Nuprl Lemma : hdf-state-transformation2
∀[init,L,G:Top]. ∀[m:ℕ].
(hdf-state(fix((λmk-hdf.(inl (λa.cbva_seq(L[a]; λg.<mk-hdf, G[g]>; m)))));init)
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if (n =z m + 1)
then mk_lambdas_fun(λg.if bag-null(select_fun_last(g;m))
then s
else select_fun_last(g;m)
fi ;m + 1)
if (n =z m) then mk_lambdas_fun(λg.⋃f∈G[g].bag-map(f;s);m)
else L[a] n
fi ; λg.<mk-hdf select_fun_last(g;m + 1), select_fun_last(g;m + 1)>; m + 2)))))\000C
init)
Proof
Definitions occuring in Statement :
hdf-state: hdf-state(X;bs),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
add: n + m,
natural_number: $n,
sqequal: s ~ t,
bag-combine: ⋃x∈bs.f[x],
bag-null: bag-null(bs),
bag-map: bag-map(f;bs),
select_fun_last: select_fun_last(g;m),
mk_lambdas_fun: mk_lambdas_fun(F;m),
cbva_seq: cbva_seq(L; F; m)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s],
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
bag-null: bag-null(bs),
hdf-state: hdf-state(X;bs),
bfalse: ff,
btrue: tt,
hdf-ap: X(a),
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
hdf-run: hdf-run(P),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
decidable: Dec(P),
nat_plus: ℕ+,
bag-map: bag-map(f;bs),
select_fun_last: select_fun_last(g;m),
sq_type: SQType(T),
subtype_rel: A ⊆r B
Latex:
\mforall{}[init,L,G:Top]. \mforall{}[m:\mBbbN{}].
(hdf-state(fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.cbva\_seq(L[a]; \mlambda{}g.<mk-hdf, G[g]> m)))));init)
\msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if (n =\msubz{} m + 1)
then mk\_lambdas\_fun(\mlambda{}g.if bag-null(select\_fun\_last(g;m))
then s
else select\_fun\_last(g;m)
fi ;m + 1)
if (n =\msubz{} m) then mk\_lambdas\_fun(\mlambda{}g.\mcup{}f\mmember{}G[g].bag-map(f;s);m)
else L[a] n
fi ; \mlambda{}g.<mk-hdf select\_fun\_last(g;m + 1)
, select\_fun\_last(g;m + 1)
> m + 2)))))
init)
Date html generated:
2016_05_16-AM-10_46_38
Last ObjectModification:
2016_01_17-AM-11_10_19
Theory : halting!dataflow
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