Nuprl Lemma : fps-compose-one
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[f:PowerSeries(X;r)]. (1(x:=f) = 1 ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-one: 1,
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,
fps-one: 1,
fps-compose: g(x:=f),
power-series: PowerSeries(X;r),
fps-coeff: f[b],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
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,
crng: CRng,
rng: Rng,
true: True,
bag-product: Πx ∈ b. f[x],
infix_ap: x f y,
empty-bag: {},
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),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bag-append: as + bs,
hd: hd(l),
bag-rep: bag-rep(n;x),
primrec: primrec(n;b;c),
length: ||as||,
tl: tl(l),
pi2: snd(t),
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
listp: A List+,
so_apply: x[s],
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
ge: i ≥ j ,
decidable: Dec(P),
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than': less_than'(a;b),
nat: ℕ
Lemmas referenced :
bag-null_wf,
bool_wf,
eqtt_to_assert,
assert-bag-null,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
bag_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
equal-empty-bag,
rng_car_wf,
rng_plus_wf,
rng_one_wf,
rng_zero_wf,
list_accum_cons_lemma,
reduce_tl_cons_lemma,
list_accum_nil_lemma,
squash_wf,
true_wf,
rng_times_one,
rng_plus_zero,
iff_weakening_equal,
crng_properties,
rng_all_properties,
rng_properties,
rng_plus_comm2,
bag-summation-is-zero,
listp_wf,
bag-parts'_wf,
infix_ap_wf,
rng_times_wf,
bag-append_wf,
hd_wf,
listp_properties,
bag-rep_wf,
length_wf_nat,
tl_wf,
list-subtype-bag,
bag-product_wf,
subtype_rel_self,
bag-member_wf,
rng_times_zero,
bag-member-parts',
decidable__lt,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
less_than_wf,
list-cases,
product_subtype_list,
reduce_hd_cons_lemma,
bag-append-comm,
length_of_nil_lemma,
length_of_cons_lemma,
primrec0_lemma,
empty_bag_append_lemma,
empty-bag_wf,
bag-union_wf,
cons_wf,
nil_wf,
bag-size_wf,
bag-size-append,
bag-size-rep,
decidable__le,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
le_wf,
bag_size_empty_lemma,
non_neg_length,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
voidElimination,
baseClosed,
isect_memberEquality,
axiomEquality,
universeEquality,
setElimination,
rename,
natural_numberEquality,
callbyvalueReduce,
sqleReflexivity,
voidEquality,
applyEquality,
imageElimination,
imageMemberEquality,
independent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
dependent_set_memberEquality,
hypothesis_subsumption,
hyp_replacement,
applyLambdaEquality,
addEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[f:PowerSeries(X;r)]. (1(x:=f) = 1)
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_59_53
Last ObjectModification:
2017_07_26-PM-06_34_05
Theory : power!series
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