series: PowerSeries(X;r),
rng_zero: 0,
fps-slice: [f]_n,
fps-coeff: f[b],
fps-single: <c>,
infix_ap: x f y,
rng_times: *,
rng_plus: +r,
fps-add: (f+g),
fps-scalar-mul: (c)*f,
fps-atom: atom(x),
atom-deq: AtomDeq,
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
action_p: IsAction(A;x;e;S;f),
subtract: n - m,
fps-product: Π(x∈b).f[x],
bag-product: Πx ∈ b. f[x],
cand: A c∧ B,
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
ident: Ident(T;op;id),
comm: Comm(T;op),
fps-exp: (f)^(n),
rng_nexp: e ↑r n,
mon_nat_op: n ⋅ e,
nat_op: n x(op;id) e,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
lt_int: i <z j,
grp_id: e,
mul_mon_of_rng: r↓xmn,
fps-rng: fps-rng(r),
fps-one: 1,
bag-rep: bag-rep(n;x)
Lemmas referenced :
KozenSilva-theorem,
int_ring_wf,
fps-add_wf,
fps-scalar-mul_wf,
subtract_wf,
subtype_rel_self,
rng_car_wf,
fps-atom_wf,
atom-deq_wf,
nat_plus_wf,
istype-int,
atom_subtype_base,
istype-void,
istype-atom,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
istype-false,
istype-le,
eqff_to_assert,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
nat_properties,
nequal-le-implies,
zero-add,
int_upper_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
upper_subtype_upper,
istype-nat,
nat_plus_subtype_nat,
bag-rep_wf,
list-subtype-bag,
exp_wf2,
equal_wf,
squash_wf,
true_wf,
istype-universe,
fps-coeff_wf,
bag_wf,
power-series_wf,
crng_wf,
iff_weakening_equal,
upto_wf,
int_seg_wf,
equal-wf-T-base,
fps-mul_wf,
fps-product_wf,
fps-exp_wf,
rng_nat_op_wf,
int_seg_properties,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
set_subtype_base,
less_than_wf,
atom-valueall-type,
fps-compose_wf,
fps-slice_wf,
integ_dom_wf,
fps-ext,
nat_wf,
bag-size_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
decidable__equal_int,
satisfiable-full-omega-tt,
bag_size_single_lemma,
nequal_wf,
assert-bag-eq,
single-bag_wf,
bag-eq_wf,
fps-compose-add,
fps-compose-scalar-mul,
fps-compose-atom,
le_wf,
equal-wf-base,
bag_size_empty_lemma,
empty-bag_wf,
neg_assert_of_eq_atom,
assert_of_eq_atom,
eq_atom_wf,
rng_nat_op-int,
fps-scalar-mul-property,
fps-add-assoc,
add-commutes,
mul-commutes,
minus-one-mul,
mul-distributes-right,
add-associates,
bag-summation-single-non-zero-no-repeats,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
lelt_wf,
strong-subtype-self,
fps-one_wf,
decidable__lt,
istype-less_than,
bag-member_wf,
fps-mul-comm,
mul_assoc_fps,
mul_one_fps,
no_repeats_upto,
bag-no-repeats-list,
false_wf,
decidable__equal_int_seg,
bag-member-list,
member_upto2,
valueall-type_wf,
deq_wf,
minus-zero,
add-zero,
one-mul,
not_wf,
fps-mul-coeff-bag-rep-simple,
bag-size-rep,
int_seg_subtype_nat,
cons_bag_empty_lemma,
primrec1_lemma,
single-bags-equal,
rng_nexp-int,
fps-exp-linear-coeff
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
hypothesisEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
isect_memberFormation_alt,
independent_isectElimination,
atomEquality,
because_Cache,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
functionIsType,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
natural_numberEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
dependent_pairFormation_alt,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
approximateComputation,
int_eqEquality,
equalityIsType1,
multiplyEquality,
addEquality,
imageElimination,
universeEquality,
imageMemberEquality,
hyp_replacement,
applyLambdaEquality,
lambdaEquality,
lambdaFormation,
dependent_pairFormation,
computeAll,
voidEquality,
isect_memberEquality,
dependent_set_memberEquality,
minusEquality,
productIsType,
isect_memberFormation,
independent_pairEquality,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[x,y:Atom].
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[n,k:\mBbbN{}\msupplus{}].
(Moessner(\mBbbZ{}-rng;x;y;((a - b)*atom(x)+(b)*atom(y));\mlambda{}i.if (i =\msubz{} 0) then 1
if (i =\msubz{} 1) then n - 1
else 0
fi ;k)[bag-rep(n;x)]
= ((a + ((k - 1) * b)) * k\^{}(n - 1)))
supposing \mneg{}(x = y)
Date html generated:
2019_10_16-AM-11_37_26
Last ObjectModification:
2018_10_18-PM-11_53_13
Theory : power!series
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