Nuprl Lemma : fps-set-to-one-add
∀[r:CRng]. ∀[f,g:PowerSeries(r)]. ∀[y:Atom]. ∀[n:ℕ]. ([(f+g)]_n(y:=1) = ([f]_n(y:=1)+[g]_n(y:=1)) ∈ PowerSeries(r))
Proof
Definitions occuring in Statement :
fps-set-to-one: [f]_n(y:=1),
fps-add: (f+g),
power-series: PowerSeries(X;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
atom: Atom,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-set-to-one: [f]_n(y:=1),
fps-add: (f+g),
fps-coeff: f[b],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
nat: ℕ,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
crng: CRng,
rng: Rng,
power-series: PowerSeries(X;r),
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
true: True,
squash: ↓T,
infix_ap: x f y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fps-ext,
fps-set-to-one_wf,
fps-add_wf,
lt_int_wf,
bag-count_wf,
atom-deq_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
bag-size_wf,
rng_plus_wf,
bag-append_wf,
bag-rep_wf,
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
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,
le_wf,
list-subtype-bag,
subtype_rel_self,
bag_wf,
power-series_wf,
crng_wf,
rng_car_wf,
rng_zero_wf,
rng_plus_zero,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
atomEquality,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
natural_numberEquality,
applyEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
setElimination,
rename,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(r)]. \mforall{}[y:Atom]. \mforall{}[n:\mBbbN{}]. ([(f+g)]\_n(y:=1) = ([f]\_n(y:=1)+[g]\_n(y:=1)))
Date html generated:
2018_05_21-PM-10_12_51
Last ObjectModification:
2017_07_26-PM-06_35_10
Theory : power!series
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