{ 
[Info:{Info:Type| 
Info} ]. [B:{B:Type| valueall-type(B)} ].
[Ps:eclass-program{i:l}(Info) List].
[F:k:
||Ps|| 
bag(eclass-program-type(Ps[k])) bag(B)].
defined-class(simple-comb-program-strict(F;B;Ps))
= simple-comb(F;
k.defined-class(Ps[k]))
supposing (f:k:||Ps|| bag(eclass-program-type(Ps[k]))
((
k:||Ps||. (
null(f k))) 
(null(F f))))
((F (k.{})) = {}) }
{ Proof }
Definitions occuring in Statement :
simple-comb-program-strict: simple-comb-program-strict(F;B;Ps),
defined-class: defined-class(prg),
eclass-program-type: eclass-program-type(prg),
eclass-program: eclass-program{i:l}(Info),
simple-comb: simple-comb(F;Xs),
eclass: EClass(A[eo; e]),
select: l[i],
length: ||as||,
null: null(as),
assert: b,
int_seg: {i..j
},
uimplies: b supposing a,
uall: [x:A]. B[x],
all: x:A. B[x],
exists: x:A. B[x],
squash: T,
implies: P Q,
and: P Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: x.A[x],
function: x:A B[x],
list: type List,
natural_number: $n,
universe: Type,
equal: s = t,
empty-bag: {},
bag: bag(T),
valueall-type: valueall-type(T)
Definitions :
all: x:A. B[x],
int_seg: {i..j},
natural_number: $n,
length: ||as||,
implies: P Q,
l_member: (x 
l),
union-list2: union-list2(eq;ll),
id-deq: IdDeq,
map: map(f;as),
lambda: x.A[x],
Id: Id,
fpf-domain: fpf-domain(f),
eclass-program-flows: eclass-program-flows(p),
select: l[i],
set: {x:A| B[x]} ,
universe: Type,
bag: bag(T),
function: x:A B[x],
member: t T,
isect: 
x:A. B[x],
uall: [x:A]. B[x],
list: type List,
eclass-program: eclass-program{i:l}(Info),
valueall-type: valueall-type(T),
equal: s = t,
uimplies: b supposing a,
prop: 
,
squash: T,
eclass-program-type: eclass-program-type(prg),
and: P Q,
product: x:A 
B[x],
assert: b,
exists: x:A. B[x],
eclass: EClass(A[eo; e]),
defined-class: defined-class(prg),
simple-comb-program-strict: simple-comb-program-strict(F;B;Ps),
simple-comb: simple-comb(F;Xs),
axiom: Ax,
apply: f a,
empty-bag: {},
so_lambda: 
x y.t[x; y],
subtype_rel: A 
r B,
uiff: uiff(P;Q),
less_than: a < b,
not: 
A,
ge: i 
j ,
le: A 
B,
strong-subtype: strong-subtype(A;B),
ifthenelse: if b then t else f fi ,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
fpf: a:A fp-> B[a],
pair: <a, b>,
top: Top,
int: 
,
subtype: S T,
rationals: 
,
real: 
,
null: null(as),
limited-type: LimitedType,
false: False,
void: Void,
lelt: i j < k,
quotient: x,y:A//B[x; y],
fpf-cap: f(x)?z,
permutation: permutation(T;L1;L2),
es-E-interface: E(X),
class-program: ClassProgram(T),
sq_stable: SqStable(P),
classrel: v X(e),
fpf-sub: f g,
true: True,
bool: 
,
fpf-join: f 
g,
mk_fpf: mk_fpf(L;f),
fpf-single: x : v,
dataflow-set-class: dataflow-set-class(x.P[x]),
spread: spread def,
event-ordering+: EO+(Info),
es-E: E,
event_ordering: EO,
cand: A c B,
nat: ,
record+: record+,
dep-isect: Error :dep-isect,
atom: Atom$n,
eq_atom: eq_atom$n(x;y),
eq_atom: x =a y,
record-select: r.x,
dataflow-program: DataflowProgram(A),
df-program-type: df-program-type(dfp),
so_lambda: x.t[x],
iff: P 
Q,
rev_implies: P Q,
deq: EqDecider(T),
combination: Combination(n;T),
listp: A List
,
modulus-of-ccontinuity: modulus-of-ccontinuity(omega;I;f),
partitions: partitions(I;p),
i-member: r I,
rleq: x y,
rnonneg: rnonneg(r),
req: x = y,
bag-member: bag-member(T;x;bs),
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
value-type: value-type(T),
no_repeats: no_repeats(T;l),
prime_ideal_p: IsPrimeIdeal(R;P),
integ_dom_p: IsIntegDom(r),
grp_leq: a b,
monoid_hom_p: IsMonHom{M1,M2}(f),
