Nuprl Lemma : fpf-join-empty
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[eq:EqDecider(A)]. (⊗ ⊕ f = f ∈ a:A fp-> B[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],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fpf: a:A fp-> B[a],
fpf-empty: ⊗,
fpf-join: f ⊕ g,
pi1: fst(t),
all: ∀x:A. B[x],
append: as @ bs,
so_lambda: so_lambda3,
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,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q
Lemmas referenced :
fpf_ap_pair_lemma,
list_ind_nil_lemma,
deq_member_nil_lemma,
filter_tt,
subtype_rel_list,
top_wf,
subtype_rel_sets_simple,
l_member_wf,
filter_wf5,
btrue_wf,
deq_wf,
fpf_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
dependent_pairEquality_alt,
isectElimination,
hypothesisEquality,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
universeIsType,
functionExtensionality_alt,
setIsType,
inhabitedIsType,
lambdaFormation_alt,
because_Cache,
functionIsType,
setElimination,
rename,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[eq:EqDecider(A)]. (\motimes{} \moplus{} f = f)
Date html generated:
2020_05_20-AM-09_02_26
Last ObjectModification:
2019_12_31-PM-05_00_05
Theory : finite!partial!functions
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