### Nuprl Lemma : fps-restrict-empty

`∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f:PowerSeries(X;r)].`
`  (fps-restrict(eq;r;f;{}) = (f[{}])*1 ∈ PowerSeries(X;r))`

Proof

Definitions occuring in Statement :  fps-restrict: `fps-restrict(eq;r;f;d)` fps-scalar-mul: `(c)*f` fps-one: `1` fps-coeff: `f[b]` power-series: `PowerSeries(X;r)` empty-bag: `{}` deq: `EqDecider(T)` 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` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` fps-one: `1` fps-coeff: `f[b]` fps-scalar-mul: `(c)*f` fps-restrict: `fps-restrict(eq;r;f;d)` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` iff: `P `⇐⇒` Q` ifthenelse: `if b then t else f fi ` squash: `↓T` prop: `ℙ` crng: `CRng` rng: `Rng` power-series: `PowerSeries(X;r)` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` rev_implies: `P `` Q` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A`
Lemmas referenced :  fps-ext fps-restrict_wf empty-bag_wf fps-scalar-mul_wf fps-coeff_wf fps-one_wf deq-sub-bag_wf bool_wf eqtt_to_assert assert-deq-sub-bag bag-null_wf assert-bag-null sub-bag-empty equal_wf squash_wf true_wf rng_car_wf rng_times_one iff_weakening_equal eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot equal-wf-T-base bag_wf sub-bag_wf rng_times_zero power-series_wf crng_wf deq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality cumulativity hypothesis productElimination independent_isectElimination lambdaFormation sqequalRule unionElimination equalityElimination equalityTransitivity equalitySymmetry because_Cache dependent_functionElimination independent_functionElimination applyEquality lambdaEquality imageElimination setElimination rename natural_numberEquality imageMemberEquality baseClosed universeEquality dependent_pairFormation promote_hyp instantiate voidElimination isect_memberEquality axiomEquality

Latex:
\mforall{}[X:Type].  \mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[f:PowerSeries(X;r)].
(fps-restrict(eq;r;f;\{\})  =  (f[\{\}])*1)

Date html generated: 2018_05_21-PM-10_10_40
Last ObjectModification: 2017_07_26-PM-06_34_26

Theory : power!series

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