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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