Nuprl Lemma : fpf-empty-join
∀[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],
fpf-cap: f(x)?z,
fpf-dom: x ∈ dom(f),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x]
Lemmas referenced :
fpf_ap_pair_lemma,
filter_nil_lemma,
append_back_nil,
append-nil,
subtype_rel_list,
top_wf,
deq-member_wf,
eqtt_to_assert,
assert-deq-member,
l_member_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
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,
because_Cache,
functionExtensionality_alt,
setElimination,
rename,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
universeIsType,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
setIsType,
functionIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[eq:EqDecider(A)]. (f \moplus{} \motimes{} = f)
Date html generated:
2020_05_20-AM-09_02_28
Last ObjectModification:
2020_01_04-PM-11_09_54
Theory : finite!partial!functions
Home
Index