### Nuprl Lemma : fps-elim-x-atom

`∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x,y:X].`
`  (atom(y)(x:=0) = if eq x y then 0 else atom(y) fi  ∈ PowerSeries(X;r))`

Proof

Definitions occuring in Statement :  fps-elim-x: `f(x:=0)` fps-atom: `atom(x)` fps-zero: `0` power-series: `PowerSeries(X;r)` deq: `EqDecider(T)` ifthenelse: `if b then t else f fi ` uall: `∀[x:A]. B[x]` apply: `f a` universe: `Type` equal: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` deq: `EqDecider(T)` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` fps-atom: `atom(x)` fps-zero: `0` fps-coeff: `f[b]` fps-elim-x: `f(x:=0)` fps-single: `<c>` fps-elim: `fps-elim(x)` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` eqof: `eqof(d)` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` crng: `CRng` rng: `Rng` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  fps-ext fps-elim-x_wf fps-atom_wf ifthenelse_wf power-series_wf fps-zero_wf bag-deq-member_wf bool_wf eqtt_to_assert assert-bag-deq-member safe-assert-deq eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot bag-eq_wf single-bag_wf assert-bag-eq bag_wf rng_zero_wf bag-member_wf rng_one_wf crng_wf deq_wf bag-member-single not_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality cumulativity hypothesis applyEquality setElimination rename productElimination independent_isectElimination lambdaFormation sqequalRule unionElimination equalityElimination equalityTransitivity equalitySymmetry dependent_pairFormation promote_hyp dependent_functionElimination instantiate independent_functionElimination voidElimination isect_memberEquality axiomEquality universeEquality hyp_replacement applyLambdaEquality

Latex:
\mforall{}[X:Type].  \mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[x,y:X].
(atom(y)(x:=0)  =  if  eq  x  y  then  0  else  atom(y)  fi  )

Date html generated: 2018_05_21-PM-09_59_27
Last ObjectModification: 2017_07_26-PM-06_33_52

Theory : power!series

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