{ 
[A,V:Type]. [B:A 
Type].
eq:EqDecider(A). f,h,g:a:A fp-> B[a] List. R:V List V 
.
fpf-union-compatible(A;V;x.B[x];eq;R;f;g)
h 
g
h fpf-union-join(eq;R;f;g)
supposing fpf-single-valued(A;eq;x.B[x];V;g)
supposing a:A. (B[a] r V) }
{ Proof }
Definitions occuring in Statement :
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf-contains: f g,
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: a:A fp-> B[a],
subtype_rel: A r B,
bool: ,
uimplies: b supposing a,
uall: [x:A]. B[x],
so_apply: x[s],
all: x:A. B[x],
implies: P Q,
function: x:A B[x],
list: type List,
universe: Type,
deq: EqDecider(T)
Definitions :
append: as @ bs,
ext-eq: A 
B,
rev_subtype_rel: A 
r B,
rev_uimplies: rev_uimplies(P;Q),
tag-by: z
T,
rev_implies: P 
Q,
iff: P Q,
record+: record+,
record: record(x.T[x]),
fset: FSet{T},
isect2: T1 
T2,
b-union: A 
B,
fpf-cap: f(x)?z,
filter: filter(P;l),
nil: [],
eqof: eqof(d),
suptype: suptype(S; T),
subtype: S T,
intensional-universe: IType,
limited-type: LimitedType,
pair: <a, b>,
union: left + right,
or: P 
Q,
guard: {T},
top: Top,
fpf-dom: x 
dom(f),
fpf-ap: f(x),
strong-subtype: strong-subtype(A;B),
le: A 
B,
ge: i 
j ,
not: 
A,
less_than: a < b,
and: P 
Q,
uiff: uiff(P;Q),
axiom: Ax,
set: {x:A| B[x]} ,
nat: 
,
exists: 
x:A. B[x],
l_member: (x l),
l_all: (xL.P[x]),
l_contains: A B,
void: Void,
false: False,
true: True,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
ifthenelse: if b then t else f fi ,
assert: 
b,
product: x:A B[x],
cand: A c B,
prop: 
,
so_lambda: 
x.t[x],
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf-contains: f g,
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g),
implies: P Q,
fpf-single-valued: fpf-single-valued(A;eq;x.B[x];V;g),
isect: x:A. B[x],
bool: ,
list: type List,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
lambda: x.A[x],
apply: f a,
so_apply: x[s],
subtype_rel: A r B,
all: x:A. B[x],
uimplies: b supposing a,
uall: [x:A]. B[x],
function: x:A B[x],
universe: Type,
member: t T,
equal: s = t,
MaAuto: Error :MaAuto,
ParallelOp: Error :ParallelOp,
RepeatFor: Error :RepeatFor,
CollapseTHEN: Error :CollapseTHEN,
Auto: Error :Auto,
CollapseTHENA: Error :CollapseTHENA,
tactic: Error :tactic,
btrue: tt,
sq_type: SQType(T),
bfalse: ff,
eq_bool: p =b q,
lt_int: i <z j,
le_int: i z j,
eq_int: (i =
j),
eq_atom: x =a y,
null: null(as),
set_blt: a <
b,
grp_blt: a < b,
infix_ap: x f y,
dcdr-to-bool: [d],
bl-all: (xL.P[x])_b,
bl-exists: (xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
eq_atom: eq_atom$n(x;y),
qeq: qeq(r;s),
q_less: q_less(r;s),
q_le: q_le(r;s),
deq-member: deq-member(eq;x;L),
deq-disjoint: deq-disjoint(eq;as;bs),
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
eq_str: Error :eq_str,
eq_id: a = b,
eq_lnk: a = b,
bimplies: p q,
band: p q,
bor: p q,
bnot: b,
int: 
,
unit: Unit,
map: map(f;as),
cons: [car / cdr],
hd: hd(l),
last: last(L),
remove-repeats: remove-repeats(eq;L),
select: l[i],
fpf-union: fpf-union(f;g;eq;R;x),
sqequal: s ~ t,
SplitOn: Error :SplitOn,
CollapseTHENM: Error :CollapseTHENM
Lemmas :
fpf-union-join-ap,
fpf-union-contains2,
nat_wf,
l_member_subtype,
eqtt_to_assert,
iff_weakening_uiff,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
not_wf,
bnot_wf,
fpf-union-join-dom,
subtype_base_sq,
bool_subtype_base,
assert_elim,
assert_wf,
l_contains_wf,
fpf-contains_wf,
fpf-trivial-subtype-top,
subtype_rel_wf,
fpf_wf,
top_wf,
member_wf,
fpf-dom_wf,
fpf-ap_wf,
l_member_wf,
fpf-union-compatible_wf,
fpf-single-valued_wf,
bool_wf,
deq_wf,
intensional-universe_wf,
assert_witness,
false_wf,
ifthenelse_wf,
true_wf,
fpf-union-join_wf,
l_all_wf,
l_contains_weakening,
list-subtype,
ext-eq_inversion,
implies_weakening_uimplies,
subtype_rel_functionality_wrt_implies,
subtype_rel_weakening,
ext-eq_weakening
\mforall{}[A,V:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}eq:EqDecider(A). \mforall{}f,h,g:a:A fp-> B[a] List. \mforall{}R:V List {}\mrightarrow{} V {}\mrightarrow{} \mBbbB{}.
fpf-union-compatible(A;V;x.B[x];eq;R;f;g) {}\mRightarrow{} h \msubseteq{}\msubseteq{} g {}\mRightarrow{} h \msubseteq{}\msubseteq{} fpf-union-join(eq;R;f;g)
supposing fpf-single-valued(A;eq;x.B[x];V;g)
supposing \mforall{}a:A. (B[a] \msubseteq{}r V)
Date html generated:
2011_08_10-AM-08_02_12
Last ObjectModification:
2011_06_18-AM-08_20_16
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