Nuprl Lemma : hdf-parallel-transformation2-2
∀[L1,L2,G1,G2,S1,S2,init1,init2:Top]. ∀[m1,m2:ℕ].
(fix((λmk-hdf,s. (inl (λa.cbva_seq(L1[s;a]; λg.<mk-hdf S1[g;s], G1[g]>; m1))))) init1
|| fix((λmk-hdf,s. (inl (λa.cbva_seq(L2[s;a]; λg.<mk-hdf S2[g;s], G2[g]>; m2))))) init2
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if n <z m1 then L1[fst(s);a] n
if n <z m1 + m2 then mk_lambdas(L2[snd(s);a] (n - m1);m1)
else mk_lambdas_fun(λg1.mk_lambdas_fun(λg2.(G1[g1] + G2[g2]);m2);m1)
fi ; λg.<mk-hdf
<S1[partial_ap_gen(g;(m1 + m2) + 1;0;m1);fst(s)]
, S2[partial_ap_gen(g;(m1 + m2) + 1;m1;m2);snd(s)]
>
, select_fun_last(g;m1 + m2)
>; (m1 + m2) + 1)))))
<init1, init2>)
Proof
Definitions occuring in Statement :
hdf-parallel: X || Y,
nat: ℕ,
ifthenelse: if b then t else f fi ,
lt_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>,
inl: inl x,
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t,
bag-append: as + bs,
select_fun_last: select_fun_last(g;m),
partial_ap_gen: partial_ap_gen(g;n;s;m),
mk_lambdas: mk_lambdas(F;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],
so_apply: x[s],
bag-append: as + bs,
lt_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),
empty-bag: {},
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,
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],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
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),
cbva_seq: cbva_seq(L; F; m),
callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m),
le_int: i ≤z j,
bnot: ¬bb,
pi1: fst(t),
mk_lambdas: mk_lambdas(F;m),
pi2: snd(t),
subtract: n - m,
mk_lambdas_fun: mk_lambdas_fun(F;m),
mk_lambdas-fun: mk_lambdas-fun(F;G;n;m),
list_ind: list_ind,
cons: [a / b],
partial_ap_gen: partial_ap_gen(g;n;s;m),
select_fun_last: select_fun_last(g;m),
select_fun_ap: select_fun_ap(g;n;m),
decidable: Dec(P),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
assert: ↑b,
compose: f o g,
nil: [],
nequal: a ≠ b ∈ T ,
eq_int: (i =z j),
subtype_rel: A ⊆r B
Latex:
\mforall{}[L1,L2,G1,G2,S1,S2,init1,init2:Top]. \mforall{}[m1,m2:\mBbbN{}].
(fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L1[s;a]; \mlambda{}g.<mk-hdf S1[g;s], G1[g]> m1))))) init1
|| fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L2[s;a]; \mlambda{}g.<mk-hdf S2[g;s], G2[g]> m2))))) init2
\msim{} fix((\mlambda{}mk-hdf,s.
(inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if n <z m1 then L1[fst(s);a] n
if n <z m1 + m2 then mk\_lambdas(L2[snd(s);a] (n - m1);m1)
else mk\_lambdas\_fun(\mlambda{}g1.mk\_lambdas\_fun(\mlambda{}g2.(G1[g1]
+ G2[g2]);m2);m1)
fi ; \mlambda{}g.<mk-hdf
<S1[partial\_ap\_gen(g;(m1 + m2) + 1;0;m1);fst(s)]
, S2[partial\_ap\_gen(g;(m1 + m2) + 1;m1;m2);snd(s)]
>
, select\_fun\_last(g;m1 + m2)
> (m1 + m2) + 1)))))
<init1, init2>)
Date html generated:
2016_05_16-AM-10_47_17
Last ObjectModification:
2016_01_17-AM-11_11_28
Theory : halting!dataflow
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