Nuprl Lemma : fps-compose-exp
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[g,f:PowerSeries(X;r)]. ∀[n:ℕ].
((g)^(n)(x:=f) = (g(x:=f))^(n) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-exp: (f)^(n),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
nat: ℕ,
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,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
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,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
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,
less_than_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-compose_wf,
fps-one_wf,
equal_wf,
squash_wf,
true_wf,
fps-exp-zero,
fps-compose-one,
iff_weakening_equal,
fps-compose-mul,
fps-exp_wf,
le_wf,
fps-mul_wf,
fps-exp-unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
unionElimination,
because_Cache,
cumulativity,
equalityTransitivity,
equalitySymmetry,
universeEquality,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[g,f:PowerSeries(X;r)]. \mforall{}[n:\mBbbN{}].
((g)\^{}(n)(x:=f) = (g(x:=f))\^{}(n))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_09_55
Last ObjectModification:
2017_07_26-PM-06_34_16
Theory : power!series
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