Nuprl Lemma : fpf-join-range
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[df:x:A fp-> Type]. ∀[f:x:A fp-> df(x)?Top]. ∀[dg:x:A fp-> Type].
∀[g:x:A fp-> dg(x)?Top].
  (f ⊕ g ∈ x:A fp-> df ⊕ dg(x)?Top) supposing 
     ((∀x:A. ((↑x ∈ dom(g)) 
⇒ (↑x ∈ dom(dg)))) and 
     (∀x:A. ((↑x ∈ dom(f)) 
⇒ (↑x ∈ dom(df)))) and 
     df || dg)
Proof
Definitions occuring in Statement : 
fpf-join: f ⊕ g
, 
fpf-compatible: f || g
, 
fpf-cap: f(x)?z
, 
fpf-dom: x ∈ dom(f)
, 
fpf: a:A fp-> B[a]
, 
deq: EqDecider(T)
, 
assert: ↑b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
top: Top
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
universe: Type
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
top: Top
, 
prop: ℙ
, 
fpf-join: f ⊕ g
, 
fpf: a:A fp-> B[a]
, 
pi1: fst(t)
, 
fpf-dom: x ∈ dom(f)
, 
fpf-cap: f(x)?z
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
false: False
, 
assert: ↑b
, 
bnot: ¬bb
, 
guard: {T}
, 
sq_type: SQType(T)
, 
or: P ∨ Q
, 
exists: ∃x:A. B[x]
, 
rev_implies: P 
⇐ Q
, 
not: ¬A
, 
iff: P 
⇐⇒ Q
, 
true: True
, 
squash: ↓T
, 
cand: A c∧ B
, 
fpf-compatible: f || g
Lemmas referenced : 
istype-universe, 
assert_wf, 
fpf-dom_wf, 
subtype-fpf2, 
fpf-cap_wf, 
top_wf, 
istype-void, 
fpf-compatible_wf, 
fpf_wf, 
deq_wf, 
deq-member_wf, 
bnot_wf, 
l_member_wf, 
filter_wf5, 
append_wf, 
not_wf, 
equal-wf-T-base, 
bool_wf, 
eqtt_to_assert, 
uiff_transitivity, 
eqff_to_assert, 
assert_of_bnot, 
equal_wf, 
fpf-ap_wf, 
assert-bnot, 
bool_subtype_base, 
subtype_base_sq, 
bool_cases_sqequal, 
fpf-join_wf, 
subtype_rel-equal, 
member_filter_2, 
assert-deq-member, 
member_append, 
iff_weakening_equal, 
fpf-join-ap, 
true_wf, 
squash_wf, 
subtype_rel_wf, 
subtype_rel_self, 
ext-eq_weakening, 
subtype_rel_weakening
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
functionIsType, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
universeIsType, 
applyEquality, 
lambdaEquality_alt, 
instantiate, 
cumulativity, 
universeEquality, 
inhabitedIsType, 
because_Cache, 
independent_isectElimination, 
lambdaFormation_alt, 
isect_memberEquality_alt, 
voidElimination, 
equalityTransitivity, 
equalitySymmetry, 
isect_memberFormation_alt, 
axiomEquality, 
dependent_pairEquality_alt, 
setEquality, 
rename, 
setElimination, 
lambdaFormation, 
lambdaEquality, 
productElimination, 
baseClosed, 
voidEquality, 
isect_memberEquality, 
unionElimination, 
equalityElimination, 
independent_functionElimination, 
dependent_functionElimination, 
promote_hyp, 
dependent_pairFormation, 
imageMemberEquality, 
natural_numberEquality, 
imageElimination, 
independent_pairFormation, 
functionEquality
Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[df:x:A  fp->  Type].  \mforall{}[f:x:A  fp->  df(x)?Top].  \mforall{}[dg:x:A  fp->  Type].
\mforall{}[g:x:A  fp->  dg(x)?Top].
    (f  \moplus{}  g  \mmember{}  x:A  fp->  df  \moplus{}  dg(x)?Top)  supposing 
          ((\mforall{}x:A.  ((\muparrow{}x  \mmember{}  dom(g))  {}\mRightarrow{}  (\muparrow{}x  \mmember{}  dom(dg))))  and 
          (\mforall{}x:A.  ((\muparrow{}x  \mmember{}  dom(f))  {}\mRightarrow{}  (\muparrow{}x  \mmember{}  dom(df))))  and 
          df  ||  dg)
Date html generated:
2019_10_16-AM-11_25_33
Last ObjectModification:
2018_10_10-PM-01_06_52
Theory : finite!partial!functions
Home
Index