Nuprl Lemma : fps-ucont-composition
∀[X:Type]
∀eq:EqDecider(X). ∀r:CRng. ∀F,G:PowerSeries(X;r) ⟶ PowerSeries(X;r).
(fps-ucont(X;eq;r;f.F[f]) ⇒ fps-ucont(X;eq;r;f.G[f]) ⇒ fps-ucont(X;eq;r;f.F o G[f]))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
compose: f o g,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: 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)↓,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
exists: ∃x:A. B[x],
so_apply: x[s],
compose: f o g,
so_lambda: λ2x.t[x],
crng: CRng,
rng: Rng,
pi1: fst(t),
uiff: uiff(P;Q),
and: P ∧ Q,
fps-restrict: fps-restrict(eq;r;f;d),
fps-coeff: f[b],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
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,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
power-series: PowerSeries(X;r),
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
sub-bag: sub-bag(T;as;bs)
Lemmas referenced :
equal-wf-base,
base_wf,
bag_wf,
fps-ucont_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
exists_wf,
all_wf,
equal_wf,
rng_car_wf,
fps-coeff_wf,
fps-restrict_wf,
bag-combine_wf,
sub-bags_wf,
fps-ext,
deq-sub-bag_wf,
bool_wf,
eqtt_to_assert,
assert-deq-sub-bag,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
sub-bag_wf,
rng_zero_wf,
squash_wf,
true_wf,
iff_weakening_equal,
sub-bag_transitivity,
bag-member-iff,
member-sub-bags,
decidable__sub-bag,
decidable-equal-deq,
bag-combine-append-left,
single-bag_wf,
bag-combine-single-left,
bag-append_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomSqleEquality,
hypothesis,
extract_by_obid,
because_Cache,
equalityTransitivity,
equalitySymmetry,
rename,
lambdaFormation,
dependent_functionElimination,
productElimination,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
promote_hyp,
setElimination,
dependent_pairFormation,
independent_functionElimination,
independent_isectElimination,
unionElimination,
equalityElimination,
instantiate,
voidElimination,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:Type]
\mforall{}eq:EqDecider(X). \mforall{}r:CRng. \mforall{}F,G:PowerSeries(X;r) {}\mrightarrow{} PowerSeries(X;r).
(fps-ucont(X;eq;r;f.F[f]) {}\mRightarrow{} fps-ucont(X;eq;r;f.G[f]) {}\mRightarrow{} fps-ucont(X;eq;r;f.F o G[f]))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_10_49
Last ObjectModification:
2017_07_26-PM-06_34_29
Theory : power!series
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