{ 
[A,B,S:Type]. [next:S 
A (S 
bag(B))]. [F:S A (S? bag(B))].
[s:S].
rec-dataflow(s;s,a.next[s;a]) 
rec-dataflow(inl s ;s,a.case s
of inl(s1) =>
F s1 a
| inr(x) =>
<s, {}>)
supposing s:S. a:A.
let hs,out = F s a
in let s2,bs = next s a
in (out = bs)
case hs
of inl(s') =>
s' = s2
| inr(z) =>
rec-dataflow-halt-state(A;B;s2;s,a.next[s;a]) }
{ Proof }
Definitions occuring in Statement :
dataflow-equiv: d1 d2,
rec-dataflow-halt-state: rec-dataflow-halt-state(A;B;s0;s,a.next[s; a]),
rec-dataflow: rec-dataflow(s0;s,m.next[s; m]),
uimplies: b supposing a,
uall: [x:A]. B[x],
so_apply: x[s1;s2],
all: x:A. B[x],
and: P Q,
unit: Unit,
apply: f a,
function: x:A B[x],
spread: spread def,
pair: <a, b>,
product: x:A B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inl: inl x ,
union: left + right,
universe: Type,
equal: s = t
Definitions :
pi1: fst(t),
corec: corec(T.F[T]),
so_lambda: 
x.t[x],
fpf: a:A fp-> B[a],
pi2: snd(t),
implies: P 
Q,
limited-type: LimitedType,
strong-subtype: strong-subtype(A;B),
le: A 
B,
ge: i 
j ,
not: 
A,
less_than: a < b,
uiff: uiff(P;Q),
subtype_rel: A 
r B,
axiom: Ax,
empty-bag: Error :empty-bag,
pair: <a, b>,
inl: inl x ,
rec-dataflow: rec-dataflow(s0;s,m.next[s; m]),
data-stream: data-stream(P;L),
list: type List,
lambda: x.A[x],
prop: 
,
and: P Q,
apply: f a,
spread: spread def,
uimplies: b supposing a,
dataflow-equiv: d1 d2,
all: x:A. B[x],
universe: Type,
uall: [x:A]. B[x],
product: x:A B[x],
function: x:A B[x],
member: t 
T,
isect: 
x:A. B[x],
bag: Error :bag,
dataflow: dataflow(A;B),
so_apply: x[s1;s2],
rec-dataflow-halt-state: rec-dataflow-halt-state(A;B;s0;s,a.next[s; a]),
equal: s = t,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
so_lambda: x y.t[x; y],
unit: Unit,
union: left + right,
sq_type: SQType(T),
Auto: Error :Auto,
CollapseTHEN: Error :CollapseTHEN,
D: Error :D,
RepUR: Error :RepUR,
CollapseTHENA: Error :CollapseTHENA,
tactic: Error :tactic,
iterate-dataflow: P*(inputs),
dataflow-ap: df(a),
iter_df_nil: iter_df_nil{iter_df_nil_compseq_tag_def:o}(P),
rec_dataflow_ap: rec_dataflow_ap_compseq_tag_def,
nil: [],
void: Void,
subtype: S T,
top: Top,
let: let,
quotient: x,y:A//B[x; y],
iter_df_cons: iter_df_cons{iter_df_cons_compseq_tag_def:o}(as; a; P),
Complete: Error :Complete,
Try: Error :Try,
cons: [car / cdr],
ParallelOp: Error :ParallelOp,
RepeatFor: Error :RepeatFor
Lemmas :
iterate-dataflow_wf,
rec-dataflow_wf,
dataflow-ap_wf,
top_wf,
pi1_wf_top,
subtype_base_sq,
dataflow_wf,
Error :bag_wf,
member_wf,
pi2_wf,
unit_wf,
rec-dataflow-halt-state_wf,
Error :empty-bag_wf,
rec-dataflow-equiv,
dataflow-equiv_wf,
subtype_rel_wf,
pi1_wf
\mforall{}[A,B,S:Type]. \mforall{}[next:S {}\mrightarrow{} A {}\mrightarrow{} (S \mtimes{} bag(B))]. \mforall{}[F:S {}\mrightarrow{} A {}\mrightarrow{} (S? \mtimes{} bag(B))]. \mforall{}[s:S].
rec-dataflow(s;s,a.next[s;a]) \mequiv{} rec-dataflow(inl s ;s,a.case s
of inl(s1) =>
F s1 a
| inr(x) =>
<s, \{\}>)
supposing \mforall{}s:S. \mforall{}a:A.
let hs,out = F s a
in let s2,bs = next s a
in (out = bs)
\mwedge{} case hs
of inl(s') =>
s' = s2
| inr(z) =>
rec-dataflow-halt-state(A;B;s2;s,a.next[s;a])
Date html generated:
2011_08_10-AM-08_22_46
Last ObjectModification:
2011_03_22-PM-01_46_26
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