Nuprl Lemma : fps-compose-atom-eq
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[f:PowerSeries(X;r)]. (atom(x)(x:=f) = (f-(f[{}])*1) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-scalar-mul: (c)*f,
fps-sub: (f-g),
fps-atom: atom(x),
fps-one: 1,
fps-coeff: f[b],
power-series: PowerSeries(X;r),
empty-bag: {},
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
crng: CRng,
comm: Comm(T;op),
rng: Rng,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
all: ∀x:A. B[x],
listp: A List+,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
power-series: PowerSeries(X;r),
so_apply: x[s],
uiff: uiff(P;Q),
fps-coeff: f[b],
fps-atom: atom(x),
fps-compose: g(x:=f),
fps-single: <c>,
fps-one: 1,
fps-scalar-mul: (c)*f,
fps-sub: (f-g),
fps-neg: -(f),
fps-add: (f+g),
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
squash: ↓T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
tlp: tlp(L),
hdp: hdp(L),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
sq_stable: SqStable(P),
nat: ℕ,
top: Top,
cons: [a / b],
bag-rep: bag-rep(n;x),
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bag-union: bag-union(bbs),
concat: concat(ll),
empty-bag: {},
bag-append: as + bs,
rev_uimplies: rev_uimplies(P;Q),
length: ||as||,
list_ind: list_ind,
nil: [],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
infix_ap: x f y,
bag-product: Πx ∈ b. f[x],
single-bag: {x},
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
rng_plus_comm,
crng_properties,
rng_properties,
rng_all_properties,
ring_p_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_times_wf,
rng_one_wf,
bag-product_wf,
bag_wf,
tl_wf,
list-subtype-bag,
listp_wf,
fps-ext,
fps-compose_wf,
fps-atom_wf,
fps-sub_wf,
fps-scalar-mul_wf,
fps-coeff_wf,
empty-bag_wf,
fps-one_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
bag-summation_wf,
squash_wf,
assoc_wf,
comm_wf,
bag-eq_wf,
bag-append_wf,
hd_wf,
listp_properties,
bag-rep_wf,
length_wf_nat,
single-bag_wf,
bool_wf,
eqtt_to_assert,
assert-bag-eq,
rng_times_one,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
rng_times_zero,
bag-parts'_wf,
true_wf,
infix_ap_wf,
bag-null_wf,
assert-bag-null,
equal-wf-T-base,
bag-summation-filter,
hdp_wf,
tlp_wf,
iff_weakening_equal,
bag-extensionality-no-repeats,
bag-filter_wf,
ifthenelse_wf,
cons_wf_listp,
cons_wf,
nil_wf,
empty-bag-no-repeats,
bag-single-no-repeats,
bag-member_wf,
decidable__equal_set,
list_wf,
decidable__equal_list,
decidable__equal_bag,
decidable-equal-deq,
less_than_wf,
length_wf,
bag-filter-no-repeats,
bag-parts'-no-repeats,
bag-member-filter,
bag-append-is-single,
sq_stable__bag-member,
bag-member-parts',
bag-member-single,
bag-size_wf,
nat_wf,
bag_size_single_lemma,
bag-size-rep,
list-cases,
reduce_tl_nil_lemma,
length_of_nil_lemma,
int_subtype_base,
product_subtype_list,
reduce_tl_cons_lemma,
reduce_hd_cons_lemma,
length_of_cons_lemma,
primrec1_lemma,
cons_bag_empty_lemma,
uiff_transitivity,
assert_wf,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
reduce_cons_lemma,
reduce_nil_lemma,
bag-append-is-empty,
l_all_iff,
l_member_wf,
cons_member,
bag-union_wf,
subtype_rel_self,
list_ind_nil_lemma,
bag-append-empty,
bag-subtype-list,
bool_cases,
bag-member-empty-iff,
empty_bag_append_lemma,
l_all_cons,
l_all_nil,
equal-empty-bag,
bag-summation-empty,
rng_plus_inv,
list_accum_cons_lemma,
list_accum_nil_lemma,
rng_minus_zero,
rng_plus_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
lambdaFormation,
cumulativity,
because_Cache,
applyEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageElimination,
productEquality,
functionExtensionality,
functionEquality,
dependent_functionElimination,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
voidElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
natural_numberEquality,
applyLambdaEquality,
intEquality,
voidEquality,
hypothesis_subsumption,
impliesFunctionality,
int_eqEquality,
computeAll,
setEquality,
inlFormation,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[f:PowerSeries(X;r)]. (atom(x)(x:=f) = (f-(f[\{\}])*1))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_06_07
Last ObjectModification:
2017_07_26-PM-06_34_09
Theory : power!series
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