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 ∈ V)
∧ (∀i:ℤ. ¬↑i ∈ dom(Estimate(x1;b)) supposing k < i ∧ i < I)
| inr(p) =>
∀i:ℤ. ¬↑i ∈ dom(Estimate(x1;b)) supposing i < I
supposing ↑b ∈ dom(Knowledge(x2;a))
Proof
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,
int-deq: IntDeq,
l_member: (x ∈ l),
list: T List,
assert: ↑b,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
spread: spread def,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
int: ℤ,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts)
Lemmas :
ts-reachable-induction,
consensus-ts5_wf,
subtype_rel_self,
consensus-state4_wf,
consensus-state5_wf,
all_wf,
l_member_wf,
isect_wf,
assert_wf,
fpf-dom_wf,
id-deq_wf,
cs-knowledge_wf,
subtype-fpf2,
subtype_top,
top_wf,
fpf-ap_wf,
le_wf,
cs-inning_wf,
less_than_wf,
int-deq_wf,
cs-estimate_wf,
consensus-state4-subtype,
not_wf,
ts-reachable_wf,
subtype_rel_wf,
ts-type_wf,
list_wf,
Id_wf,
sq_stable__all,
fpf_wf,
set_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
sq_stable__uall,
false_wf,
true_wf,
sq_stable__and,
sq_stable__le,
sq_stable__less_than,
sq_stable_from_decidable,
decidable__true,
sq_stable__equal,
sq_stable__not,
squash_wf,
member-less_than,
decidable__false,
decide_bfalse_lemma,
deq_member_nil_lemma,
subtype_rel-deq,
sq_stable__l_member,
decidable__equal_Id,
mk_fpf_wf,
list-subtype,
infix_ap_wf,
ts-rel_wf,
subtype_rel-equal,
assert_witness,
member_wf,
and_wf,
assert_elim,
atom2_subtype_base,
decidable__le,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
le_transitivity,
le_weakening,
fpf-join-dom,
fpf-single_wf,
equal-wf-base-T,
int_subtype_base,
fpf-single-dom,
or_wf,
fpf-join-ap-sq,
equal-wf-T-base,
bnot_wf,
uiff_transitivity,
assert_of_bnot,
less_than_transitivity1,
less_than_irreflexivity,
deq_wf,
fpf_ap_single_lemma,
it_wf,
assert_functionality_wrt_uiff,
eq_id_wf,
fpf-single-dom-sq,
assert-eq-id,
iff_transitivity,
iff_weakening_uiff,
equal_functionality_wrt_subtype_rel2,
fpf-join-ap,
iff_weakening_equal
\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:
2015_07_17-AM-11_41_53
Last ObjectModification:
2015_07_16-AM-10_19_03
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