Nuprl Lemma : hdf-parallel-transformation1
∀[L1,L2:Base]. ∀[m1,m2:ℕ+].
(fix((λmk-hdf.(inl (λa.simple-cbva-seq(L1[a];λout.<mk-hdf, out>;m1)))))
|| fix((λmk-hdf.(inl (λa.simple-cbva-seq(L2[a];λout.<mk-hdf, out>;m2)))))
~ fix((λmk-hdf.(inl (λa.simple-cbva-seq(λn.if n <z m1 then L1[a] n
if n <z m1 + m2 then mk_lambdas(L2[a] (n - m1);m1)
else mk_lambdas(λout1.mk_lambdas(λout2.(out1 + out2);m2 - 1);m1 - 1)
fi ;λout.<mk-hdf, out>;(m1 + m2) + 1))))))
Proof
Definitions occuring in Statement :
hdf-parallel: X || Y,
nat_plus: ℕ+,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
so_apply: x[s],
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,
base: Base,
sqequal: s ~ t,
bag-append: as + bs,
simple-cbva-seq: simple-cbva-seq(L;F;m),
mk_lambdas: mk_lambdas(F;m)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ifthenelse: if b then t else f fi ,
hdf-parallel: X || Y,
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),
btrue: tt,
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],
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
decidable: Dec(P),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
empty-bag: {},
nil: [],
lt_int: i <z j,
bfalse: ff,
subtract: n - m,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
true: True,
eq_int: (i =z j),
ge: i ≥ j ,
sq_type: SQType(T),
it: ⋅
Latex:
\mforall{}[L1,L2:Base]. \mforall{}[m1,m2:\mBbbN{}\msupplus{}].
(fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.simple-cbva-seq(L1[a];\mlambda{}out.<mk-hdf, out>m1)))))
|| fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.simple-cbva-seq(L2[a];\mlambda{}out.<mk-hdf, out>m2)))))
\msim{} fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.simple-cbva-seq(\mlambda{}n.if n <z m1 then L1[a] n
if n <z m1 + m2 then mk\_lambdas(L2[a] (n - m1);m1)
else mk\_lambdas(\mlambda{}out1.mk\_lambdas(\mlambda{}out2.(out1
+ out2);m2
- 1);m1 - 1)
fi ;\mlambda{}out.<mk-hdf, out>(m1 + m2) + 1))))))
Date html generated:
2016_05_16-AM-10_46_48
Last ObjectModification:
2016_01_17-AM-11_11_14
Theory : halting!dataflow
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