{ 
[Info:Type]. [es:EO+(Info)]. [X:EClass(Top)]. [f:sys-antecedent(es;X)].
[e:E(X)].
(cut(X;f;{e})
= if e 
prior(X)
then if f e = e
then {e}
else {e} 
cut(X;f;{f e})
fi cut(X;f;{prior(X)(e)})
if f e = e then {e}
else {e} cut(X;f;{f e})
fi ) }
{ Proof }
Definitions occuring in Statement :
cut-of: cut(X;f;s),
es-cut: Cut(X;f),
es-prior-interface: prior(X),
sys-antecedent: sys-antecedent(es;Sys),
es-E-interface: E(X),
eclass-val: X(e),
in-eclass: e X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-eq-E: e = e',
es-eq: es-eq(es),
ifthenelse: if b then t else f fi ,
uall: [x:A]. B[x],
top: Top,
apply: f a,
universe: Type,
equal: s = t,
fset-singleton: {x},
fset-union: x y
Definitions :
fset: FSet{T},
pair: <a, b>,
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
decide: case b of inl(x) => s[x] | inr(y) => t[y],
assert: 
b,
le: A 
B,
ge: i 
j ,
not: 
A,
less_than: a < b,
uimplies: b supposing a,
product: x:A 
B[x],
and: P 
Q,
uiff: uiff(P;Q),
axiom: Ax,
eclass-val: X(e),
es-eq-E: e = e',
es-eq: es-eq(es),
fset-union: x y,
es-prior-interface: prior(X),
in-eclass: e X,
fset-singleton: {x},
cut-of: cut(X;f;s),
es-cut: Cut(X;f),
set: {x:A| B[x]} ,
es-E-interface: E(X),
union: left + right,
sys-antecedent: sys-antecedent(es;Sys),
subtype: S 
T,
subtype_rel: A r B,
atom: Atom,
apply: f a,
es-base-E: es-base-E(es),
token: "$token",
ifthenelse: if b then t else f fi ,
record-select: r.x,
top: Top,
event_ordering: EO,
es-E: E,
lambda: 
x.A[x],
dep-isect: Error :dep-isect,
eq_atom: x =a y,
eq_atom: eq_atom$n(x;y),
record+: record+,
all: x:A. B[x],
function: x:A 
B[x],
isect: 
x:A. B[x],
uall: [x:A]. B[x],
eclass: EClass(A[eo; e]),
so_lambda: 
x y.t[x; y],
universe: Type,
member: t T,
event-ordering+: EO+(Info),
equal: s = t,
list: type List,
nil: [],
es-interface-pred: X-pred,
cons: [car / cdr],
prop: 
,
tl: tl(l),
hd: hd(l),
l_member: (x l),
implies: P 
Q,
fset-member: a s,
so_lambda: x.t[x],
l_all: (xL.P[x]),
fset-closed: (s closed under fs),
false: False,
intensional-universe: IType,
guard: {T},
sq_type: SQType(T),
record: record(x.T[x]),
limited-type: LimitedType,
void: Void,
bfalse: ff,
btrue: tt,
iff: P 
Q,
eq_bool: p =b q,
lt_int: i <z j,
le_int: i z j,
eq_int: (i =
j),
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'),
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,
bool: 
,
cand: A c B,
es-causle: e c e',
f-subset: xs ys,
filter: filter(P;l),
fpf-cap: f(x)?z,
set-equal: set-equal(T;x;y),
quotient: x,y:A//B[x; y],
deq: EqDecider(T),
true: True,
es-cut-add: c+e,
isl: isl(x),
can-apply: can-apply(f;x),
rev_implies: P Q,
fset-filter: {x s | P[x]},
fset-intersection: a b,
fset-remove: fset-remove(eq;y;s),
fset-add: fset-add(eq;x;s),
exists: x:A. B[x],
or: P Q,
empty-fset: {},
cond-class: [X?Y],
so_apply: x[s],
eq_knd: a = b,
fpf-dom: x dom(f),
mem_empty: mem_empty{mem_empty_compseq_tag_def:o}(a; eq),
union_empty: union_empty{union_empty_compseq_tag_def:o}(b; eq),
MaAuto: Error :MaAuto,
CollapseTHEN: Error :CollapseTHEN,
squash: 
T,
Knd: Knd,
IdLnk: IdLnk,
Id: Id,
l_disjoint: l_disjoint(T;l1;l2),
cs-not-completed: in state s, a has not completed inning i,
cs-archived: by state s, a archived v in inning i,
cs-passed: by state s, a passed inning i without archiving a value,
cs-inning-committed: in state s, inning i has committed v,
cs-inning-committable: in state s, inning i could commit v ,
