Nuprl Lemma : State-loc-comb-fun-eq
∀[Info,B,A:Type]. ∀[f:Id ⟶ A ⟶ B ⟶ B]. ∀[init:Id ⟶ bag(B)]. ∀[X:EClass(A)]. ∀[es:EO+(Info)]. ∀[e:E].
(State-loc-comb(init;f;X)(e)
= if e ∈b X
then if first(e)
then f loc(e) X(e) sv-bag-only(init loc(e))
else f loc(e) X(e) State-loc-comb(init;f;X)(pred(e))
fi
if first(e) then sv-bag-only(init loc(e))
else State-loc-comb(init;f;X)(pred(e))
fi
∈ B) supposing
(single-valued-classrel(es;X;A) and
(∀l:Id. single-valued-bag(init l;B)) and
(∀l:Id. (1 ≤ #(init l))))
Proof
Definitions occuring in Statement :
State-loc-comb: State-loc-comb(init;f;X),
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,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
classfun-res: X@e,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}[Info,B,A:Type]. \mforall{}[f:Id {}\mrightarrow{} A {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[init:Id {}\mrightarrow{} bag(B)]. \mforall{}[X:EClass(A)]. \mforall{}[es:EO+(Info)].
\mforall{}[e:E].
(State-loc-comb(init;f;X)(e)
= if e \mmember{}\msubb{} X
then if first(e)
then f loc(e) X(e) sv-bag-only(init loc(e))
else f loc(e) X(e) State-loc-comb(init;f;X)(pred(e))
fi
if first(e) then sv-bag-only(init loc(e))
else State-loc-comb(init;f;X)(pred(e))
fi ) supposing
(single-valued-classrel(es;X;A) 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-10_02_58
Last ObjectModification:
2016_01_17-PM-11_07_13
Theory : classrel!lemmas
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