Nuprl Lemma : fps-set-to-one-slice
∀[r:CRng]. ∀[y:Atom]. ∀[n,k:ℕ]. ∀[f:PowerSeries(r)].
([[f]_k]_n(y:=1) = if (k =z n) then [f]_n(y:=1) else 0 fi ∈ PowerSeries(r))
Proof
Definitions occuring in Statement :
fps-set-to-one: [f]_n(y:=1),
fps-slice: [f]_n,
fps-zero: 0,
power-series: PowerSeries(X;r),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
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,
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-zero: 0,
fps-set-to-one: [f]_n(y:=1),
fps-coeff: f[b],
fps-slice: [f]_n,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
bor: p ∨bq,
crng: CRng,
rng: Rng,
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,
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
nequal: a ≠ b ∈ T
Lemmas referenced :
fps-ext,
fps-set-to-one_wf,
fps-slice_wf,
ifthenelse_wf,
eq_int_wf,
power-series_wf,
fps-zero_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
lt_int_wf,
bag-count_wf,
atom-deq_wf,
assert_of_lt_int,
nat_wf,
rng_zero_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
bag-size_wf,
bag-size-append,
bag-size-rep,
subtract_wf,
nat_properties,
decidable__le,
full-omega-unsat,
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,
bag-append_wf,
bag-rep_wf,
list-subtype-bag,
neg_assert_of_eq_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
add-is-int-iff,
subtract-is-int-iff,
false_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
atomEquality,
setElimination,
rename,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
lambdaEquality,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
approximateComputation,
int_eqEquality,
intEquality,
independent_pairFormation,
addEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
axiomEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[y:Atom]. \mforall{}[n,k:\mBbbN{}]. \mforall{}[f:PowerSeries(r)].
([[f]\_k]\_n(y:=1) = if (k =\msubz{} n) then [f]\_n(y:=1) else 0 fi )
Date html generated:
2018_05_21-PM-10_13_10
Last ObjectModification:
2018_05_19-PM-04_16_44
Theory : power!series
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