group_p: IsGroup(T;op;id;inv),
monoid_p: IsMonoid(T;op;id),
monot: monot(T;x,y.R[x; y];f),
cancel: Cancel(T;S;op),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
action_p: IsAction(A;x;e;S;f),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
bilinear: BiLinear(T;pl;tm),
inverse: Inverse(T;op;id;inv),
comm: Comm(T;op),
assoc: Assoc(T;op),
ident: Ident(T;op;id),
coprime: CoPrime(a,b),
uconnex: uconnex(T; x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
eqfun_p: IsEqFun(T;eq),
inject: Inj(A;B;f),
inv_funs: InvFuns(A;B;f;g),
uni_sat: a = !x:T. Q[x],
decidable: Dec(P),
path-goes-thru: x-f*-y thru i,
cut-order: a (X;f) b,
collect-event: collect-event(es;X;n;v.num[v];L.P[L];e),
same-thread: same-thread(es;p;e;e'),
es-r-immediate-pred: es-r-immediate-pred(es;R;e';e),
es-fset-loc: i locs(s),
existse-between3: e(e1,e2].P[e],
existse-between2: e[e1,e2].P[e],
alle-between2: e[e1,e2].P[e],
existse-between1: e[e1,e2).P[e],
alle-between1: e[e1,e2).P[e],
alle-le: ee'.P[e],
alle-lt: e<e'.P[e],
existse-le: ee'.P[e],
existse-before: e<e'.P[e],
es-causle: e c e',
es-le: e loc e' ,
es-locl: (e <loc e'),
es-causl: (e < e'),
infix_ap: x f y,
cs-precondition: state s may consider v in inning i,
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
cs-inning-committable: in state s, inning i could commit v ,
cs-inning-committed: in state s, inning i has committed v,
cs-passed: by state s, a passed inning i without archiving a value,
cs-archived: by state s, a archived v in inning i,
cs-not-completed: in state s, a has not completed inning i,
l_disjoint: l_disjoint(T;l1;l2),
fset-closed: (s closed under fs),
f-subset: xs ys,
fset-member: a s,
p-outcome: Outcome,
i-closed: i-closed(I),
i-finite: i-finite(I),
sq_exists: x:{A| B[x]},
q-rel: q-rel(r;x),
qless: r < s,
qle: r s,
fun-connected: y is f*(x),
l_all: (xL.P[x]),
l_exists: (xL. P[x]),
prime: prime(a),
reducible: reducible(a),
cmp-le: cmp-le(cmp;x;y),
l_contains: A B,
or: P 
Q,
union: left + right,
consensus-rcv: consensus-rcv(V;A),
consensus-state3: consensus-state3(T),
MaName: MaName,
Knd: Knd,
IdLnk: IdLnk,
fset: FSet{T},
dstype: dstype(TypeNames; d; a),
nat_plus: ,
unit: Unit,
rng_car: |r|,
grp_lt: a < b,
grp_car: |g|,
set_lt: a <p b,
set_leq: a b,
set_car: |p|,
assoced: a ~ b,
divides: b | a,
atom: Atom,
dataflow-history-val: dataflow-history-val(es;e;x.P[x]),
fpf_ap_pair: fpf_ap_pair{fpf_ap_pair_compseq_tag_def:o}(x; eq; f; d),
fpf-dom: x dom(f),
bnot: 
b,
deq-member: deq-member(eq;x;L),
es-loc: loc(e),
bor: p q,
band: p q,
bimplies: p q,
es-ble: e loc e',
es-bless: e <loc e',
es-eq-E: e = e',
eq_lnk: a = b,
eq_id: a = b,
name_eq: name_eq(x;y),
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
deq-disjoint: deq-disjoint(eq;as;bs),
btrue: tt,
bfalse: ff,
le_int: i z j,
lt_int: i <z j,
last: last(L),
data-stream: data-stream(P;L),
df-program-meaning: df-program-meaning(dfp),
parallel-df-program-case2: parallel-df-program-case2(B;F;dfps),
pi1: fst(t),
fpf-ap: f(x),
null-df-program: null-df-program(B),
es-info: info(e),
es-le-before: loc(e),
AssertBY: Error :AssertBY,
CollapseTHEN: Error :CollapseTHEN,
Auto: Error :Auto,
token: "$token",
es-base-E: es-base-E(es),
filter: filter(P;l),
guard: {T},
so_apply: x[s],
sq_type: SQType(T),
nil: [],
cons: [car / cdr],
eqof: eqof(d),
suptype: suptype(S; T),
sqequal: s ~ t,
b-union: A 
B,
isect2: T1 T2,
dataflow: dataflow(A;B),
stream: stream(A),
tuple-type: tuple-type(L),
