Nuprl Lemma : fpf-union-contains2
∀[A,V:Type]. ∀[B:A ⟶ Type].
  ∀eq:EqDecider(A). ∀f,g:x:A fp-> B[x] List.
    ∀x:A. ∀R:(V List) ⟶ V ⟶ 𝔹.  (fpf-union-compatible(A;V;x.B[x];eq;R;f;g) 
⇒ g(x)?[] ⊆ fpf-union(f;g;eq;R;x)) 
    supposing fpf-single-valued(A;eq;x.B[x];V;g) 
  supposing ∀a:A. (B[a] ⊆r V)
Proof
Definitions occuring in Statement : 
fpf-single-valued: fpf-single-valued(A;eq;x.B[x];V;g)
, 
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g)
, 
fpf-union: fpf-union(f;g;eq;R;x)
, 
fpf-cap: f(x)?z
, 
fpf: a:A fp-> B[a]
, 
l_contains: A ⊆ B
, 
nil: []
, 
list: T List
, 
deq: EqDecider(T)
, 
bool: 𝔹
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
subtype_rel: A ⊆r B
, 
fpf-single-valued: fpf-single-valued(A;eq;x.B[x];V;g)
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
top: Top
, 
fpf-union: fpf-union(f;g;eq;R;x)
, 
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 
, 
band: p ∧b q
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
assert: ↑b
, 
false: False
, 
l_contains: A ⊆ B
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
decidable: Dec(P)
, 
cand: A c∧ B
, 
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g)
, 
label: ...$L... t
Lemmas referenced : 
equal_wf, 
l_member_wf, 
fpf-ap_wf, 
list_wf, 
assert_wf, 
fpf-dom_wf, 
subtype-fpf2, 
top_wf, 
bool_wf, 
eqtt_to_assert, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
l_contains_weakening, 
l_contains_nil, 
nil_wf, 
fpf-union-compatible_wf, 
fpf-single-valued_wf, 
fpf_wf, 
deq_wf, 
all_wf, 
subtype_rel_wf, 
l_all_iff, 
append_wf, 
filter_wf5, 
subtype_rel_list, 
subtype_rel_dep_function, 
subtype_rel_transitivity, 
subtype_rel_self, 
set_wf, 
member_append, 
decidable__assert, 
member_filter, 
or_wf, 
l_member_subtype, 
not_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
cut, 
introduction, 
sqequalRule, 
sqequalHypSubstitution, 
lambdaEquality, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
axiomEquality, 
hypothesis, 
rename, 
lambdaFormation, 
extract_by_obid, 
isectElimination, 
cumulativity, 
applyEquality, 
functionExtensionality, 
independent_isectElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
because_Cache, 
unionElimination, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
dependent_pairFormation, 
promote_hyp, 
instantiate, 
independent_functionElimination, 
functionEquality, 
universeEquality, 
setElimination, 
setEquality, 
inrFormation, 
independent_pairFormation, 
inlFormation, 
productEquality, 
addLevel, 
orFunctionality, 
hyp_replacement, 
applyLambdaEquality
Latex:
\mforall{}[A,V:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].
    \mforall{}eq:EqDecider(A).  \mforall{}f,g:x:A  fp->  B[x]  List.
        \mforall{}x:A.  \mforall{}R:(V  List)  {}\mrightarrow{}  V  {}\mrightarrow{}  \mBbbB{}.
            (fpf-union-compatible(A;V;x.B[x];eq;R;f;g)  {}\mRightarrow{}  g(x)?[]  \msubseteq{}  fpf-union(f;g;eq;R;x)) 
        supposing  fpf-single-valued(A;eq;x.B[x];V;g) 
    supposing  \mforall{}a:A.  (B[a]  \msubseteq{}r  V)
Date html generated:
2018_05_21-PM-09_18_31
Last ObjectModification:
2018_02_09-AM-10_17_10
Theory : finite!partial!functions
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