Step
*
of Lemma
hdf-memory-base4-3
∀[F1,F2,F3,F4,f1,f2,f3,f4,s:Top]. ∀[hdr1,hdr2,hdr3,hdr4:Name].
(hdf-memory((λx,s. f1[x;s]) o hdf-base(a.if name_eq(fst(a);hdr1) then [F1[a]] else [] fi )
|| (λx,s. f2[x;s]) o hdf-base(a.if name_eq(fst(a);hdr2) then [F2[a]] else [] fi )
|| (λx,s. f3[x;s]) o hdf-base(a.if name_eq(fst(a);hdr3) then [F3[a]] else [] fi ) || (λx,s. f4[x;s])
o hdf-base(a.if name_eq(fst(a);hdr4) then [F4[a]] else [] fi );[s])
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if name_eq(fst(a);hdr1) then f1[F1[a];s]
if name_eq(fst(a);hdr2) then f2[F2[a];s]
if name_eq(fst(a);hdr3) then f3[F3[a];s]
if name_eq(fst(a);hdr4) then f4[F4[a];s]
else s
fi ; λg.<mk-hdf (g (λx.x)), [s]>; 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)))
BY
{ ((UnivCD THENA Auto)
THEN (RWO "hdf-base-transformation1" 0 THENA Auto)
THEN (RWO "hdf-compose1-transformation2" 0 THENA Auto)
THEN Repeat ((RWO "hdf-parallel-transformation2-1" 0 THENA Auto))
THEN Reduce 0
THEN (RWO "hdf-memory-transformation2" 0 THENA Auto)
THEN RepUR ``cbva_seq mk_lambdas_fun select_fun_ap select_fun_last mk_lambdas bag-map`` 0
THEN Repeat ((RecUnfold `callbyvalueall_seq` 0 THEN Reduce 0))
THEN Repeat ((RecUnfold `mk_lambdas-fun` 0 THEN Reduce 0))
THEN Repeat (((RWO "primrec-unroll" 0 THENA Auto) THEN Reduce 0))
THEN (RWO "map-ifthenelse" 0 THENA Auto)
THEN HdfStateTransStart `s') }
1
1. hdr1 : Name
2. hdr2 : Name
3. hdr3 : Name
4. hdr4 : Name
5. ¬(hdr3 = hdr4 ∈ Name)
6. ¬(hdr2 = hdr4 ∈ Name)
7. ¬(hdr2 = hdr3 ∈ Name)
8. ¬(hdr1 = hdr4 ∈ Name)
9. ¬(hdr1 = hdr3 ∈ Name)
10. ¬(hdr1 = hdr2 ∈ Name)
11. F4 : Base
12. f4 : Base
13. F3 : Base
14. f3 : Base
15. F2 : Base
16. f2 : Base
17. F1 : Base
18. f1 : Base
19. j : ℤ
20. 0 < j
21. ∀s:Base
(λmk-hdf,s. (inl (λa.let v ←─ if name_eq(fst(a);hdr1) then [λs.f1[F1[a];s]] else [] fi
in let v1 ←─ if name_eq(fst(a);hdr2) then [λs.f2[F2[a];s]] else [] fi
in let v2 ←─ if name_eq(fst(a);hdr3) then [λs.f3[F3[a];s]] else [] fi
in let v3 ←─ if name_eq(fst(a);hdr4) then [λs.f4[F4[a];s]] else [] fi
in let v4 ←─ v2 + v3
in let v5 ←─ v1 + v4
in let v6 ←─ v + v5
in let v7 ←─ ∪f∈v6.map(f;s)
in let v8 ←─ if bag-null(v7) then s else v7 fi
in <mk-hdf v8, s>))^j - 1
⊥
[s] ≤ fix((λmk-hdf,s. (inl (λa.let v ←─ if name_eq(fst(a);hdr1) then f1[F1[a];s]
if name_eq(fst(a);hdr2) then f2[F2[a];s]
if name_eq(fst(a);hdr3) then f3[F3[a];s]
if name_eq(fst(a);hdr4) then f4[F4[a];s]
else s
fi
in <mk-hdf v, [s]>))))
s)
22. s : Base@i
23. a : Base
⊢ let v ←─ if name_eq(fst(a);hdr1) then [λs.f1[F1[a];s]] else [] fi
