Nuprl Lemma : cs-ref-map-step
∀[V:Type]
({∃v1,v2:V. (¬(v1 = v2 ∈ V))}
⇒ (∀A:Id List. ∀W:{a:Id| (a ∈ A)} List List.
∀f:ConsensusState ⟶ (consensus-state3(V) List)
(cs-ref-map-constraints(V;A;W;f)
⇒ (∀x,y:ts-reachable(consensus-ts4(V;A;W)).
((x ts-rel(consensus-ts4(V;A;W)) y)
⇒ {(||f x|| ≤ ||f y||)
∧ (∃i:ℤ
((∀j:ℕ||f x||. f y[j] = f x[j] ∈ consensus-state3(V) supposing ¬(j = i ∈ ℤ))
∧ (||f y|| = (||f x|| + 1) ∈ ℤ) ∧ (f y[||f x||] = INITIAL ∈ consensus-state3(V))
supposing ||f x|| < ||f y||))})))
supposing ||W|| ≥ 1 ))
Proof
Definitions occuring in Statement :
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
consensus-ts4: consensus-ts4(V;A;W),
consensus-state4: ConsensusState,
cs-initial: INITIAL,
consensus-state3: consensus-state3(T),
Id: Id,
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
infix_ap: x f y,
ge: i ≥ j ,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T,
ts-reachable: ts-reachable(ts),
ts-rel: ts-rel(ts)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
prop: ℙ,
consensus-ts4: consensus-ts4(V;A;W),
ts-rel: ts-rel(ts),
pi2: snd(t),
pi1: fst(t),
infix_ap: x f y,
ts-reachable: ts-reachable(ts),
ts-init: ts-init(ts),
ts-type: ts-type(ts),
guard: {T},
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
ts-stable-rel: ts-stable-rel(ts;x,y.R[x; y]),
consensus-state4: ConsensusState,
consensus-rel: CR[x,y],
less_than: a < b,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
Id: Id,
sq_type: SQType(T),
fpf-domain: fpf-domain(f),
fpf-single: x : v,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
less_than': less_than'(a;b),
cs-inning-committable: in state s, inning i could commit v ,
cs-archived: by state s, a archived v in inning i,
cs-not-completed: in state s, a has not completed inning i,
cs-inning-committed: in state s, inning i has committed v,
nat: ℕ,
assert: ↑b,
cons: [a / b]
Latex:
\mforall{}[V:Type]
(\{\mexists{}v1,v2:V. (\mneg{}(v1 = v2))\}
{}\mRightarrow{} (\mforall{}A:Id List. \mforall{}W:\{a:Id| (a \mmember{} A)\} List List.
\mforall{}f:ConsensusState {}\mrightarrow{} (consensus-state3(V) List)
(cs-ref-map-constraints(V;A;W;f)
{}\mRightarrow{} (\mforall{}x,y:ts-reachable(consensus-ts4(V;A;W)).
((x ts-rel(consensus-ts4(V;A;W)) y)
{}\mRightarrow{} \{(||f x|| \mleq{} ||f y||)
\mwedge{} (\mexists{}i:\mBbbZ{}
((\mforall{}j:\mBbbN{}||f x||. f y[j] = f x[j] supposing \mneg{}(j = i))
\mwedge{} (||f y|| = (||f x|| + 1)) \mwedge{} (f y[||f x||] = INITIAL)
supposing ||f x|| < ||f y||))\})))
supposing ||W|| \mgeq{} 1 ))
Date html generated:
2016_05_16-PM-00_11_54
Last ObjectModification:
2016_01_17-PM-03_57_58
Theory : event-ordering
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