Step
*
3
1
2
1
1
1
1
of Lemma
pW-rec_wf
1. P : Type
2. A : P ⟶ Type
3. B : p:P ⟶ A[p] ⟶ Type
4. C : p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P
5. pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]) ∈ Type
6. pW ∈ P ⟶ Type
7. Q : par:P ⟶ (pW par) ⟶ ℙ
8. ind : par:P
⟶ a:A[par]
⟶ f:(b:B[par;a] ⟶ (pW C[par;a;b]))
⟶ (b:B[par;a] ⟶ Q[C[par;a;b];f b])
⟶ Q[par;pW-sup(a;f)]
9. par : P
10. a : A[par]
11. w1 : b:B[par;a] ⟶ (pco-W C[par;a;b])
12. ∀path:Path. (StepAgree(path 0;par;<a, w1>) ⇒ (↓∃n:ℕ. Barred(pcw-partial(path;n))))
13. param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) ∈ n:ℕ
⟶ (ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]))
⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
⟶ ℙ
14. ∀[pp:n:ℕ × (ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]))]. (Barred(pp) ∈ ℙ)
15. n : ℕ
16. ¬0 < n
17. s : ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
18. ∀%6:ℕn. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) %6 s (s %6))
19. ∀t:{t:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])| param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) n s \000Ct}
(let q,w',d = t in
case d
of inl(b) =>
let a,f = w'
in let ind(p,a,f,G) = ind[p;a;f;G] in
letrec F(p,w) = let a,f=w in
ind(p,a,f,λb.F(C[p;a;b],f(b)) in
F(C[q;a;b];f b)
| inr(z) =>
Ax ∈ let q,w',d = t in
case d of inl(b) => let a,f = w' in Q[C[q;a;b];f b] | inr(z) => True)
20. <a, w1> ∈ pW par
21. w1 ∈ b:B[par;a] ⟶ (pW C[par;a;b])
22. b : B[par;a]
⊢ let ind(p,a,f,G) = ind p a f G in
letrec F(p,w) = let a,f=w in
ind(p,a,f,λb.F(C p a b,f(b)) in
F(C par a b;w1 b) ∈ Q[C[par;a;b];w1 b]
BY
{ TACTIC:(InstHyp [⌜<par, <a, w1>, inl b>⌝] (-4)⋅ THENM (Reduce (-1) THEN Auto)) }
1
.....wf.....
1. P : Type
2. A : P ⟶ Type
3. B : p:P ⟶ A[p] ⟶ Type
4. C : p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P
5. pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]) ∈ Type
6. pW ∈ P ⟶ Type
7. Q : par:P ⟶ (pW par) ⟶ ℙ
8. ind : par:P
⟶ a:A[par]
⟶ f:(b:B[par;a] ⟶ (pW C[par;a;b]))
⟶ (b:B[par;a] ⟶ Q[C[par;a;b];f b])
⟶ Q[par;pW-sup(a;f)]
9. par : P
10. a : A[par]
11. w1 : b:B[par;a] ⟶ (pco-W C[par;a;b])
12. ∀path:Path. (StepAgree(path 0;par;<a, w1>) ⇒ (↓∃n:ℕ. Barred(pcw-partial(path;n))))
13. param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) ∈ n:ℕ
⟶ (ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]))
⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
⟶ ℙ
14. ∀[pp:n:ℕ × (ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]))]. (Barred(pp) ∈ ℙ)
15. n : ℕ
16. ¬0 < n
17. s : ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
18. ∀%6:ℕn. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) %6 s (s %6))
19. ∀t:{t:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])| param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) n s \000Ct}
(let q,w',d = t in
case d
of inl(b) =>
let a,f = w'
in let ind(p,a,f,G) = ind[p;a;f;G] in
letrec F(p,w) = let a,f=w in
ind(p,a,f,λb.F(C[p;a;b],f(b)) in
F(C[q;a;b];f b)
| inr(z) =>
Ax ∈ let q,w',d = t in
case d of inl(b) => let a,f = w' in Q[C[q;a;b];f b] | inr(z) => True)
20. <a, w1> ∈ pW par
21. w1 ∈ b:B[par;a] ⟶ (pW C[par;a;b])
22. b : B[par;a]
⊢ <par, <a, w1>, inl b> ∈ {t:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])|
param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) n s t}
Latex:
Latex:
1. P : Type
2. A : P {}\mrightarrow{} Type
3. B : p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type
4. C : p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P
5. pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]) \mmember{} Type
6. pW \mmember{} P {}\mrightarrow{} Type
7. Q : par:P {}\mrightarrow{} (pW par) {}\mrightarrow{} \mBbbP{}
8. ind : par:P
{}\mrightarrow{} a:A[par]
{}\mrightarrow{} f:(b:B[par;a] {}\mrightarrow{} (pW C[par;a;b]))
{}\mrightarrow{} (b:B[par;a] {}\mrightarrow{} Q[C[par;a;b];f b])
{}\mrightarrow{} Q[par;pW-sup(a;f)]
9. par : P
10. a : A[par]
11. w1 : b:B[par;a] {}\mrightarrow{} (pco-W C[par;a;b])
12. \mforall{}path:Path. (StepAgree(path 0;par;<a, w1>) {}\mRightarrow{} (\mdownarrow{}\mexists{}n:\mBbbN{}. Barred(pcw-partial(path;n))))
13. param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) \mmember{} n:\mBbbN{}
{}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]))
{}\mrightarrow{} pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
{}\mrightarrow{} \mBbbP{}
14. \mforall{}[pp:n:\mBbbN{} \mtimes{} (\mBbbN{}n {}\mrightarrow{} pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b]))]. (Barred(pp) \mmember{} \mBbbP{})
15. n : \mBbbN{}
16. \mneg{}0 < n
17. s : \mBbbN{}n {}\mrightarrow{} pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
18. \mforall{}\%6:\mBbbN{}n. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) \%6 s (s \%6))
19. \mforall{}t:\{t:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])|
param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;<a, w1>) n s t\}
(let q,w',d = t in
case d
of inl(b) =>
let a,f = w'
in let ind(p,a,f,G) = ind[p;a;f;G] in
letrec F(p,w) = let a,f=w in
ind(p,a,f,\mlambda{}b.F(C[p;a;b],f(b)) in
F(C[q;a;b];f b)
| inr(z) =>
Ax \mmember{} let q,w',d = t in
case d of inl(b) => let a,f = w' in Q[C[q;a;b];f b] | inr(z) => True)
20. <a, w1> \mmember{} pW par
21. w1 \mmember{} b:B[par;a] {}\mrightarrow{} (pW C[par;a;b])
22. b : B[par;a]
\mvdash{} let ind(p,a,f,G) = ind p a f G in
letrec F(p,w) = let a,f=w in
ind(p,a,f,\mlambda{}b.F(C p a b,f(b)) in
F(C par a b;w1 b) \mmember{} Q[C[par;a;b];w1 b]
By
Latex:
TACTIC:(InstHyp [\mkleeneopen{}<par, <a, w1>, inl b>\mkleeneclose{}] (-4)\mcdot{} THENM (Reduce (-1) THEN Auto))
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