Step
*
2
1
1
1
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. p1 : copath(a.B[a];w)
10. p2 : copath(a.B[a];w')
11. sg-reachable(coW-game(a.B[a];w;w');<copath-cons(t;()), copath-cons(b;())>;<p1, p2>)
12. <p1, p2> ∈ {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'))}
⊢ <copath-tl(p1), copath-tl(p2)>
∈ {p:Pos(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))|
sg-reachable(coW-game(a.B[a];coW-item(w;t);coW-item(w';b));InitialPos(coW-game(a.B[a];coW-item(w;t);...));p)}
BY
{ ((MemTypeHD (-1) THENA Auto) THEN Thin (-2) THEN Reduce -1 THEN (MemTypeCD THENW Auto)) }
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. p1 : copath(a.B[a];w)
10. p2 : copath(a.B[a];w')
11. sg-reachable(coW-game(a.B[a];w;w');<copath-cons(t;()), copath-cons(b;())>;<p1, p2>)
12. (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'))
⊢ <copath-tl(p1), copath-tl(p2)> ∈ Pos(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))
2
.....set predicate.....
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. p1 : copath(a.B[a];w)
10. p2 : copath(a.B[a];w')
11. sg-reachable(coW-game(a.B[a];w;w');<copath-cons(t;()), copath-cons(b;())>;<p1, p2>)
12. (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'))
⊢ sg-reachable(coW-game(a.B[a];coW-item(w;t);coW-item(w';b));InitialPos(coW-game(a.B[a];coW-item(w;t);...));<...
, ...
>)
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. p1 : copath(a.B[a];w)
10. p2 : copath(a.B[a];w')
11. sg-reachable(coW-game(a.B[a];w;w');<copath-cons(t;()), copath-cons(b;())><p1, p2>)
12. <p1, p2> \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{} <copath-tl(p1), copath-tl(p2)>
\mmember{} \{p:Pos(coW-game(a.B[a];coW-item(w;t);coW-item(w';b)))|
sg-reachable(coW-game(a.B[a];coW-item(w;t);coW-item(w';b));InitialPos(...);p)\}
By
Latex:
((MemTypeHD (-1) THENA Auto) THEN Thin (-2) THEN Reduce -1 THEN (MemTypeCD THENW Auto))
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