Nuprl Lemma : fps-set-to-one-neg
∀[r:CRng]. ∀[f:PowerSeries(r)]. ∀[y:Atom]. ∀[n:ℕ]. ([-(f)]_n(y:=1) = -([f]_n(y:=1)) ∈ PowerSeries(r))
Proof
Definitions occuring in Statement :
fps-set-to-one: [f]_n(y:=1)
,
fps-neg: -(f)
,
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-neg: -(f)
,
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
,
squash: ↓T
,
prop: ℙ
,
crng: CRng
,
rng: Rng
,
true: True
,
guard: {T}
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
sq_type: SQType(T)
,
bnot: ¬bb
,
assert: ↑b
,
false: False
,
not: ¬A
,
power-series: PowerSeries(X;r)
,
ge: i ≥ j
,
decidable: Dec(P)
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
top: Top
Lemmas referenced :
fps-ext,
fps-set-to-one_wf,
fps-neg_wf,
lt_int_wf,
bag-count_wf,
atom-deq_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_wf,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_zero_wf,
rng_minus_zero,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
bag-size_wf,
rng_minus_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
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,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
voidElimination,
dependent_set_memberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[f:PowerSeries(r)]. \mforall{}[y:Atom]. \mforall{}[n:\mBbbN{}]. ([-(f)]\_n(y:=1) = -([f]\_n(y:=1)))
Date html generated:
2018_05_21-PM-10_12_43
Last ObjectModification:
2017_07_26-PM-06_35_05
Theory : power!series
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