Nuprl Lemma : fps-compose-compose
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g,h:PowerSeries(X;r)]. ∀[x:X]. (f(x:=g)(x:=h) = f(x:=g(x:=h)) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
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,
compose: f o g,
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),
fps-compose: g(x:=f),
bag-product: Πx ∈ b. f[x],
bag-rep: bag-rep(n;x),
bag-append: as + bs,
bag-parts': bag-parts'(eq;bs;x),
bag-summation: Σ(x∈b). f[x],
bag-null: bag-null(bs),
null: null(as),
nil: [],
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bag-parts: bag-parts(eq;bs),
bag-partitions: bag-partitions(eq;bs),
bag-splits: bag-splits(b),
list_ind: list_ind,
single-bag: {x},
cons: [a / b],
bag-to-set: bag-to-set(eq;bs),
bag-remove-repeats: bag-remove-repeats(eq;bs),
list-to-set: list-to-set(eq;L),
l-union: as ⋃ bs,
reduce: reduce(f;k;as),
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
bag-combine: ⋃x∈bs.f[x],
bag-union: bag-union(bbs),
concat: concat(ll),
bag-map: bag-map(f;bs),
map: map(f;as),
append: as @ bs,
pi1: fst(t),
bag-accum: bag-accum(v,x.f[v; x];init;bs),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
power-series: PowerSeries(X;r),
infix_ap: x f y
Lemmas referenced :
fps-linear-ucont-equal,
fps-compose_wf,
power-series_wf,
equal_wf,
squash_wf,
true_wf,
fps-compose-add,
fps-add_wf,
iff_weakening_equal,
rng_car_wf,
fps-compose-scalar-mul,
fps-scalar-mul_wf,
bag_wf,
crng_wf,
valueall-type_wf,
fps-ucont-composition,
fps-compose-ucont,
bag_to_squash_list,
list_induction,
fps-single_wf,
list-subtype-bag,
list_wf,
fps-ext,
nil_wf,
fps-one_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,
fps-compose-one,
deq_wf,
single-bag_wf,
cons-bag-as-append,
fps-mul-single,
fps-mul_wf,
fps-compose-mul,
safe-assert-deq,
fps-atom_wf,
fps-coeff_wf,
fps-sub_wf,
fps-compose-atom-eq,
fps-compose-sub,
list_accum_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
list_accum_nil_lemma,
list_ind_nil_lemma,
length_of_nil_lemma,
primrec0_lemma,
rng_plus_wf,
rng_times_one,
rng_plus_zero,
fps-compose-atom-neq
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,
independent_pairFormation,
lambdaFormation,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
voidEquality,
voidElimination,
unionElimination,
equalityElimination,
dependent_pairFormation,
instantiate,
equalityUniverse,
levelHypothesis,
callbyvalueReduce,
sqleReflexivity,
functionExtensionality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g,h:PowerSeries(X;r)]. \mforall{}[x:X].
(f(x:=g)(x:=h) = f(x:=g(x:=h)))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_11_40
Last ObjectModification:
2017_07_26-PM-06_34_45
Theory : power!series
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