Step
*
2
1
2
3
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 : Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
13. y : Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
14. 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)
BY
{ ((RWO "sg-legal1-normalize" 0 THENA Auto)
THEN (RWO "sg-legal1-change-init" (-1) THENA Auto)
THEN MoveToConcl (-1)
THEN ((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 (GenConcl ⌜y
= q
∈ {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 RepUR ``sg-legal1 coW-game`` 0
THEN All Reduce
THEN (D 0 THENA Auto)
THEN (RWO "length-copath-tl" 0 THENA Auto)
THEN ParallelLast
THEN Auto
THEN (FLemma `copathAgree-tl` [-3] THEN Auto)
THEN NthHypEq (-1)
THEN RepeatFor 2 (EqCDA)
THEN Auto) }
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. x : Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
13. y : Pos(coW-game(a.B[a];w;w')@<copath-cons(t;()), copath-cons(b;())>)
14. Legal1(x;y)
\mvdash{} 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)
By
Latex:
((RWO "sg-legal1-normalize" 0 THENA Auto)
THEN (RWO "sg-legal1-change-init" (-1) THENA Auto)
THEN MoveToConcl (-1)
THEN ((GenConcl \mkleeneopen{}x = p\mkleeneclose{}\mcdot{} THENA Auto) THEN Thin (-1) THEN D -1 THEN D -2)
THEN (GenConcl \mkleeneopen{}y = q\mkleeneclose{}\mcdot{} THENA Auto)
THEN Thin (-1)
THEN D -1
THEN D -2
THEN RepUR ``sg-legal1 coW-game`` 0
THEN All Reduce
THEN (D 0 THENA Auto)
THEN (RWO "length-copath-tl" 0 THENA Auto)
THEN ParallelLast
THEN Auto
THEN (FLemma `copathAgree-tl` [-3] THEN Auto)
THEN NthHypEq (-1)
THEN RepeatFor 2 (EqCDA)
THEN Auto)
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