Nuprl Lemma : consensus-ts5-true-knowledge
[V:Type]
A:Id List. W:{a:Id| (a 
A)} List List. x:ts-reachable(consensus-ts5(V;A;W)).
let x1,x2 = x
in a,b:{a:Id| (a A)} .
let I,z = Knowledge(x2;a)(b)
in (I 
Inning(x1;b))
case z
of inl(p) =>
let k,v = p
in (k < I)
(
k dom(Estimate(x1;b)))
(Estimate(x1;b)(k) = v)
(i:
. 
i dom(Estimate(x1;b)) supposing (k < i) (i < I))
| inr(p) =>
i:. i dom(Estimate(x1;b)) supposing i < I
supposing b dom(Knowledge(x2;a))
Proof not projected
Definitions occuring in Statement :
consensus-ts5: consensus-ts5(V;A;W),
cs-knowledge: Knowledge(x;a),
cs-estimate: Estimate(s;a),
cs-inning: Inning(s;a),
fpf-ap: f(x),
fpf-dom: x dom(f),
id-deq: IdDeq,
Id: Id,
assert: b,
uimplies: b supposing a,
uall: [x:A]. B[x],
le: A B,
all: x:A. B[x],
not: A,
and: P Q,
less_than: a < b,
set: {x:A| B[x]} ,
spread: spread def,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
list: type List,
int: ,
universe: Type,
equal: s = t,
l_member: (x l),
int-deq: IntDeq,
ts-reachable: ts-reachable(ts)
Definitions :
true: True,
cand: A c B,
pi1: fst(t),
ts-type: ts-type(ts),
so_lambda: 
x.t[x],
member: t T,
implies: P 
Q,
prop: 
,
and: P Q,
uimplies: b supposing a,
consensus-ts5: consensus-ts5(V;A;W),
all: x:A. B[x],
uall: [x:A]. B[x],
consensus-state4: ConsensusState,
bfalse: ff,
false: False,
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: b,
not: A,
le: A B,
sq_stable: SqStable(P),
reduce: reduce(f;k;as),
deq-member: deq-member(eq;x;L),
fpf-empty: 
,
pi2: snd(t),
cs-estimate: Estimate(s;a),
cs-inning: Inning(s;a),
cs-knowledge: Knowledge(x;a),
fpf-dom: x dom(f),
ts-init: ts-init(ts),
infix_ap: x f y,
Id: Id,
squash: 
T,
guard: {T},
eqof: eqof(d),
fpf-single: x : v,
fpf-ap: f(x),
top: Top,
uiff: uiff(P;Q),
so_apply: x[s],
ts-reachable: ts-reachable(ts),
iff: P 
Q,
unit: Unit,
bool: 
,
consensus-rel-add-knowledge-step: consensus-rel-add-knowledge-step(V;A;W;x1;x2;y1;y2;a),
consensus-rel-knowledge-archive-step: consensus-rel-knowledge-archive-step(V;A;W;x1;x2;y1;y2;a),
consensus-rel-knowledge-inning-step: consensus-rel-knowledge-inning-step(V;A;W;x1;x2;y1;y2;a),
or: P 
Q,
consensus-rel-knowledge-step: consensus-rel-knowledge-step(V;A;W;x1;x2;y1;y2;a),
exists: 
x:A. B[x],
consensus-rel-knowledge: consensus-rel-knowledge(V;A;W;x;y),
ts-rel: ts-rel(ts),
decidable: Dec(P),
sq_type: SQType(T),
it: 
,
eq_id: a = b
Lemmas :
ts-type_wf,
ts-reachable_wf,
not_wf,
cs-estimate_wf,
subtype-top,
subtype-fpf2,
int-deq_wf,
cs-inning_wf,
le_wf,
fpf-ap_wf,
fpf_wf,
top_wf,
fpf-trivial-subtype-top,
cs-knowledge_wf,
id-deq_wf,
fpf-dom_wf,
assert_wf,
l_member_wf,
Id_wf,
all_wf,
consensus-state5_wf,
consensus-state4_wf,
consensus-ts5_wf,
ts-reachable-induction,
subtype_rel_simple_product,
subtype_rel_self,
subtype_rel_function,
sq_stable__all,
false_wf,
sq_stable__uall,
assert_of_bnot,
eqff_to_assert,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
iff_weakening_uiff,
bool_wf,
squash_wf,
sq_stable__not,
sq_stable__equal,
decidable__lt,
decidable__le,
sq_stable_from_decidable,
sq_stable__and,
true_wf,
list-subtype,
mk_fpf_wf,
strong-subtype-self,
strong-subtype-set3,
strong-subtype-deq-subtype,
ts-rel_wf,
assert_witness,
decidable__equal_Id,
atom2_subtype_base,
subtype_base_sq,
deq_wf,
equal_wf,
not_functionality_wrt_uiff,
assert-eq-id,
fpf-single-dom-sq,
fpf-join-ap-sq,
eq_id_wf,
fpf-single_wf
\mforall{}[V:Type]
\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List. \mforall{}x:ts-reachable(consensus-ts5(V;A;W)).
let x1,x2 = x
in \mforall{}a,b:\{a:Id| (a \mmember{} A)\} .
let I,z = Knowledge(x2;a)(b)
in (I \mleq{} Inning(x1;b))
\mwedge{} case z
of inl(p) =>
let k,v = p
in (k < I)
\mwedge{} (\muparrow{}k \mmember{} dom(Estimate(x1;b)))
\mwedge{} (Estimate(x1;b)(k) = v)
\mwedge{} (\mforall{}i:\mBbbZ{}. \mneg{}\muparrow{}i \mmember{} dom(Estimate(x1;b)) supposing (k < i) \mwedge{} (i < I))
| inr(p) =>
\mforall{}i:\mBbbZ{}. \mneg{}\muparrow{}i \mmember{} dom(Estimate(x1;b)) supposing i < I
supposing \muparrow{}b \mmember{} dom(Knowledge(x2;a))
Date html generated:
2012_01_23-PM-12_06_20
Last ObjectModification:
2011_12_14-PM-06_37_36
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