Step
*
2
1
2
6
1
of Lemma
coW-game-step-isom
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. ∀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))
11. ∀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))
12. ∀x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>).
(Legal1(x;y)
⇒ Legal1((λp.let u,v = p in <copath-tl(u), copath-tl(v)>) x;(λp.let u,v = p in <copath-tl(u), copath-tl(v)>) y))
13. ∀x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>).
(Legal2(x;y)
⇒ Legal2((λp.let u,v = p in <copath-tl(u), copath-tl(v)>) x;(λp.let u,v = p in <copath-tl(u), copath-tl(v)>) y))
14. x : Pos(coW-game(a.B[a];w;w'))
15. sg-reachable(coW-game(a.B[a];w;w');<copath-cons(t;()), copath-cons(b;())>;x)
16. x ∈ {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'))}
⊢ x
= ((λp.let u,v = p in <copath-cons(t;u), copath-cons(b;v)>) ((λp.let u,v = p in <copath-tl(u), copath-tl(v)>) x))
∈ Pos(coW-game(a.B[a];w;w'))
BY
{ ((GenConcl ⌜x
= p
∈ {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'))} ⌝⋅
THENA Auto
)
THEN Thin (-1)
THEN D -1
THEN D -2
THEN All Reduce) }
1
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. ∀x,y:Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))).
(Legal1(x;y)
⇒ Legal1(let u,v = x
in <copath-cons(t;u), copath-cons(b;v)>;let u,v = y
in <copath-cons(t;u), copath-cons(b;v)>))
11. ∀x,y:Pos(sg-normalize(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))).
(Legal2(x;y)
⇒ Legal2(let u,v = x
in <copath-cons(t;u), copath-cons(b;v)>;let u,v = y
in <copath-cons(t;u), copath-cons(b;v)>))
12. ∀x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>).
(Legal1(x;y) ⇒ Legal1(let u,v = x in <copath-tl(u), copath-tl(v)>;let u,v = y in <copath-tl(u), copath-tl(v)>))
13. ∀x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>).
(Legal2(x;y) ⇒ Legal2(let u,v = x in <copath-tl(u), copath-tl(v)>;let u,v = y in <copath-tl(u), copath-tl(v)>))
14. x : Pos(coW-game(a.B[a];w;w'))
15. sg-reachable(coW-game(a.B[a];w;w');<copath-cons(t;()), copath-cons(b;())>;x)
16. x ∈ {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'))}
17. p1 : copath(a.B[a];w)
18. p2 : copath(a.B[a];w')
19. (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'))
⊢ <p1, p2> = <copath-cons(t;copath-tl(p1)), copath-cons(b;copath-tl(p2))> ∈ Pos(coW-game(a.B[a];w;w'))
Latex:
Latex:
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. \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))
11. \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))
12. \mforall{}x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>).
(Legal1(x;y)
{}\mRightarrow{} Legal1((\mlambda{}p.let u,v = p in <copath-tl(u), copath-tl(v)>) x;(\mlambda{}p.let u,v = p
in <copath-tl(u)
, copath-tl(v)
>)
y))
13. \mforall{}x,y:Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>).
(Legal2(x;y)
{}\mRightarrow{} Legal2((\mlambda{}p.let u,v = p in <copath-tl(u), copath-tl(v)>) x;(\mlambda{}p.let u,v = p
in <copath-tl(u)
, copath-tl(v)
>)
y))
14. x : Pos(coW-game(a.B[a];w;w'))
15. sg-reachable(coW-game(a.B[a];w;w');<copath-cons(t;()), copath-cons(b;())>x)
16. x \mmember{} \{p:copath(a.B[a];w) \mtimes{} copath(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)\}
\mvdash{} x
= ((\mlambda{}p.let u,v = p
in <copath-cons(t;u), copath-cons(b;v)>)
((\mlambda{}p.let u,v = p in <copath-tl(u), copath-tl(v)>) x))
By
Latex:
((GenConcl \mkleeneopen{}x = p\mkleeneclose{}\mcdot{} THENA Auto) THEN Thin (-1) THEN D -1 THEN D -2 THEN All Reduce)
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