Nuprl Lemma : hdf-compose0-transformation1
∀[f,L,S,G,s:Top]. ∀[m:ℕ].
(hdf-compose0(f;fix((λmk-hdf,s. (inl (λa.cbva_seq(L[s;a]; λg.<mk-hdf S[s;a;g], G[s;a;g]>; m))))) s)
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if (n =z m) then mk_lambdas_fun(λg.⋃b∈G[s;a;g].f b;m) else L[s;a] n fi ;
λg.<mk-hdf S[s;a;partial_ap(g;m + 1;m)], select_fun_last(g;m)>; m + 1)))))
s)
Proof
Definitions occuring in Statement :
hdf-compose0: hdf-compose0(f;X),
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[s1;s2;s3],
so_apply: x[s1;s2],
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],
select_fun_last: select_fun_last(g;m),
partial_ap: partial_ap(g;n;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[s1;s2;s3],
so_apply: x[s1;s2],
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
hdf-compose0: hdf-compose0(f;X),
bfalse: ff,
btrue: tt,
hdf-halted: hdf-halted(P),
hdf-ap: X(a),
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
hdf-run: hdf-run(P),
hdf-halt: hdf-halt(),
isr: isr(x),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
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,
so_lambda: λ2x y.t[x; y],
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
decidable: Dec(P),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
compose: f o g,
select_fun_last: select_fun_last(g;m),
subtype_rel: A ⊆r B
Latex:
\mforall{}[f,L,S,G,s:Top]. \mforall{}[m:\mBbbN{}].
(hdf-compose0(f;fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L[s;a]; \mlambda{}g.<mk-hdf S[s;a;g], G[s;a;g]> m))))) s\000C)
\msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if (n =\msubz{} m)
then mk\_lambdas\_fun(\mlambda{}g.\mcup{}b\mmember{}G[s;a;g].f b;m)
else L[s;a] n
fi ; \mlambda{}g.<mk-hdf S[s;a;partial\_ap(g;m + 1;m)]
, select\_fun\_last(g;m)
> m + 1)))))
s)
Date html generated:
2016_05_16-AM-10_46_20
Last ObjectModification:
2016_01_17-AM-11_10_04
Theory : halting!dataflow
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