Nuprl Lemma : fps-linear-ucont-equal

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[F,G:PowerSeries(X;r) ⟶ PowerSeries(X;r)].`
`    F = G ∈ (PowerSeries(X;r) ⟶ PowerSeries(X;r)) `
`    supposing fps-ucont(X;eq;r;f.F[f])`
`    ∧ fps-ucont(X;eq;r;f.G[f])`
`    ∧ (∀f,g:PowerSeries(X;r).  (F[(f+g)] = (F[f]+F[g]) ∈ PowerSeries(X;r)))`
`    ∧ (∀f,g:PowerSeries(X;r).  (G[(f+g)] = (G[f]+G[g]) ∈ PowerSeries(X;r)))`
`    ∧ (∀c:|r|. ∀f:PowerSeries(X;r).  (F[(c)*f] = (c)*F[f] ∈ PowerSeries(X;r)))`
`    ∧ (∀c:|r|. ∀f:PowerSeries(X;r).  (G[(c)*f] = (c)*G[f] ∈ PowerSeries(X;r)))`
`    ∧ (∀b:bag(X). (F[<b>] = G[<b>] ∈ PowerSeries(X;r))) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-ucont: `fps-ucont(X;eq;r;f.G[f])` fps-scalar-mul: `(c)*f` fps-add: `(f+g)` fps-single: `<c>` power-series: `PowerSeries(X;r)` bag: `bag(T)` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T` crng: `CRng` rng_car: `|r|`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` uiff: `uiff(P;Q)` all: `∀x:A. B[x]` fps-ucont: `fps-ucont(X;eq;r;f.G[f])` exists: `∃x:A. B[x]` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` crng: `CRng` rng: `Rng` true: `True` squash: `↓T` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` cand: `A c∧ B` comm: `Comm(T;op)` infix_ap: `x f y` assoc: `Assoc(T;op)` bag-summation: `Σ(x∈b). f[x]` 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]` cons-bag: `x.b` monoid_p: `IsMonoid(T;op;id)` ident: `Ident(T;op;id)` fps-restrict: `fps-restrict(eq;r;f;d)` fps-coeff: `f[b]` 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` sub-bag: `sub-bag(T;as;bs)`
Lemmas referenced :  fps-ext power-series_wf bag_wf fps-ucont_wf all_wf equal_wf fps-add_wf rng_car_wf fps-scalar-mul_wf fps-single_wf crng_wf deq_wf valueall-type_wf squash_wf true_wf iff_weakening_equal bag-append_wf fps-coeff_wf sub-bags_wf fps-restrict-summation fps-add-comm mon_assoc_fps bag_to_squash_list list_induction bag-summation_wf fps-zero_wf list-subtype-bag subtype_rel_self list_wf list_accum_nil_lemma empty-bag_wf rng_zero_wf fps-scalar-mul-zero single-bag_wf cons-bag-as-append bag-summation-append abmonoid_comm_fps mon_ident_fps and_wf bag-summation-single fps-restrict_wf 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 sub-bag_transitivity bag-append-comm
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin functionExtensionality rename extract_by_obid isectElimination hypothesisEquality applyEquality cumulativity hypothesis independent_isectElimination lambdaFormation dependent_functionElimination because_Cache productEquality sqequalRule lambdaEquality setElimination isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality universeEquality natural_numberEquality imageElimination imageMemberEquality baseClosed independent_functionElimination independent_pairFormation promote_hyp hyp_replacement applyLambdaEquality voidElimination voidEquality equalityUniverse levelHypothesis independent_pairEquality dependent_set_memberEquality unionElimination equalityElimination dependent_pairFormation instantiate

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[F,G:PowerSeries(X;r)  {}\mrightarrow{}  PowerSeries(X;r)].
F  =  G
supposing  fps-ucont(X;eq;r;f.F[f])
\mwedge{}  fps-ucont(X;eq;r;f.G[f])
\mwedge{}  (\mforall{}f,g:PowerSeries(X;r).    (F[(f+g)]  =  (F[f]+F[g])))
\mwedge{}  (\mforall{}f,g:PowerSeries(X;r).    (G[(f+g)]  =  (G[f]+G[g])))
\mwedge{}  (\mforall{}c:|r|.  \mforall{}f:PowerSeries(X;r).    (F[(c)*f]  =  (c)*F[f]))
\mwedge{}  (\mforall{}c:|r|.  \mforall{}f:PowerSeries(X;r).    (G[(c)*f]  =  (c)*G[f]))
\mwedge{}  (\mforall{}b:bag(X).  (F[<b>]  =  G[<b>]))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-10_11_01
Last ObjectModification: 2017_07_26-PM-06_34_32

Theory : power!series

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