Nuprl Lemma : fpf-join-ap-sq
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> Top]. ∀[g:Top]. ∀[x:A]. (f ⊕ g(x) ~ if x ∈ dom(f) then f(x) else g(x) fi )
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),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
fpf-ap: f(x),
fpf-join: f ⊕ g,
pi2: snd(t),
fpf-cap: f(x)?z,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ
Lemmas referenced :
fpf-dom_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
top_wf,
fpf_wf,
deq_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseClosed,
lambdaEquality,
universeEquality,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f:a:A fp-> Top]. \mforall{}[g:Top]. \mforall{}[x:A].
(f \moplus{} g(x) \msim{} if x \mmember{} dom(f) then f(x) else g(x) fi )
Date html generated:
2018_05_21-PM-09_21_50
Last ObjectModification:
2018_02_09-AM-10_18_30
Theory : finite!partial!functions
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