### Nuprl Lemma : fpf-join-ap

`∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f,g:a:A fp-> B[a]]. ∀[x:A].`
`  f ⊕ g(x) = if x ∈ dom(f) then f(x) else g(x) fi  ∈ B[x] supposing ↑x ∈ dom(f ⊕ g)`

Proof

Definitions occuring in Statement :  fpf-join: `f ⊕ g` fpf-ap: `f(x)` fpf-dom: `x ∈ dom(f)` fpf: `a:A fp-> B[a]` deq: `EqDecider(T)` assert: `↑b` ifthenelse: `if b then t else f fi ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` fpf-dom: `x ∈ dom(f)` deq-member: `x ∈b L` reduce: `reduce(f;k;as)` list_ind: list_ind pi1: `fst(t)` fpf-ap: `f(x)` pi2: `snd(t)` fpf-join: `f ⊕ g` fpf-cap: `f(x)?z` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` top: `Top`
Lemmas referenced :  fpf-ap_wf fpf-join_wf assert_wf fpf-dom_wf top_wf subtype-fpf2 deq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule hypothesis lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaEquality applyEquality independent_isectElimination because_Cache lambdaFormation isect_memberEquality voidElimination voidEquality axiomEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}[A:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[f,g:a:A  fp->  B[a]].  \mforall{}[x:A].
f  \moplus{}  g(x)  =  if  x  \mmember{}  dom(f)  then  f(x)  else  g(x)  fi    supposing  \muparrow{}x  \mmember{}  dom(f  \moplus{}  g)

Date html generated: 2018_05_21-PM-09_21_45
Last ObjectModification: 2018_02_09-AM-10_18_26

Theory : finite!partial!functions

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