Nuprl Lemma : fps-compose-ucont
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[g:PowerSeries(X;r)]. ∀x:X. fps-ucont(X;eq;r;f.f(x:=g)) supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
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],
all: ∀x:A. B[x],
universe: Type,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
valueall-type: valueall-type(T),
has-value: (a)↓,
all: ∀x:A. B[x],
crng: CRng,
comm: Comm(T;op),
and: P ∧ Q,
cand: A c∧ B,
rng: Rng,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
fps-restrict: fps-restrict(eq;r;f;d),
fps-compose: g(x:=f),
fps-coeff: f[b],
tlp: tlp(L),
hdp: hdp(L),
so_lambda: λ2x.t[x],
power-series: PowerSeries(X;r),
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
prop: ℙ,
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
listp: A List+,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
cons: [a / b],
top: Top,
length: ||as||,
list_ind: list_ind,
nil: [],
bag-union: bag-union(bbs),
concat: concat(ll),
bag-append: as + bs,
nat: ℕ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bag-size: #(bs),
reduce: reduce(f;k;as),
colength: colength(L),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
select: L[n],
sub-bag: sub-bag(T;as;bs)
Lemmas referenced :
rng_plus_comm,
sq_stable__monoid_p,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
crng_properties,
rng_properties,
crng_all_properties,
bag-append_wf,
bag-rep_wf,
bag-size_wf,
list-subtype-bag,
bag-summation-equal,
listp_wf,
bag_wf,
bag-parts'_wf,
infix_ap_wf,
rng_times_wf,
hdp_wf,
length_wf_nat,
tlp_wf,
bag-product_wf,
rng_one_wf,
deq-sub-bag_wf,
eqtt_to_assert,
assert-deq-sub-bag,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
sub-bag_wf,
bag-member_wf,
fps-coeff_wf,
fps-compose_wf,
fps-restrict_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
istype-universe,
bag-member-parts',
sub-bag-union-of-list,
hd_wf,
listp_properties,
hd_member,
list-cases,
null_nil_lemma,
reduce_tl_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
istype-void,
sub-bag-rep,
tl_wf,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
length_of_cons_lemma,
reduce_cons_lemma,
equal_wf,
squash_wf,
true_wf,
istype-nat,
subtype_rel_self,
iff_weakening_equal,
bag-size-append,
l_all_wf2,
not_wf,
equal-wf-T-base,
l_member_wf,
list_wf,
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,
le_witness_for_triv,
istype-false,
nil_wf,
colength-cons-not-zero,
colength_wf_list,
istype-le,
subtract-1-ge-0,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
cons_wf,
length-append,
add_nat_plus,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
false_wf,
length_wf,
bag-size-zero,
empty-bag_wf,
l_all_cons,
bag-union_wf,
non_neg_length,
bag-append-assoc2,
bag-append-ac,
bag-append-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
axiomSqleEquality,
hypothesis,
isectIsTypeImplies,
inhabitedIsType,
rename,
lambdaFormation_alt,
extract_by_obid,
setElimination,
independent_pairFormation,
because_Cache,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
dependent_pairFormation_alt,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
universeIsType,
unionElimination,
equalityElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
functionIsType,
universeEquality,
natural_numberEquality,
hypothesis_subsumption,
hyp_replacement,
applyLambdaEquality,
intEquality,
setIsType,
intWeakElimination,
approximateComputation,
int_eqEquality,
functionIsTypeImplies,
voidEquality,
dependent_set_memberEquality_alt,
baseApply,
closedConclusion,
sqequalBase,
pointwiseFunctionality,
productIsType,
addEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[g:PowerSeries(X;r)]. \mforall{}x:X. fps-ucont(X;eq;r;f.f(x:=g))
supposing valueall-type(X)
Date html generated:
2019_10_16-AM-11_36_06
Last ObjectModification:
2018_11_26-PM-03_09_19
Theory : power!series
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