Step
*
2
2
2
of Lemma
prec-ext
1. P : Type
2. a : Atom ⟶ P ⟶ ((P + P + Type) List)
3. pcorec(lbl,p.a[lbl;p]) ≡ ptuple(lbl,p.a[lbl;p];pcorec(lbl,p.a[lbl;p]))
4. i : P
5. pcorec(lbl,p.a[lbl;p]) i ≡ labl:{lbl:Atom| 0 < ||a[lbl;i]||} × tuple-type(map(λx.case x
of inl(y) =>
case y
of inl(p) =>
pcorec(lbl,p.a[lbl;p]) p
| inr(p) =>
(pcorec(lbl,p.a[lbl;p]) p) List
| inr(E) =>
E;a[labl;i]))
6. labl : {lbl:Atom| 0 < ||a[lbl;i]||}
7. u : P + P + Type
8. v : (P + P + Type) List
9. ∀x1:tuple-type(map(λx.case x
of inl(y) =>
case y of inl(p) => prec(lbl,p.a[lbl;p];p) | inr(p) => prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E;v))
(1 + add-sz(pcorec-size(lbl,p.a[lbl;p]);v;x1))↓
10. null(v) = ff
11. x1 : case u
of inl(y) =>
case y of inl(p) => prec(lbl,p.a[lbl;p];p) | inr(p) => prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E × tuple-type(map(λx.case x
of inl(y) =>
case y of inl(p) => prec(lbl,p.a[lbl;p];p) | inr(p) => prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E;v))
⊢ (1
+ let u@0,v@0 = x1
in case u
of inl(z) =>
case z of inl(p) => pcorec-size(lbl,p.a[lbl;p]) p u@0 | inr(p) => l_sum(map(pcorec-size(lbl,p.a[lbl;p]) p;u@0))
| inr(E) =>
0
+ tuple-sum(λc,u. case c
of inl(z) =>
case z
of inl(p) =>
pcorec-size(lbl,p.a[lbl;p]) p u
| inr(p) =>
l_sum(map(pcorec-size(lbl,p.a[lbl;p]) p;u))
| inr(E) =>
0;v;v@0))↓
BY
{ (D -1 THEN Reduce 0 THEN (D -4 With ⌜x3⌝ THENA Auto) THEN Fold `add-sz` 0) }
1
1. P : Type
2. a : Atom ⟶ P ⟶ ((P + P + Type) List)
3. pcorec(lbl,p.a[lbl;p]) ≡ ptuple(lbl,p.a[lbl;p];pcorec(lbl,p.a[lbl;p]))
4. i : P
5. pcorec(lbl,p.a[lbl;p]) i ≡ labl:{lbl:Atom| 0 < ||a[lbl;i]||} × tuple-type(map(λx.case x
of inl(y) =>
case y
of inl(p) =>
pcorec(lbl,p.a[lbl;p]) p
| inr(p) =>
(pcorec(lbl,p.a[lbl;p]) p) List
| inr(E) =>
E;a[labl;i]))
6. labl : {lbl:Atom| 0 < ||a[lbl;i]||}
7. u : P + P + Type
8. v : (P + P + Type) List
9. null(v) = ff
10. x2 : case u
of inl(y) =>
case y of inl(p) => prec(lbl,p.a[lbl;p];p) | inr(p) => prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E
11. x3 : tuple-type(map(λx.case x
of inl(y) =>
case y of inl(p) => prec(lbl,p.a[lbl;p];p) | inr(p) => prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E;v))
12. (1 + add-sz(pcorec-size(lbl,p.a[lbl;p]);v;x3))↓
⊢ (1
+ case u
of inl(z) =>
case z of inl(p) => pcorec-size(lbl,p.a[lbl;p]) p x2 | inr(p) => l_sum(map(pcorec-size(lbl,p.a[lbl;p]) p;x2))
| inr(E) =>
0
+ add-sz(pcorec-size(lbl,p.a[lbl;p]);v;x3))↓
Latex:
Latex:
1. P : Type
2. a : Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)
3. pcorec(lbl,p.a[lbl;p]) \mequiv{} ptuple(lbl,p.a[lbl;p];pcorec(lbl,p.a[lbl;p]))
4. i : P
5. pcorec(lbl,p.a[lbl;p]) i \mequiv{} labl:\{lbl:Atom| 0 < ||a[lbl;i]||\} \mtimes{} tuple-type(map(\mlambda{}x.case x
of inl(y) =>
case y
of inl(p) =>
pcorec(lbl,p.a[lbl;p]) p
| inr(p) =>
(pcorec(lbl,p.a[lbl;p]) p) List
| inr(E) =>
E;a[labl;i]))
6. labl : \{lbl:Atom| 0 < ||a[lbl;i]||\}
7. u : P + P + Type
8. v : (P + P + Type) List
9. \mforall{}x1:tuple-type(map(\mlambda{}x.case x
of inl(y) =>
case y
of inl(p) =>
prec(lbl,p.a[lbl;p];p)
| inr(p) =>
prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E;v))
(1 + add-sz(pcorec-size(lbl,p.a[lbl;p]);v;x1))\mdownarrow{}
10. null(v) = ff
11. x1 : case u
of inl(y) =>
case y of inl(p) => prec(lbl,p.a[lbl;p];p) | inr(p) => prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E \mtimes{} tuple-type(map(\mlambda{}x.case x
of inl(y) =>
case y
of inl(p) =>
prec(lbl,p.a[lbl;p];p)
| inr(p) =>
prec(lbl,p.a[lbl;p];p) List
| inr(E) =>
E;v))
\mvdash{} (1
+ let u@0,v@0 = x1
in case u
of inl(z) =>
case z
of inl(p) =>
pcorec-size(lbl,p.a[lbl;p]) p u@0
| inr(p) =>
l\_sum(map(pcorec-size(lbl,p.a[lbl;p]) p;u@0))
| inr(E) =>
0
+ tuple-sum(\mlambda{}c,u. case c
of inl(z) =>
case z
of inl(p) =>
pcorec-size(lbl,p.a[lbl;p]) p u
| inr(p) =>
l\_sum(map(pcorec-size(lbl,p.a[lbl;p]) p;u))
| inr(E) =>
0;v;v@0))\mdownarrow{}
By
Latex:
(D -1 THEN Reduce 0 THEN (D -4 With \mkleeneopen{}x3\mkleeneclose{} THENA Auto) THEN Fold `add-sz` 0)
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