Nuprl Lemma : hdf-state-base
∀[f,C,F,s:Top].
(hdf-state((λx,s. f[x;s]) o hdf-base(a.if C[a] then [F[a]] else [] fi );[s])
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if C[a] then f[F[a];s] else s fi ; λg.<mk-hdf (g (λx.x)), g (λx.[x])>; 1))))) s\000C)
Proof
Definitions occuring in Statement :
hdf-state: hdf-state(X;bs),
hdf-compose1: f o X,
hdf-base: hdf-base(m.F[m]),
cons: [a / b],
nil: [],
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
so_apply: x[s],
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
natural_number: $n,
sqequal: s ~ t,
cbva_seq: cbva_seq(L; F; m)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
top: Top,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
cbva_seq: cbva_seq(L; F; m),
select_fun_last: select_fun_last(g;m),
bag-map: bag-map(f;bs),
mk_lambdas_fun: mk_lambdas_fun(F;m),
select_fun_ap: select_fun_ap(g;n;m),
subtract: n - m,
mk_lambdas: mk_lambdas(F;m),
all: ∀x:A. B[x],
callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
eq_int: (i =z j),
mk_lambdas-fun: mk_lambdas-fun(F;G;n;m),
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
cons: [a / b],
nil: [],
it: ⋅,
empty-bag: {},
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
has-valueall: has-valueall(a),
bag-null: bag-null(bs),
uiff: uiff(P;Q),
has-value: (a)↓,
bag-combine: ⋃x∈bs.f[x],
bag-union: bag-union(bbs),
concat: concat(ll),
reduce: reduce(f;k;as),
list_ind: list_ind,
map: map(f;as),
null: null(as)
Latex:
\mforall{}[f,C,F,s:Top].
(hdf-state((\mlambda{}x,s. f[x;s]) o hdf-base(a.if C[a] then [F[a]] else [] fi );[s])
\msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if C[a] then f[F[a];s] else s fi ; \mlambda{}g.<mk-hdf (g (\mlambda{}x.x))
, g (\mlambda{}x.[x])
> 1)))))
s)
Date html generated:
2016_05_16-AM-10_48_20
Last ObjectModification:
2016_01_17-AM-11_08_09
Theory : halting!dataflow
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