Nuprl Lemma : fps-exp-add
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[n,m:ℕ]. ∀[f:PowerSeries(X;r)]. ((f)^(n + m) = ((f)^(n)*(f)^(m)) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-exp: (f)^(n),
fps-mul: (f*g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
nat: ℕ,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
add: n + m,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
power-series_wf,
subtract-1-ge-0,
istype-nat,
crng_wf,
deq_wf,
valueall-type_wf,
istype-universe,
fps-exp_wf,
decidable__le,
intformnot_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
istype-le,
equal_wf,
mul_one_fps,
iff_weakening_equal,
squash_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
true_wf,
fps-mul_wf,
fps-exp-zero,
subtype_rel_self,
fps-exp-unroll,
decidable__lt,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
general_arith_equation1,
fps-mul-assoc,
fps-mul-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
because_Cache,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
addEquality,
unionElimination,
applyEquality,
imageElimination,
productElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[f:PowerSeries(X;r)].
((f)\^{}(n + m) = ((f)\^{}(n)*(f)\^{}(m)))
supposing valueall-type(X)
Date html generated:
2020_05_20-AM-09_05_49
Last ObjectModification:
2020_01_01-AM-11_52_10
Theory : power!series
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