Nuprl Lemma : hdf-state-base4-2
∀[C1,C2,C3,C4,F1,F2,F3,F4,f1,f2,f3,f4,s:Top]. ∀[hdr1,hdr2,hdr3,hdr4:Name].
(hdf-state((λx,s. f1[x;s]) o hdf-base(a.if name_eq(fst(a);hdr1) ∧b C1[a] then [F1[a]] else [] fi )
|| (λx,s. f2[x;s]) o hdf-base(a.if name_eq(fst(a);hdr2) ∧b C2[a] then [F2[a]] else [] fi )
|| (λx,s. f3[x;s]) o hdf-base(a.if name_eq(fst(a);hdr3) ∧b C3[a] then [F3[a]] else [] fi ) || (λx,s. f4[\000Cx;s])
o hdf-base(a.if name_eq(fst(a);hdr4) ∧b C4[a] then [F4[a]] else [] fi );[s])
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if name_eq(fst(a);hdr1) ∧b C1[a] then f1[F1[a];s]
if name_eq(fst(a);hdr2) ∧b C2[a] then f2[F2[a];s]
if name_eq(fst(a);hdr3) ∧b C3[a] then f3[F3[a];s]
if name_eq(fst(a);hdr4) ∧b C4[a] then f4[F4[a];s]
else s
fi ; λg.<mk-hdf (g (λx.x)), g (λx.[x])>; 1)))))
s) supposing
((¬(hdr1 = hdr2 ∈ Name)) and
(¬(hdr1 = hdr3 ∈ Name)) and
(¬(hdr1 = hdr4 ∈ Name)) and
(¬(hdr2 = hdr3 ∈ Name)) and
(¬(hdr2 = hdr4 ∈ Name)) and
(¬(hdr3 = hdr4 ∈ Name)))
Proof
Definitions occuring in Statement :
hdf-parallel: X || Y,
hdf-state: hdf-state(X;bs),
hdf-compose1: f o X,
hdf-base: hdf-base(m.F[m]),
name_eq: name_eq(x;y),
name: Name,
cons: [a / b],
nil: [],
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
not: ¬A,
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
natural_number: $n,
sqequal: s ~ t,
equal: s = t ∈ T,
cbva_seq: cbva_seq(L; F; m)
Lemmas :
callbyvalueall-ifthenelse,
int_subtype_base,
list_subtype_base,
atom_subtype_base,
ifthenelse_sqle,
band-sqequal-inl,
name_eq-is-inl,
callbyvalueall-single,
not-name_eq-implies-sq-bfalse,
equal_wf,
name_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
bag-combine-unit-left-top,
cbv_sqle,
evalall-sqequal,
null_cons_lemma,
has-valueall_wf_base,
exists_wf,
base_wf,
sqequal-wf-base
\mforall{}[C1,C2,C3,C4,F1,F2,F3,F4,f1,f2,f3,f4,s:Top]. \mforall{}[hdr1,hdr2,hdr3,hdr4:Name].
(hdf-state((\mlambda{}x,s. f1[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr1) \mwedge{}\msubb{} C1[a] then [F1[a]] else [] fi )
|| (\mlambda{}x,s. f2[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr2) \mwedge{}\msubb{} C2[a] then [F2[a]] else [] f\000Ci )
|| (\mlambda{}x,s. f3[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr3) \mwedge{}\msubb{} C3[a]
then [F3[a]]
else []
fi ) || (\mlambda{}x,s. f4[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr4) \mwedge{}\msubb{} C4[a]
then [F4[a]]
else []
fi );[s]) \msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if name\_eq(fst(a);hdr1) \mwedge{}\msubb{} C1[a]
then f1[F1[a];s]
if name\_eq(fst(a);hdr2) \mwedge{}\msubb{} C2[a]
then f2[F2[a];s]
if name\_eq(fst(a);hdr3) \mwedge{}\msubb{} C3[a]
then f3[F3[a];s]
if name\_eq(fst(a);hdr4) \mwedge{}\msubb{} C4[a]
then f4[F4[a];s]
else s
fi ; \mlambda{}g.<mk-hdf (g (\mlambda{}x.x))
, g (\mlambda{}x.[x])
> 1)))))
s) supposing
((\mneg{}(hdr1 = hdr2)) and
(\mneg{}(hdr1 = hdr3)) and
(\mneg{}(hdr1 = hdr4)) and
(\mneg{}(hdr2 = hdr3)) and
(\mneg{}(hdr2 = hdr4)) and
(\mneg{}(hdr3 = hdr4)))
Date html generated:
2015_07_17-AM-08_15_49
Last ObjectModification:
2015_05_03-PM-11_20_00
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