in let v1 ←─ if name_eq(fst(a);hdr2) then [λs.f2[F2[a];s]] else [] fi
in let v2 ←─ if name_eq(fst(a);hdr3) then [λs.f3[F3[a];s]] else [] fi
in let v3 ←─ if name_eq(fst(a);hdr4) then [λs.f4[F4[a];s]] else [] fi
in let v4 ←─ v2 + v3
in let v5 ←─ v1 + v4
in let v6 ←─ v + v5
in let v7 ←─ ∪f∈v6.[f s]
in let v8 ←─ if bag-null(v7) then [s] else v7 fi
in <λmk-hdf,s. (inl (λa.let v ←─ if name_eq(fst(a);hdr1) then [λs.f1[F1[a];s]] else [] fi
in let v1 ←─ if name_eq(fst(a);hdr2) then [λs.f2[F2[a];s]] else [] fi
in let v2 ←─ if name_eq(fst(a);hdr3)
then [λs.f3[F3[a];s]]
else []
fi
in let v3 ←─ if name_eq(fst(a);hdr4)
then [λs.f4[F4[a];s]]
else []
fi
in let v4 ←─ v2 + v3
in let v5 ←─ v1 + v4
in let v6 ←─ v + v5
in let v7 ←─ ∪f∈v6.map(f;s)
in let v8 ←─ if bag-null(v7) then s else v7 fi
in <mk-hdf v8, s>))^j - 1
⊥
v8
, [s]
> ≤ let v ←─ if name_eq(fst(a);hdr1) then f1[F1[a];s]
if name_eq(fst(a);hdr2) then f2[F2[a];s]
if name_eq(fst(a);hdr3) then f3[F3[a];s]
if name_eq(fst(a);hdr4) then f4[F4[a];s]
else s
fi
in <fix((λmk-hdf,s. (inl (λa.let v ←─ if name_eq(fst(a);hdr1) then f1[F1[a];s]
if name_eq(fst(a);hdr2) then f2[F2[a];s]
if name_eq(fst(a);hdr3) then f3[F3[a];s]
if name_eq(fst(a);hdr4) then f4[F4[a];s]
else s
fi
in <mk-hdf v, [s]>))))
v
, [s]
>
2
1. hdr1 : Name
2. hdr2 : Name
3. hdr3 : Name
4. hdr4 : Name
5. ¬(hdr3 = hdr4 ∈ Name)
6. ¬(hdr2 = hdr4 ∈ Name)
7. ¬(hdr2 = hdr3 ∈ Name)
8. ¬(hdr1 = hdr4 ∈ Name)
9. ¬(hdr1 = hdr3 ∈ Name)
10. ¬(hdr1 = hdr2 ∈ Name)
11. F4 : Base
12. f4 : Base
13. F3 : Base
14. f3 : Base
15. F2 : Base
16. f2 : Base
17. F1 : Base
18. f1 : Base
19. j : ℤ
20. 0 < j
21. ∀s:Base
(λmk-hdf,s. (inl (λa.let v ←─ if name_eq(fst(a);hdr1) then f1[F1[a];s]
if name_eq(fst(a);hdr2) then f2[F2[a];s]
if name_eq(fst(a);hdr3) then f3[F3[a];s]
if name_eq(fst(a);hdr4) then f4[F4[a];s]
else s
fi
in <mk-hdf v, [s]>))^j - 1
⊥
s ≤ fix((λmk-hdf,s. (inl (λa.let v ←─ if name_eq(fst(a);hdr1) then [λs.f1[F1[a];s]] else [] fi
in let v1 ←─ if name_eq(fst(a);hdr2) then [λs.f2[F2[a];s]] else [] fi
in let v2 ←─ if name_eq(fst(a);hdr3) then [λs.f3[F3[a];s]] else [] fi
in let v3 ←─ if name_eq(fst(a);hdr4) then [λs.f4[F4[a];s]] else [] fi
in let v4 ←─ v2 + v3
in let v5 ←─ v1 + v4
in let v6 ←─ v + v5
in let v7 ←─ ∪f∈v6.map(f;s)
in let v8 ←─ if bag-null(v7) then s else v7 fi
in <mk-hdf v8, s>))))
[s])
22. s : Base@i
23. a : Base
⊢ let v ←─ if name_eq(fst(a);hdr1) then f1[F1[a];s]
if name_eq(fst(a);hdr2) then f2[F2[a];s]
if name_eq(fst(a);hdr3) then f3[F3[a];s]
if name_eq(fst(a);hdr4) then f4[F4[a];s]
else s
fi
in <λmk-hdf,s. (inl (λa.let v ←─ if name_eq(fst(a);hdr1) then f1[F1[a];s]
