Nuprl Lemma : fpf-join-empty-sq
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[eq:EqDecider(A)]. (⊗ ⊕ f ~ <fst(f), λa.((snd(f)) a)>)
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-empty: ⊗,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
fpf: a:A fp-> B[a],
fpf-empty: ⊗,
fpf-join: f ⊕ g,
pi1: fst(t),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
pi2: snd(t),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
fpf-cap: f(x)?z,
fpf-dom: x ∈ dom(f),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
fpf_ap_pair_lemma,
list_ind_nil_lemma,
deq_member_nil_lemma,
filter_tt,
subtype_rel_list,
top_wf,
deq_wf,
fpf_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
sqequalRule,
cut,
lemma_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
because_Cache,
functionEquality,
cumulativity,
universeEquality,
isect_memberFormation,
introduction,
sqequalAxiom
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[eq:EqDecider(A)].
(\motimes{} \moplus{} f \msim{} <fst(f), \mlambda{}a.((snd(f)) a)>)
Date html generated:
2018_05_21-PM-09_21_09
Last ObjectModification:
2018_02_09-AM-10_18_11
Theory : finite!partial!functions
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