Nuprl Lemma : KozenSilva-lemma
∀[r:CRng]. ∀[x,y:Atom]. ∀[h:PowerSeries(r)]. ∀[n,m:ℕ].
[([h]_n(y:=1)*Δ(x,y))]_m = ([h]_n(y:=(atom(x)+atom(y)))*((atom(x)+atom(y)))^(m - n)) ∈ PowerSeries(r)
supposing (n ≤ m) ∧ (¬(x = y ∈ Atom))
Proof
Definitions occuring in Statement :
fps-set-to-one: [f]_n(y:=1),
fps-pascal: Δ(x,y),
fps-compose: g(x:=f),
fps-exp: (f)^(n),
fps-slice: [f]_n,
fps-mul: (f*g),
fps-add: (f+g),
fps-atom: atom(x),
power-series: PowerSeries(X;r),
atom-deq: AtomDeq,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
not: ¬A,
and: P ∧ Q,
subtract: n - m,
atom: Atom,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
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],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
cand: A c∧ B,
crng: CRng,
rng: Rng,
compose: f o g,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
ident: Ident(T;op;id),
comm: Comm(T;op),
single-bag: {x},
bag-filter: [x∈b|p[x]],
int_seg: {i..j-},
l_member!: (x ∈! l),
le: A ≤ B,
lelt: i ≤ j < k,
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
l_all: (∀x∈L.P[x]),
fps-summation: fps-summation(r;b;x.f[x]),
fps-atom: atom(x),
nequal: a ≠ b ∈ T ,
empty-bag: {},
fps-add: (f+g),
fps-coeff: f[b],
fps-single: <c>,
bag-eq: bag-eq(eq;as;bs),
bag-count: (#x in bs),
bag-all: bag-all(x.p[x];bs),
count: count(P;L),
bag-map: bag-map(f;bs),
bag-reduce: bag-reduce(x,y.f[x; y];zero;bs),
lt_int: i <z j,
band: p ∧b q,
fps-slice: [f]_n,
fps-mul: (f*g)
Lemmas referenced :
le_wf,
not_wf,
equal-wf-base,
atom_subtype_base,
nat_wf,
power-series_wf,
crng_wf,
fps-linear-ucont-equal,
atom-valueall-type,
atom-deq_wf,
fps-slice_wf,
fps-mul_wf,
fps-set-to-one_wf,
fps-pascal_wf,
fps-compose_wf,
fps-add_wf,
fps-atom_wf,
fps-exp_wf,
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_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_wf,
rng_car_wf,
bag_wf,
fps-ucont-composition,
fps-slice-ucont,
fps-mul-ucont,
fps-set-to-one-ucont,
fps-compose-ucont,
equal_wf,
fps-add-slice,
iff_weakening_equal,
squash_wf,
true_wf,
valueall-type_wf,
deq_wf,
fps-set-to-one-add,
mul_over_plus_fps,
mul_comm_fps,
fps-compose-add,
fps-set-to-one-scalar-mul,
fps-scalar-mul-slice,
fps-scalar-mul-property,
fps-scalar-mul_wf,
fps-compose-scalar-mul,
eq_int_wf,
bag-size_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
fps-set-to-one-single,
fps-single-slice,
fps-mul-slice,
fps-single_wf,
bag-co-restrict_wf,
fps-zero_wf,
mon_assoc_fps,
abmonoid_comm_fps,
mon_ident_fps,
bag-restrict-size-bound,
fps-summation_wf,
filter_is_singleton,
upto_wf,
subtype_rel_list,
int_seg_wf,
length_upto,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
select_upto,
non_neg_length,
length_wf_nat,
lelt_wf,
select_wf,
less_than_wf,
length_wf,
all_wf,
decidable__equal_int,
assert_wf,
int_seg_properties,
list-subtype-bag,
bag-summation-single,
fps-pascal-slice,
fps-compose-single,
rng_zero_wf,
reduce_nil_lemma,
reduce_cons_lemma,
map_nil_lemma,
map_cons_lemma,
rng_plus_zero,
fps-mul-assoc,
bag-restrict_wf,
fps-exp-add,
bag-summation-filter,
bag-summation_wf,
bag-summation-equal,
bag-member_wf,
ifthenelse_wf,
mul_zero_fps,
fps-zero-slice,
fps-compose-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
productEquality,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
atomEquality,
applyEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality,
dependent_set_memberEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
lambdaFormation,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
cumulativity,
equalityElimination,
promote_hyp,
instantiate,
independent_pairEquality,
hyp_replacement,
applyLambdaEquality,
addEquality,
functionEquality,
equalityUniverse,
levelHypothesis,
functionExtensionality
Latex:
\mforall{}[r:CRng]. \mforall{}[x,y:Atom]. \mforall{}[h:PowerSeries(r)]. \mforall{}[n,m:\mBbbN{}].
[([h]\_n(y:=1)*\mDelta{}(x,y))]\_m = ([h]\_n(y:=(atom(x)+atom(y)))*((atom(x)+atom(y)))\^{}(m - n))
supposing (n \mleq{} m) \mwedge{} (\mneg{}(x = y))
Date html generated:
2018_05_21-PM-10_13_40
Last ObjectModification:
2017_07_26-PM-06_35_24
Theory : power!series
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