Nuprl Lemma : process-ordered-message_wf
∀[M:Type]. ∀[nL:n:ℕ × {L:({n + 1...} × M) List| sorted-by(λx,y. fst(x) < fst(y);L)} ]. ∀[km:ℕ × M].
(process-ordered-message(nL;km) ∈ {tr:({fst(nL)...} × M) List × n:{fst(nL)...} × (({n + 1...} × M) List)|
let out,n',L' = tr in
sorted-by(λx,y. fst(x) < fst(y);L')
∧ (0 < ||out|| ⇒ (((fst(km)) = (fst(nL)) ∈ ℤ) ∧ (hd(out) = km ∈ (ℕ × M))))
∧ (((fst(nL)) = (fst(km)) ∈ ℤ)
⇒ ([km / (snd(nL))] = (out @ L') ∈ (({fst(nL)...} × M) List)))
∧ (fst(nL) < fst(km)
⇒ (insert-ordered-message(snd(nL);km)
= (out @ L')
∈ (({(fst(nL)) + 1...} × M) List)))
∧ (fst(km) < fst(nL)
⇒ ((↑null(out)) ∧ (L' = (snd(nL)) ∈ (({(fst(nL)) + 1...} × M) List))))} )
Proof
Definitions occuring in Statement :
process-ordered-message: process-ordered-message(nL;km),
insert-ordered-message: insert-ordered-message(L;x),
sorted-by: sorted-by(R;L),
hd: hd(l),
length: ||as||,
null: null(as),
append: as @ bs,
cons: [a / b],
list: T List,
int_upper: {i...},
nat: ℕ,
assert: ↑b,
less_than: a < b,
spreadn: spread3,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
product: x:A × B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
process-ordered-message: process-ordered-message(nL;km),
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
int_upper: {i...},
so_apply: x[s],
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
has-value: (a)↓,
spreadn: spread3,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
guard: {T},
cons: [a / b],
nat_plus: ℕ+,
uiff: uiff(P;Q),
split-maximal-consecutive: split-maximal-consecutive(g;L),
find-maximal-consecutive: find-maximal-consecutive(g;L),
list_accum: list_accum,
tl: tl(l),
nil: [],
it: ⋅,
eval_list: eval_list(t),
list_ind: list_ind,
firstn: firstn(n;as),
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
subtract: n - m,
bor: p ∨bq,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
iff: P ⇐⇒ Q,
l_member: (x ∈ l),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
listp: A List+,
rev_implies: P ⇐ Q
Latex:
\mforall{}[M:Type]. \mforall{}[nL:n:\mBbbN{} \mtimes{} \{L:(\{n + 1...\} \mtimes{} M) List| sorted-by(\mlambda{}x,y. fst(x) < fst(y);L)\} ]. \mforall{}[km:\mBbbN{} \mtimes{} M].
(process-ordered-message(nL;km) \mmember{} \{tr:(\{fst(nL)...\} \mtimes{} M) List
\mtimes{} n:\{fst(nL)...\}
\mtimes{} ((\{n + 1...\} \mtimes{} M) List)|
let out,n',L' = tr in
sorted-by(\mlambda{}x,y. fst(x) < fst(y);L')
\mwedge{} (0 < ||out|| {}\mRightarrow{} (((fst(km)) = (fst(nL))) \mwedge{} (hd(out) = km)))
\mwedge{} (((fst(nL)) = (fst(km))) {}\mRightarrow{} ([km / (snd(nL))] = (out @ L')))
\mwedge{} (fst(nL) < fst(km)
{}\mRightarrow{} (insert-ordered-message(snd(nL);km) = (out @ L')))
\mwedge{} (fst(km) < fst(nL) {}\mRightarrow{} ((\muparrow{}null(out)) \mwedge{} (L' = (snd(nL)))))\} )
Date html generated:
2016_05_17-PM-00_57_22
Last ObjectModification:
2016_01_18-AM-07_51_11
Theory : event-logic-applications
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