Nuprl Lemma : CLK_stric_inc
∀MsgType:ValueAllType. ∀f:CLK_headers_type{i:l}(MsgType). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀clock1,clock2:ℤ.
((∃a:MsgType × ℤ. ∃e:E. ((e1 <loc e) ∧ e ≤loc e2 ∧ a ∈ CLK_msg'base(MsgType;f)(e)))
⇒ (e1 <loc e2)
⇒ clock1 ∈ CLK_Clock(MsgType;f)(e1)
⇒ clock2 ∈ CLK_Clock(MsgType;f)(e2)
⇒ clock1 < clock2)
Proof
Definitions occuring in Statement :
CLK_Clock: CLK_Clock(MsgType;f),
CLK_msg'base: CLK_msg'base(MsgType;f),
CLK_headers_type: CLK_headers_type{i:l}(MsgType),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-E: E,
vatype: ValueAllType,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
CLK_headers_type: CLK_headers_type{i:l}(MsgType),
l_all: (∀x∈L.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
listp: A List+,
name: Name,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
length: ||as||,
list_ind: list_ind,
CLK_headers: CLK_headers(),
cons: [a / b],
nil: [],
it: ⋅,
true: True,
select: L[n],
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
CLK_headers_fun: CLK_headers_fun(MsgType),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
btrue: tt,
null: null(as),
CLK_Clock: CLK_Clock(MsgType;f),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
State1: State1(init;tr;X),
exists: ∃x:A. B[x],
trans: Trans(T;x,y.E[x; y]),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
CLK_msg'base: CLK_msg'base(MsgType;f),
encodes-msg-type: hdr encodes T,
cand: A c∧ B,
CLK_upd_clock: CLK_upd_clock(MsgType),
imax: imax(a;b),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Latex:
\mforall{}MsgType:ValueAllType. \mforall{}f:CLK\_headers\_type\{i:l\}(MsgType). \mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E.
\mforall{}clock1,clock2:\mBbbZ{}.
((\mexists{}a:MsgType \mtimes{} \mBbbZ{}. \mexists{}e:E. ((e1 <loc e) \mwedge{} e \mleq{}loc e2 \mwedge{} a \mmember{} CLK\_msg'base(MsgType;f)(e)))
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} clock1 \mmember{} CLK\_Clock(MsgType;f)(e1)
{}\mRightarrow{} clock2 \mmember{} CLK\_Clock(MsgType;f)(e2)
{}\mRightarrow{} clock1 < clock2)
Date html generated:
2016_05_17-PM-02_44_48
Last ObjectModification:
2016_01_18-AM-09_50_25
Theory : lamport!s!clocks
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