Nuprl Lemma : loop-class-memory-fun-eq
∀[Info,B:Type]. ∀[X:EClass(B ⟶ B)]. ∀[init:Id ⟶ bag(B)]. ∀[es:EO+(Info)]. ∀[e:E].
  (loop-class-memory(X;init)(e)
     = if first(e) then sv-bag-only(init loc(e))
       if pred(e) ∈b X then X(pred(e)) loop-class-memory(X;init)(pred(e))
       else loop-class-memory(X;init)(pred(e))
       fi 
     ∈ B) supposing 
     (single-valued-classrel(es;X;B ⟶ B) and 
     (∀l:Id. single-valued-bag(init l;B)) and 
     (∀l:Id. (1 ≤ #(init l))))
Proof
Definitions occuring in Statement : 
loop-class-memory: loop-class-memory(X;init)
, 
classfun: X(e)
, 
single-valued-classrel: single-valued-classrel(es;X;T)
, 
member-eclass: e ∈b X
, 
eclass: EClass(A[eo; e])
, 
event-ordering+: EO+(Info)
, 
es-first: first(e)
, 
es-pred: pred(e)
, 
es-loc: loc(e)
, 
es-E: E
, 
Id: Id
, 
ifthenelse: if b then t else f fi 
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
le: A ≤ B
, 
all: ∀x:A. B[x]
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
natural_number: $n
, 
universe: Type
, 
equal: s = t ∈ T
, 
sv-bag-only: sv-bag-only(b)
, 
single-valued-bag: single-valued-bag(b;T)
, 
bag-size: #(bs)
, 
bag: bag(T)
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
prop: ℙ
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
assert: ↑b
, 
false: False
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
nat: ℕ
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
loop-class-memory: loop-class-memory(X;init)
, 
primed-class-opt: Prior(X)?b
, 
classfun: X(e)
, 
eclass: EClass(A[eo; e])
, 
sq_exists: ∃x:{A| B[x]}
, 
not: ¬A
, 
decidable: Dec(P)
, 
le: A ≤ B
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
classfun-res: X@e
, 
es-locl: (e <loc e')
, 
cand: A c∧ B
, 
es-functional-class-at: X is functional at e
, 
squash: ↓T
, 
true: True
, 
rev_uimplies: rev_uimplies(P;Q)
, 
es-E: E
, 
es-base-E: es-base-E(es)
Latex:
\mforall{}[Info,B:Type].  \mforall{}[X:EClass(B  {}\mrightarrow{}  B)].  \mforall{}[init:Id  {}\mrightarrow{}  bag(B)].  \mforall{}[es:EO+(Info)].  \mforall{}[e:E].
    (loop-class-memory(X;init)(e)
          =  if  first(e)  then  sv-bag-only(init  loc(e))
              if  pred(e)  \mmember{}\msubb{}  X  then  X(pred(e))  loop-class-memory(X;init)(pred(e))
              else  loop-class-memory(X;init)(pred(e))
              fi  )  supposing 
          (single-valued-classrel(es;X;B  {}\mrightarrow{}  B)  and 
          (\mforall{}l:Id.  single-valued-bag(init  l;B))  and 
          (\mforall{}l:Id.  (1  \mleq{}  \#(init  l))))
Date html generated:
2016_05_17-AM-11_17_27
Last ObjectModification:
2016_01_18-AM-00_16_37
Theory : process-model
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