Nuprl Lemma : hdf-parallel-transformation2-3
∀[L,G,H,S,init,out:Top]. ∀[m:ℕ].
(inl (λa.<inr Ax , out>) || fix((λmk-hdf,s. (inl (λa.cbva_seq(L[s;a]; λg.<case H[g;s]
of inl() =>
mk-hdf S[g;s]
| inr() =>
inr Ax
, G[g]
>; m)))))
init ~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if (n =z m)
then case fst(s)
of inl(x) =>
mk_lambdas_fun(λg.(out + G[g]);m)
| inr(x) =>
mk_lambdas_fun(λg.G[g];m)
else L[snd(s);a] n
fi ; λg.<case H[partial_ap(g;m + 1;m);snd(s)]
of inl() =>
mk-hdf
<inr Ax
, S[partial_ap(g;m + 1;m);snd(s)]
>
| inr() =>
inr Ax
, select_fun_last(g;m)
>; m + 1)))))
<inl Ax, init>)
Proof
Definitions occuring in Statement :
hdf-parallel: X || Y,
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],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inr: inr x ,
inl: inl x,
add: n + m,
natural_number: $n,
sqequal: s ~ t,
axiom: Ax,
bag-append: as + bs,
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,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
hdf-parallel: X || Y,
bfalse: ff,
btrue: tt,
hdf-halted: hdf-halted(P),
band: p ∧b q,
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],
so_apply: x[s1;s2],
empty-bag: {},
nil: [],
it: ⋅,
pi1: fst(t),
pi2: snd(t),
select_fun_last: select_fun_last(g;m),
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
fun_exp: f^n,
primrec: primrec(n;b;c),
partial_ap: partial_ap(g;n;m),
mk_lambdas_fun: mk_lambdas_fun(F;m),
mk_lambdas-fun: mk_lambdas-fun(F;G;n;m),
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
mk_lambdas: mk_lambdas(F;m),
subtract: n - m,
cbva_seq: cbva_seq(L; F; m),
callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m),
bag-append: as + bs,
append: as @ bs,
list_ind: list_ind,
cons: [a / b],
select_fun_ap: select_fun_ap(g;n;m),
decidable: Dec(P),
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
sq_type: SQType(T),
assert: ↑b,
compose: f o g,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B
Latex:
\mforall{}[L,G,H,S,init,out:Top]. \mforall{}[m:\mBbbN{}].
(inl (\mlambda{}a.<inr Ax , out>) || fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L[s;a]; \mlambda{}g.<case H[g;s]
of inl() =>
mk-hdf S[g;s]
| inr() =>
inr Ax
, G[g]
> m)))))
init
\msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if (n =\msubz{} m)
then case fst(s)
of inl(x) =>
mk\_lambdas\_fun(\mlambda{}g.(out + G[g]);m)
| inr(x) =>
mk\_lambdas\_fun(\mlambda{}g.G[g];m)
else L[snd(s);a] n
fi ; \mlambda{}g.<case H[partial\_ap(g;m + 1;m);snd(s)]
of inl() =>
mk-hdf <inr Ax , S[partial\_ap(g;m + 1;m);snd(s)]>
| inr() =>
inr Ax
, select\_fun\_last(g;m)
> m + 1)))))
<inl Ax, init>)
Date html generated:
2016_05_16-AM-10_47_27
Last ObjectModification:
2016_01_17-AM-11_11_32
Theory : halting!dataflow
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