Nuprl Lemma : fps-compose-mul
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[g,f,h:PowerSeries(X;r)].
((g*h)(x:=f) = (g(x:=f)*h(x:=f)) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-mul: (f*g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
infix_ap: x f y,
prop: ℙ,
group_p: IsGroup(T;op;id;inv),
ring_p: IsRing(T;plus;zero;neg;times;one),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
true: True,
top: Top,
pi2: snd(t),
pi1: fst(t),
fps-coeff: f[b],
fps-mul: (f*g),
fps-compose: g(x:=f),
uiff: uiff(P;Q),
so_apply: x[s],
power-series: PowerSeries(X;r),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
listp: A List+,
rng: Rng,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
comm: Comm(T;op),
crng: CRng,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
compose: f o g,
spreadn: spread3,
or: P ∨ Q,
sq_or: a ↓∨ b,
false: False,
not: ¬A,
hdp: hdp(L),
tlp: tlp(L),
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
cons: [a / b],
it: ⋅,
nil: [],
list_ind: list_ind,
length: ||as||,
less_than': less_than'(a;b),
le: A ≤ B,
int_iseg: {i...j},
sq_type: SQType(T),
label: ...$L... t,
bag-append: as + bs,
bag-product: Πx ∈ b. f[x],
bag-member: x ↓∈ bs,
subtract: n - m,
inject: Inj(A;B;f),
respects-equality: respects-equality(S;T),
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
cons-bag: x.b,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
less_than: a < b,
ifthenelse: if b then t else f fi ,
btrue: tt,
unit: Unit,
bool: 𝔹
Lemmas referenced :
bag-summation-ring-linear1,
subtype_rel_self,
comm_wf,
assoc_wf,
true_wf,
squash_wf,
bag-summation-product,
bag-map_wf,
bag-combine_wf,
iff_weakening_equal,
bag-product_wf,
equal_wf,
bag-double-summation2,
istype-void,
pi1_wf_top,
bag-parts'_wf,
length_wf_nat,
bag-rep_wf,
hd_wf,
bag-append_wf,
bag-partitions_wf,
infix_ap_wf,
rng_zero_wf,
rng_plus_wf,
bag-summation_wf,
istype-universe,
valueall-type_wf,
deq_wf,
crng_wf,
power-series_wf,
fps-mul_wf,
fps-compose_wf,
fps-ext,
list-subtype-bag,
tl_wf,
rng_one_wf,
rng_times_wf,
rng_car_wf,
bag-product_wf,
rng_properties,
crng_properties,
crng_all_properties,
listp_wf,
bag_wf,
listp_properties,
rng_plus_comm,
subtype_rel_product,
list_wf,
istype-top,
pi2_wf,
top_wf,
bag-summation-reindex,
bag-summation-equal2,
bag-member_wf,
bag-map-combine,
compose_wf,
bag-map-trivial,
bag-map-map,
nth_tl_wf,
bag-co-restrict_wf,
bag-restrict_wf,
bag-size_wf,
firstn_wf,
bag-co-restrict-property,
bag-member-append,
reduce_hd_cons_lemma,
cons-listp,
not_wf,
bag-parts'_wf2,
hdp_wf,
tlp_wf,
reduce_tl_cons_lemma,
bag-restrict-rep,
bag-restrict-append,
bag-co-restrict-rep,
bag-co-restrict-append,
bag-subtype-list,
bag-append-empty,
bag-size-rep,
bag-restrict-disjoint,
empty_bag_append_lemma,
nth_tl_append,
istype-less_than,
istype-le,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
intformless_wf,
decidable__lt,
false_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
subtract-is-int-iff,
decidable__le,
nat_properties,
subtract_nat_wf,
length_tl,
istype-int,
le_wf,
length_wf,
subtype_rel_list,
firstn_append,
firstn_all,
bag-co-restrict-disjoint,
cons_wf,
non_neg_length,
length_of_cons_lemma,
product_subtype_list,
list-cases,
append_wf,
bag-member-partitions,
bag-member-map,
bag-member-parts',
bag-member-combine,
equal-wf-T-base,
l_member_wf,
l_all_wf2,
bag_qinc,
less_than_wf,
subtype_rel_set,
bag-union_wf,
bag-size-append,
istype-nat,
length_firstn,
subtract_wf,
decidable__equal_int,
length_nth_tl,
int_subtype_base,
set_subtype_base,
nat_wf,
subtype_base_sq,
bag-rep-size-restrict,
append_firstn_lastn_sq,
bag-append-comm,
bag-restrict-split,
bag-summation-append,
crng_times_ac_1,
rng_times_assoc,
bag-extensionality-no-repeats,
decidable-equal-deq,
decidable__equal_bag,
decidable__equal_list,
decidable__equal_set,
decidable__equal_product,
strong-subtype-self,
strong-subtype-set3,
strong-subtype-deq-subtype,
bag-deq_wf,
list-deq_wf,
product-deq_wf,
bag-combine-no-repeats,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
istype-false,
length_of_nil_lemma,
bag-map-no-repeats,
no-repeats-bag-partitions,
bag-parts'-no-repeats,
bag-subtype,
respects-equality-trivial,
respects-equality-set-trivial,
respects-equality-product,
bag-member-product,
general-append-cancellation,
cons_wf_listp,
bag-no-repeats_wf,
bag-settype,
subtype_rel_bag,
bag-combine-no-repeats2,
bag-product-no-repeats,
bag-no-repeats-settype,
sq_stable__bag-member,
bag-append-union,
cons-bag-as-append,
bag-union-single,
single-bag_wf,
bag-append-ac,
bag-append-assoc2,
l_contains-nth_tl,
l_contains-firstn,
l_all-l_contains,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
bool_wf,
ifthenelse_wf,
general_length_nth_tl,
add-is-int-iff,
l_all_append,
bag-rep-add,
length-append
Rules used in proof :
functionIsType,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
imageElimination,
natural_numberEquality,
voidElimination,
independent_pairEquality,
dependent_functionElimination,
productIsType,
productEquality,
universeEquality,
instantiate,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
inhabitedIsType,
lambdaEquality_alt,
sqequalRule,
independent_isectElimination,
applyEquality,
because_Cache,
productElimination,
independent_pairFormation,
universeIsType,
lambdaFormation_alt,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :memTop,
equalityIstype,
applyLambdaEquality,
hyp_replacement,
setIsType,
unionElimination,
dependent_set_memberEquality_alt,
setEquality,
addEquality,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
hypothesis_subsumption,
intEquality,
sqequalBase,
cumulativity,
dependent_pairEquality_alt,
minusEquality,
inlFormation_alt,
equalityElimination,
lambdaFormation,
lambdaEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[g,f,h:PowerSeries(X;r)].
((g*h)(x:=f) = (g(x:=f)*h(x:=f)))
supposing valueall-type(X)
Date html generated:
2020_05_20-AM-09_07_02
Last ObjectModification:
2020_01_10-PM-02_36_13
Theory : power!series
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