if name_eq(fst(a);hdr2) then f2[F2[a];s]
if name_eq(fst(a);hdr3) then f3[F3[a];s]
if name_eq(fst(a);hdr4) then f4[F4[a];s]
else s
fi
in <mk-hdf v, [s]>))^j - 1
⊥
v
, [s]
> ≤ let v ←─ if name_eq(fst(a);hdr1) then [λs.f1[F1[a];s]] else [] fi
in let v1 ←─ if name_eq(fst(a);hdr2) then [λs.f2[F2[a];s]] else [] fi
in let v2 ←─ if name_eq(fst(a);hdr3) then [λs.f3[F3[a];s]] else [] fi
in let v3 ←─ if name_eq(fst(a);hdr4) then [λs.f4[F4[a];s]] else [] fi
in let v4 ←─ v2 + v3
in let v5 ←─ v1 + v4
in let v6 ←─ v + v5
in let v7 ←─ ∪f∈v6.[f s]
in let v8 ←─ if bag-null(v7) then [s] else v7 fi
in <fix((λmk-hdf,s. (inl (λa.let v ←─ if name_eq(fst(a);hdr1)
then [λs.f1[F1[a];s]]
else []
fi
in let v1 ←─ if name_eq(fst(a);hdr2)
then [λs.f2[F2[a];s]]
else []
fi
in let v2 ←─ if name_eq(fst(a);hdr3)
then [λs.f3[F3[a];s]]
else []
fi
in let v3 ←─ if name_eq(fst(a);hdr4)
then [λs.f4[F4[a];s]]
else []
fi
in let v4 ←─ v2 + v3
in let v5 ←─ v1 + v4
in let v6 ←─ v + v5
in let v7 ←─ ∪f∈v6.map(f;s)
in let v8 ←─ if bag-null(v7)
then s
else v7
fi
in <mk-hdf v8, s>))))
v8
, [s]
>
Latex:
\mforall{}[F1,F2,F3,F4,f1,f2,f3,f4,s:Top]. \mforall{}[hdr1,hdr2,hdr3,hdr4:Name].
(hdf-memory((\mlambda{}x,s. f1[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr1) then [F1[a]] else [] fi )
|| (\mlambda{}x,s. f2[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr2) then [F2[a]] else [] fi )
|| (\mlambda{}x,s. f3[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr3) then [F3[a]] else [] fi )
|| (\mlambda{}x,s. f4[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr4)
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)
then f1[F1[a];s]
if name\_eq(fst(a);hdr2)
then f2[F2[a];s]
if name\_eq(fst(a);hdr3)
then f3[F3[a];s]
if name\_eq(fst(a);hdr4)
then f4[F4[a];s]
else s
fi ; \mlambda{}g.<mk-hdf (g (\mlambda{}x.x))
, [s]
> 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)))
By
((UnivCD THENA Auto)
THEN (RWO "hdf-base-transformation1" 0 THENA Auto)
THEN (RWO "hdf-compose1-transformation2" 0 THENA Auto)
THEN Repeat ((RWO "hdf-parallel-transformation2-1" 0 THENA Auto))
THEN Reduce 0
THEN (RWO "hdf-memory-transformation2" 0 THENA Auto)
THEN RepUR ``cbva\_seq mk\_lambdas\_fun select\_fun\_ap select\_fun\_last mk\_lambdas bag-map`` 0
THEN Repeat ((RecUnfold `callbyvalueall\_seq` 0 THEN Reduce 0))
THEN Repeat ((RecUnfold `mk\_lambdas-fun` 0 THEN Reduce 0))
THEN Repeat (((RWO "primrec-unroll" 0 THENA Auto) THEN Reduce 0))
THEN (RWO "map-ifthenelse" 0 THENA Auto)
THEN HdfStateTransStart `s')
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