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
cs-precondition: state s may consider v in inning i,
es-causl: (e < e'),
es-locl: (e <loc e'),
es-le: e loc e' ,
existse-before: e<e'.P[e],
existse-le: ee'.P[e],
alle-lt: e<e'.P[e],
alle-le: ee'.P[e],
alle-between1: e[e1,e2).P[e],
existse-between1: e[e1,e2).P[e],
alle-between2: e[e1,e2].P[e],
existse-between2: e[e1,e2].P[e],
existse-between3: e(e1,e2].P[e],
es-fset-loc: i locs(s),
es-r-immediate-pred: es-r-immediate-pred(es;R;e';e),
same-thread: same-thread(es;p;e;e'),
collect-event: collect-event(es;X;n;v.num[v];L.P[L];e),
cut-order: a (X;f) b,
decidable: Dec(P),
uni_sat: a = !x:T. Q[x],
inv_funs: InvFuns(A;B;f;g),
inject: Inj(A;B;f),
eqfun_p: IsEqFun(T;eq),
refl: Refl(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
uconnex: uconnex(T; x,y.R[x; y]),
coprime: CoPrime(a,b),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
comm: Comm(T;op),
inverse: Inverse(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
action_p: IsAction(A;x;e;S;f),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
cancel: Cancel(T;S;op),
monot: monot(T;x,y.R[x; y];f),
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
monoid_hom_p: IsMonHom{M1,M2}(f),
grp_leq: a b,
integ_dom_p: IsIntegDom(r),
prime_ideal_p: IsPrimeIdeal(R;P),
no_repeats: no_repeats(T;l),
value-type: value-type(T),
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
req: x = y,
rnonneg: rnonneg(r),
rleq: x y,
i-member: r I,
partitions: partitions(I;p),
modulus-of-ccontinuity: modulus-of-ccontinuity(omega;I;f),
fpf-sub: f g,
sq_stable: SqStable(P)
Lemmas :
sq_stable_from_decidable,
decidable__fset-closed,
uiff_inversion,
fset-member_witness,
f-subset_weakening,
deq_wf,
squash_wf,
empty-fset_wf,
empty-fset_wf-cut,
member-fset-union,
fset-member_wf-cut,
es-cut-union,
es-eq_wf,
fset-union-associative,
rev_implies_wf,
iff_wf,
cut-of-closed,
f-singleton-subset,
f-union-subset,
f-subset-union,
member-fset-singleton,
subtype_base_sq,
bool_subtype_base,
assert_elim,
true_wf,
deq-member_wf,
eclass-val_wf,
es-cut-add_wf,
set-equal_wf,
f-subset_antisymmetry,
f-subset_wf,
eclass-val_wf2,
assert_wf,
not_wf,
bool_wf,
subtype_rel_wf,
es-prior-interface_wf,
es-interface-subtype_rel2,
es-prior-interface_wf1,
es-prior-interface_wf0,
in-eclass_wf,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
iff_weakening_uiff,
eqtt_to_assert,
cut-of-property,
bnot_wf,
assert-es-eq-E-2,
not_functionality_wrt_uiff,
es-eq-E_wf,
es-E-interface-subtype_rel,
intensional-universe_wf,
false_wf,
fset-closed_wf,
fset-member_wf,
l_all_wf,
l_member_wf,
es-eq_wf-interface,
es-interface-pred_wf2,
member_wf,
es-cut_wf,
ifthenelse_wf,
fset_wf,
es-E-interface_wf,
fset-union_wf,
fset-singleton_wf,
cut-of_wf,
sys-antecedent_wf,
event-ordering+_wf,
event-ordering+_inc,
subtype_rel_self,
es-base-E_wf,
es-E_wf,
top_wf,
eclass_wf
\mforall{}[Info:Type]. \mforall{}[es:EO+(Info)]. \mforall{}[X:EClass(Top)]. \mforall{}[f:sys-antecedent(es;X)]. \mforall{}[e:E(X)].
(cut(X;f;\{e\})
= if e \mmember{}\msubb{} prior(X) then if f e = e then \{e\} else \{e\} \mcup{} cut(X;f;\{f e\}) fi \mcup{} cut(X;f;\{prior(X)(e)\})
if f e = e then \{e\}
else \{e\} \mcup{} cut(X;f;\{f e\})
fi )
Date html generated:
2011_08_16-PM-05_53_42
Last ObjectModification:
2011_06_20-AM-01_37_55
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