Nuprl Lemma : cs-ref-map-equal
∀[V:Type]. ∀[A:Id List]. ∀[W:{a:Id| (a ∈ A)} List List]. ∀[f:ConsensusState ⟶ (consensus-state3(V) List)].
∀[x,y:ConsensusState]. ∀[i:ℕ||f x||].
(f x[i] = f y[i] ∈ consensus-state3(V)) supposing
((∀v:V
((in state x, inning i could commit v ⇐⇒ in state y, inning i could commit v )
∧ (in state x, inning i has committed v ⇐⇒ in state y, inning i has committed v))) and
i < ||f y||)
supposing cs-ref-map-constraints(V;A;W;f)
Proof
Definitions occuring in Statement :
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
cs-inning-committable: in state s, inning i could commit v ,
cs-inning-committed: in state s, inning i has committed v,
consensus-state4: ConsensusState,
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],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
cs-ref-map-constraints: cs-ref-map-constraints(V;A;W;f),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B
Latex:
\mforall{}[V:Type]. \mforall{}[A:Id List]. \mforall{}[W:\{a:Id| (a \mmember{} A)\} List List].
\mforall{}[f:ConsensusState {}\mrightarrow{} (consensus-state3(V) List)].
\mforall{}[x,y:ConsensusState]. \mforall{}[i:\mBbbN{}||f x||].
(f x[i] = f y[i]) supposing
((\mforall{}v:V
((in state x, inning i could commit v \mLeftarrow{}{}\mRightarrow{} in state y, inning i could commit v )
\mwedge{} (in state x, inning i has committed v \mLeftarrow{}{}\mRightarrow{} in state y, inning i has committed v))) and
i < ||f y||)
supposing cs-ref-map-constraints(V;A;W;f)
Date html generated:
2016_05_16-PM-00_06_47
Last ObjectModification:
2016_01_17-PM-03_54_16
Theory : event-ordering
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