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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