{ 
[A,B,M:Type]. [dfs:bag(dataflow(M;bag(B)))]. [ms:M List].
[P:dataflow(M;bag(A))]. [Q:A 
dataflow(M;bag(B))].
(bind-dataflow-aux(P;dfs;a.Q[a])*(ms)
= bind-dataflow-aux(P*(ms);bag-map(
df.df*(ms);dfs)
+ 
n
upto(||ms||).bag-map(a.Q[a]*(nth_tl(n;ms));
snd(P*(firstn(n;ms))(ms[n])));a.Q[a])) }
{ Proof }
Definitions occuring in Statement :
bind-dataflow-aux: bind-dataflow-aux(P;dfs;a.Q[a]),
iterate-dataflow: P*(inputs),
dataflow-ap: df(a),
dataflow: dataflow(A;B),
select: l[i],
firstn: firstn(n;as),
nth_tl: nth_tl(n;as),
length: ||as||,
uall: [x:A]. B[x],
so_apply: x[s],
pi2: snd(t),
lambda: x.A[x],
function: x:A B[x],
list: type List,
universe: Type,
equal: s = t,
bag-combine: xbs.f[x],
bag-append: as + bs,
bag-map: bag-map(f;bs),
bag: bag(T),
upto: upto(n)
Definitions :
union: left + right,
or: P 
Q,
assert: 
b,
divides: b | a,
assoced: a ~ b,
set_leq: a 
b,
set_lt: a <p b,
grp_lt: a < b,
cand: A c
B,
l_member: (x l),
l_contains: A 
B,
cmp-le: cmp-le(cmp;x;y),
inject: Inj(A;B;f),
reducible: reducible(a),
prime: prime(a),
squash: 
T,
l_exists: (
xL. P[x]),
l_all: (xL.P[x]),
infix_ap: x f y,
fun-connected: y is f*(x),
qle: r s,
qless: r < s,
q-rel: q-rel(r;x),
sq_exists: x:{A| B[x]},
product: x:A 
B[x],
exists: x:A. B[x],
i-finite: i-finite(I),
i-closed: i-closed(I),
p-outcome: Outcome,
fset-member: a s,
f-subset: xs ys,
fset-closed: (s closed under fs),
l_disjoint: l_disjoint(T;l1;l2),
decidable: Dec(P),
void: Void,
top: Top,
uimplies: b supposing a,
set: {x:A| B[x]} ,
real: 
,
grp_car: |g|,
subtype: S T,
int: 
,
nat: 
,
le: A B,
prop: 
,
guard: {T},
implies: P 
Q,
all: x:A. B[x],
quotient: x,y:A//B[x; y],
primrec: primrec(n;b;c),
corec: corec(T.F[T]),
pair: <a, b>,
bool: 
,
axiom: Ax,
select: l[i],
firstn: firstn(n;as),
dataflow-ap: df(a),
pi2: snd(t),
nth_tl: nth_tl(n;as),
upto: upto(n),
bag-combine: xbs.f[x],
lambda: x.A[x],
bag-map: bag-map(f;bs),
bag-append: as + bs,
apply: f a,
so_apply: x[s],
bind-dataflow-aux: bind-dataflow-aux(P;dfs;a.Q[a]),
iterate-dataflow: P*(inputs),
member: t T,
isect: 
x:A. B[x],
uall: [x:A]. B[x],
so_lambda: 
x.t[x],
equal: s = t,
function: x:A B[x],
list: type List,
dataflow: dataflow(A;B),
bag: bag(T),
universe: Type,
ge: i 
j ,
false: False,
not: 
A,
minus: -n,
add: n + m,
subtract: n - m,
length: ||as||,
natural_number: $n,
less_than: a < b,
nil: [],
last: last(L),
cons: [car / cdr],
append: as @ bs,
sqequal: s ~ t,
fpf: a:A fp-> B[a],
iter_df_nil: iter_df_nil{iter_df_nil_compseq_tag_def:o}(P),
iter_df_cons: iter_df_cons{iter_df_cons_compseq_tag_def:o}(as; a; P),
tl: tl(l),
hd: hd(l),
strong-subtype: strong-subtype(A;B),
and: P Q,
uiff: uiff(P;Q),
subtype_rel: A r B,
base: Base,
data-stream: data-stream(P;L),
true: True,
rev_implies: P 
Q,
iff: P Q,
pi1: fst(t),
map: map(f;as),
rationals: 
,
int_iseg: {i...j},
lelt: i j < k,
int_seg: {i..j
},
rec_dataflow_ap: rec_dataflow_ap_compseq_tag_def,
let: let,
single-bag: {x},
compose: f o g,
label: ...$L... t,
stream: stream(A),
b-union: A B,
fpf-sub: f g,
deq: EqDecider(T),
ma-state: State(ds),
fpf-cap: f(x)?z,
permutation: permutation(T;L1;L2),
bfalse: ff,
limited-type: LimitedType,
btrue: tt,
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,
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_id: a = b,
eq_lnk: a = b,
bimplies: p q,
band: p q,
bor: p q,
bnot: b,
unit: Unit,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
bag-union: bag-union(bbs)
Lemmas :
bag-subtype-list,
nth_tl_append,
firstn_append,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
select_append_back,
bag-combine-single-left,
ifthenelse_wf,
nth_tl-append,
bool_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
le_int_wf,
bnot_wf,
lt_int_wf,
firstn-append,
bnot_of_le_int,
bag-map-combine,
select_append_front,
bag-combine-append-left,
permutation_wf,
subtype_rel_wf,
subtype_rel_bag,
int_seg_properties,
length_nil,
length_cons,
length_append,
bag-map-map,
bag-map-append,
single-bag_wf,
bag-append-assoc,
upto_add_1,
length-append,
bag_qinc,
bag-combine_wf,
int_seg_wf,
length_firstn,
non_neg_length,
true_wf,
squash_wf,
dataflow-ap_wf,
pi1_wf_top,
rev_implies_wf,
iff_wf,
assert_wf,
not_wf,
false_wf,
pos_length2,
pi1_wf,
bag-append_wf,
bag-map_wf,
upto_wf,
append_wf,
select_wf,
pi2_wf,
nth_tl_wf,
bind-dataflow-aux_wf,
iterate-dataflow_wf,
firstn_wf,
last_wf,
iterate-dataflow-append,
firstn_all,
firstn_decomp,
ge_wf,
nat_properties,
nat_ind_tp,
bag_wf,
dataflow_wf,
uall_wf,
guard_wf,
le_wf,
nat_wf,
comp_nat_ind_tp,
length_wf1,
length_wf_nat,
top_wf,
member_wf,
decidable__lt
\mforall{}[A,B,M:Type]. \mforall{}[dfs:bag(dataflow(M;bag(B)))]. \mforall{}[ms:M List]. \mforall{}[P:dataflow(M;bag(A))].
\mforall{}[Q:A {}\mrightarrow{} dataflow(M;bag(B))].
(bind-dataflow-aux(P;dfs;a.Q[a])*(ms)
= bind-dataflow-aux(P*(ms);bag-map(\mlambda{}df.df*(ms);dfs)
+ \mcup{}n\mmember{}upto(||ms||).bag-map(\mlambda{}a.Q[a]*(nth\_tl(n;ms));snd(P*(firstn(n;ms))(ms[n])));a.Q[a]))
Date html generated:
2011_08_16-AM-09_49_30
Last ObjectModification:
2011_05_09-PM-06_25_20
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