record: record(x.T[x]),
tag-by: zT,
label: ...$L... t,
ma-state: State(ds),
list_ind: list_ind def,
pi2: snd(t),
q_le: q_le(r;s),
q_less: q_less(r;s),
qeq: qeq(r;s),
eq_type: eq_type(T;T'),
b-exists: (i<n.P[i])_b,
bl-exists: (xL.P[x])_b,
bl-all: (xL.P[x])_b,
dcdr-to-bool: [d],
grp_blt: a < b,
set_blt: a < b,
eq_int: (i =
j),
eq_bool: p =b q,
eq_knd: a = b,
in-eclass: e X,
ext-eq: A 
B,
tactic: Error :tactic,
D: Error :D,
THENM: Error :THENM,
Unfold: Error :Unfold,
MaAuto: Error :MaAuto,
append: as @ bs,
es-before: before(e),
hd: hd(l),
tl: tl(l),
CollapseTHENA: Error :CollapseTHENA,
ParallelOp: Error :ParallelOp,
RepUR: Error :RepUR,
intensional-universe: IType,
eq_term: a == b,
corec: corec(T.F[T]),
primrec: primrec(n;b;c),
locl: locl(a),
rcv: rcv(l,tg),
gt: i > j,
multiply: n * m,
iseg: l1 l2,
proper-iseg: L1 < L2,
es-interface-prior-vals: X(e),
RepeatFor: Error :RepeatFor
Lemmas :
bfalse_wf,
bool_subtype_base,
es-le-before-not-null,
null-map,
data-stream-null-df-program,
last-map,
es-le_wf,
es-le-before_wf2,
length-map-sq,
length-data-stream,
l_member_length,
equal-nil-sq-nil,
pos-length,
subtype_rel_list,
last_wf,
non_null_iff_length,
data-stream_wf,
df-program-meaning_wf_null,
df-program-meaning_wf,
dataflow_wf,
null-bag,
equal-empty-bag,
intensional-universe_wf,
sq_stable__equal,
sq_stable__all,
sq_stable_wf,
es-locl_wf,
es-before_wf3,
append_wf,
es-before_wf,
length_append,
length_cons,
length_nil,
member_null,
pos_length2,
ext-eq_weakening,
subtype_rel_weakening,
uiff_transitivity,
select-map,
map_length,
non_neg_length,
subtype_rel_function,
subtype_rel_dep_function,
subtype_rel_bag,
length-map,
le_wf,
last-stream-parallel-df-program-case2-meaning,
null-df-program_wf,
unit_wf,
member-fpf-dom,
true_wf,
fpf-dom_wf,
false_wf,
subtype_base_sq,
ifthenelse_wf,
pi1_wf_top,
fpf-ap_wf,
es-base-E_wf,
subtype_rel_self,
es-le-before_wf,
es-info_wf,
deq-member_wf,
bnot_wf,
not_functionality_wrt_iff,
assert_of_bnot,
eqff_to_assert,
not_wf,
assert-deq-member,
eqtt_to_assert,
iff_weakening_uiff,
iff_transitivity,
es-loc_wf,
bool_wf,
union-list2_wf,
select_member,
decidable__l_member,
sq_stable_from_decidable,
member_map,
map_wf,
id-deq_wf,
member-union-list2,
df-program-type_wf,
fpf-trivial-subtype-top,
fpf_wf,
dataflow-program_wf,
eclass-program-flows_wf,
fpf-domain_wf,
nat_wf,
l_member_wf,
Id_wf,
event-ordering+_inc,
es-E_wf,
event-ordering+_wf,
sq_stable__subtype_rel,
int_seg_properties,
bag-subtype-list,
permutation_wf,
length_wf_nat,
subtype_rel_wf,
member_wf,
top_wf,
empty-bag_wf,
null_wf3,
squash_wf,
valueall-type_wf,
assert_wf,
eclass_wf,
simple-comb-program-strict_wf,
defined-class_wf,
int_seg_wf,
length_wf1,
eclass-program-type_wf,
bag_wf,
eclass-program_wf,
select_wf
\mforall{}[Info:\{Info:Type| \mdownarrow{}Info\} ]. \mforall{}[B:\{B:Type| valueall-type(B)\} ]. \mforall{}[Ps:eclass-program\{i:l\}(Info) List].
\mforall{}[F:k:\mBbbN{}||Ps|| {}\mrightarrow{} bag(eclass-program-type(Ps[k])) {}\mrightarrow{} bag(B)].
defined-class(simple-comb-program-strict(F;B;Ps)) = simple-comb(F;\mlambda{}k.defined-class(Ps[k]))
supposing (\mforall{}f:k:\mBbbN{}||Ps|| {}\mrightarrow{} bag(eclass-program-type(Ps[k]))
((\mexists{}k:\mBbbN{}||Ps||. (\muparrow{}null(f k))) {}\mRightarrow{} (\muparrow{}null(F f))))
\mwedge{} ((F (\mlambda{}k.\{\})) = \{\})
Date html generated:
2011_08_16-PM-06_23_45
Last ObjectModification:
2011_06_13-PM-12_20_39
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