Nuprl Lemma : state-class1-fun-eq
∀[Info,B,A:Type]. ∀[init:Id ⟶ B]. ∀[tr:Id ⟶ A ⟶ B ⟶ B]. ∀[X:EClass(A)]. ∀[es:EO+(Info)]. ∀[e:E].
state-class1(init;tr;X)(e)
= if e ∈b X then if first(e) then tr loc(e) X@e (init loc(e)) else tr loc(e) X@e state-class1(init;tr;X)(pred(e)) fi
if first(e) then init loc(e)
else state-class1(init;tr;X)(pred(e))
fi
∈ B
supposing single-valued-classrel(es;X;A)
Proof
Definitions occuring in Statement :
state-class1: state-class1(init;tr;X),
classfun-res: X@e,
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],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
state-class1: state-class1(init;tr;X),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
btrue: tt,
true: True,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}[Info,B,A:Type]. \mforall{}[init:Id {}\mrightarrow{} B]. \mforall{}[tr:Id {}\mrightarrow{} A {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[X:EClass(A)]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E].
state-class1(init;tr;X)(e)
= if e \mmember{}\msubb{} X
then if first(e)
then tr loc(e) X@e (init loc(e))
else tr loc(e) X@e state-class1(init;tr;X)(pred(e))
fi
if first(e) then init loc(e)
else state-class1(init;tr;X)(pred(e))
fi
supposing single-valued-classrel(es;X;A)
Date html generated:
2016_05_17-AM-11_17_51
Last ObjectModification:
2015_12_29-PM-05_14_04
Theory : process-model
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