Step
*
2
1
2
8
of Lemma
coW-game-step-isom
.....wf.....
1. A : 𝕌'
2. B : A ⟶ Type
3. w : coW(A;a.B[a])
4. w' : coW(A;a.B[a])
5. t : coW-dom(a.B[a];w)
6. b : coW-dom(a.B[a];w')
7. λp.let u,v = p
in <copath-cons(t;u), copath-cons(b;v)> ∈ Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))
⟶ Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
8. Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>) ⊆r {p:copath(a.B[a];w) × copath(a.B[a];w')|
let p1,p2 = p
in (0 < copath-length(p1) ∧ (copath-hd(p1) = t ∈ coW-dom(a.B[a];w)))
∧ 0 < copath-length(p2)
∧ (copath-hd(p2) = b ∈ coW-dom(a.B[a];w'))}
9. λp.let u,v = p
in <copath-tl(u), copath-tl(v)> ∈ Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>) ⟶ Pos(sg-norm\000Calize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))
10. g : Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>) ⟶ Pos(sg-normalize(coW-game(a.B[a];coW-item(w\000C;t);coW-item(w';b))))
⊢ istype((∀x,y:Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))).
(Legal1(x;y)
⇒ Legal1((λp.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) x;(λp.let u,v = p
in <copath-cons(t;u)
, copath-cons(b;v)
>)
y)))
∧ (∀x,y:Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))).
(Legal2(x;y)
⇒ Legal2((λp.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) x;(λp.let u,v = p
in <copath-cons(t;u), copath-cons(b;v)>)
y)))
∧ (∀x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>). (Legal1(x;y) ⇒ Legal1(g x;g y)))
∧ (∀x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>). (Legal2(x;y) ⇒ Legal2(g x;g y)))
∧ (∀x:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
(((λp.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) (g x))
= x
∈ Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)))
∧ (∀x:Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))
((g ((λp.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) x))
= x
∈ Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))))
∧ (((λp.let u,v = p
in <copath-cons(t;u), copath-cons(b;v)>)
InitialPos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))))
= InitialPos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
∈ Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>))
∧ ((g InitialPos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>))
= InitialPos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))
∈ Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))))
BY
{ ((Assert respects-equality(copath(a.B[a];w) × copath(a.B[a];w');Pos(coW-game(a.B[a];w;w')@<copath-cons(t;())
, copath-cons(b;())
>)) BY
(RepUR ``sg-pos sg-change-init`` 0 THEN Auto))
THEN Auto
) }
Latex:
Latex:
.....wf.....
1. A : \mBbbU{}'
2. B : A {}\mrightarrow{} Type
3. w : coW(A;a.B[a])
4. w' : coW(A;a.B[a])
5. t : coW-dom(a.B[a];w)
6. b : coW-dom(a.B[a];w')
7. \mlambda{}p.let u,v = p
in <copath-cons(t;u), copath-cons(b;v)>
\mmember{} Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))
{}\mrightarrow{} Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
8. Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>) \msubseteq{}r \{p:copath(a.B[a];w) \mtimes{} copath\000C(a.B[a];w')|
let p1,p2 = p
in (0 < copath-length(p1) \mwedge{} (copath-hd(p1) = t))
\mwedge{} 0 < copath-length(p2)
\mwedge{} (copath-hd(p2) = b)\}
9. \mlambda{}p.let u,v = p
in <copath-tl(u), copath-tl(v)> \mmember{} Pos(coW-game(a.B[a];w;w')@<copath-cons(t;())
, copath-cons(b;())
>) {}\mrightarrow{} Pos(sg-normalize(coW-game(a.\000CB[a];coW-item(w;t);coW-item(w';b))))
10. g : Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
{}\mrightarrow{} Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))
\mvdash{} istype((\mforall{}x,y:Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))).
(Legal1(x;y)
{}\mRightarrow{} Legal1((\mlambda{}p.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) x;
(\mlambda{}p.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) y)))
\mwedge{} (\mforall{}x,y:Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))).
(Legal2(x;y)
{}\mRightarrow{} Legal2((\mlambda{}p.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) x;(\mlambda{}p.let u,v = p
in <copath-cons(t;u)
, copath-cons(b;v)
>)
y)))
\mwedge{} (\mforall{}x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>). (Legal1(x;y) {}\mRightarrow{} Legal1(g\000C x;g y)))
\mwedge{} (\mforall{}x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>). (Legal2(x;y) {}\mRightarrow{} Legal2(g\000C x;g y)))
\mwedge{} (\mforall{}x:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
(((\mlambda{}p.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) (g x)) = x))
\mwedge{} (\mforall{}x:Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))
((g ((\mlambda{}p.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) x)) = x))
\mwedge{} (((\mlambda{}p.let u,v = p
in <copath-cons(t;u), copath-cons(b;v)>)
InitialPos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))))
= InitialPos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>))
\mwedge{} ((g InitialPos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>))
= InitialPos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b))))))
By
Latex:
((Assert respects-equality(copath(a.B[a];w)
\mtimes{} copath(a.B[a];w');Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)) BY
(RepUR ``sg-pos sg-change-init`` 0 THEN Auto))
THEN Auto
)
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