### Nuprl Lemma : fps-add-ucont-general

`∀[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[f]+G[f])))`

Proof

Definitions occuring in Statement :  fps-ucont: `fps-ucont(X;eq;r;f.G[f])` fps-add: `(f+g)` power-series: `PowerSeries(X;r)` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` so_apply: `x[s]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` fps-ucont: `fps-ucont(X;eq;r;f.G[f])` all: `∀x:A. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` fps-coeff: `f[b]` fps-add: `(f+g)` rng_plus: `+r` pi1: `fst(t)` pi2: `snd(t)` infix_ap: `x f y` top: `Top` prop: `ℙ` so_lambda: `λ2x.t[x]` crng: `CRng` rng: `Rng` so_apply: `x[s]` true: `True` squash: `↓T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` fps-restrict: `fps-restrict(eq;r;f;d)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` iff: `P `⇐⇒` Q` ifthenelse: `if b then t else f fi ` power-series: `PowerSeries(X;r)` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` sub-bag: `sub-bag(T;as;bs)`
Lemmas referenced :  bag-append_wf top_wf power-series_wf all_wf equal_wf rng_car_wf fps-coeff_wf fps-add_wf fps-restrict_wf bag_wf fps-ucont_wf crng_wf deq_wf rng_plus_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 subtype_rel_self iff_weakening_equal bag-append-assoc2 bag-append-comm
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality productElimination dependent_pairFormation cut introduction extract_by_obid isectElimination hypothesis sqequalRule isect_memberEquality voidElimination voidEquality lambdaEquality setElimination rename applyEquality functionEquality universeEquality because_Cache natural_numberEquality imageElimination independent_isectElimination unionElimination equalityElimination equalityTransitivity equalitySymmetry independent_functionElimination promote_hyp instantiate cumulativity imageMemberEquality baseClosed

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[f]+G[f])))

Date html generated: 2018_05_21-PM-10_11_06
Last ObjectModification: 2018_05_19-PM-04_15_35

Theory : power!series

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