Nuprl Lemma : fps-compose-identity
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f:PowerSeries(X;r)]. ∀[x:X]. (f(x:=atom(x)) = f ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-atom: atom(x),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
crng: CRng,
rng: Rng,
fps-compose: g(x:=f),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
fps-one: 1,
fps-coeff: f[b],
fps-single: <c>,
empty-bag: {},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
cons-bag: x.b,
top: Top,
fps-atom: atom(x),
deq: EqDecider(T),
eqof: eqof(d),
bag-eq: bag-eq(eq;as;bs),
band: p ∧b q,
bag-all: bag-all(x.p[x];bs),
bag-reduce: bag-reduce(x,y.f[x; y];zero;bs),
reduce: reduce(f;k;as),
list_ind: list_ind,
bag-map: bag-map(f;bs),
map: map(f;as),
nil: [],
single-bag: {x},
cons: [a / b],
lt_int: i <z j,
bag-count: (#x in bs),
count: count(P;L),
rng_zero: 0,
pi1: fst(t),
pi2: snd(t),
fps-sub: (f-g)
Lemmas referenced :
fps-linear-ucont-equal,
fps-compose_wf,
fps-atom_wf,
power-series_wf,
fps-compose-ucont,
fps-id-ucont,
equal_wf,
squash_wf,
true_wf,
fps-compose-add,
fps-add_wf,
iff_weakening_equal,
fps-compose-scalar-mul,
fps-scalar-mul_wf,
rng_car_wf,
bag_wf,
crng_wf,
deq_wf,
valueall-type_wf,
bag_to_squash_list,
list_induction,
fps-single_wf,
list-subtype-bag,
list_wf,
fps-compose-one,
fps-one_wf,
fps-ext,
nil_wf,
bag-eq_wf,
empty-bag_wf,
bool_wf,
eqtt_to_assert,
assert-bag-eq,
bag-null_wf,
assert-bag-null,
rng_one_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
rng_zero_wf,
single-bag_wf,
cons-bag-as-append,
fps-mul-single,
fps-compose-mul,
fps-mul_wf,
safe-assert-deq,
fps-compose-atom-neq,
fps-compose-atom-eq,
fps-sub_wf,
fps-scalar-mul-zero,
neg_id_fps,
mon_ident_fps
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
dependent_functionElimination,
independent_pairFormation,
lambdaFormation,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
setElimination,
rename,
applyLambdaEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality,
promote_hyp,
hyp_replacement,
voidEquality,
voidElimination,
unionElimination,
equalityElimination,
dependent_pairFormation,
instantiate,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f:PowerSeries(X;r)]. \mforall{}[x:X]. (f(x:=atom(x)) = f)
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_11_49
Last ObjectModification:
2017_07_26-PM-06_34_46
Theory : power!series
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