Step
*
3
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. w : pW par
11. param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) ∈ 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
12. ∀[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)
13. n : ℕ
14. s : ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
15. ∀%6:ℕn. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) %6 s (s %6))
16. ∀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;w) 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,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)
17. 0 < n
18. p : P
19. w1 : pco-W p
20. x : B[p;fst(w1)]
21. (s (n - 1)) = <p, w1, inl x> ∈ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
⊢ let a,f = w1
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[p;a;x];f x) ∈ let a,f = w1
in Q[C[p;a;x];f x]
BY
{ TACTIC:(pcoWD (-3)
THEN D -3
THEN Reduce 0
THEN (InstLemma `param-co-W_wf` [⌜P⌝;⌜A⌝;⌜B⌝;⌜C⌝]⋅ THENA Auto)
THEN (GenConclTerm ⌜C[p;a;x]⌝⋅ THENA Auto)
THEN (GenConcl ⌜(w2 x) = z ∈ (pco-W v)⌝⋅ THENA Auto)
THEN pcoWD (-2)
THEN D -2
THEN RecUnfold `pW-rec` 0
THEN Reduce 0
THEN Fold `pW-sup` 0
THEN RepUR ``so_apply`` 0) }
1
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. w : pW par
11. param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) ∈ 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
12. ∀[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)
13. n : ℕ
14. s : ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
15. ∀%6:ℕn. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) %6 s (s %6))
16. ∀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;w) 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,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)
17. 0 < n
18. p : P
19. a : A[p]
20. w2 : b:B[p;a] ⟶ (pco-W C[p;a;b])
21. x : B[p;fst(<a, w2>)]
22. (s (n - 1)) = <p, <a, w2>, inl x> ∈ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
23. pco-W ∈ P ⟶ Type
24. v : P
25. C[p;a;x] = v ∈ P
26. a1 : A[v]
27. z1 : b:B[v;a1] ⟶ (pco-W C[v;a1;b])
28. (w2 x) = <a1, z1> ∈ (pco-W v)
⊢ ind v a1 z1
(λb.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 v a1 b;z1 b)) ∈ Q v pW-sup(a1;z1)
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. w : pW par
11. param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) \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{}
12. \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{})
13. n : \mBbbN{}
14. s : \mBbbN{}n {}\mrightarrow{} pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
15. \mforall{}\%6:\mBbbN{}n. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) \%6 s (s \%6))
16. \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;w) 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)
17. 0 < n
18. p : P
19. w1 : pco-W p
20. x : B[p;fst(w1)]
21. (s (n - 1)) = <p, w1, inl x>
\mvdash{} let a,f = w1
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[p;a;x];f x) \mmember{} let a,f = w1
in Q[C[p;a;x];f x]
By
Latex:
TACTIC:(pcoWD (-3)
THEN D -3
THEN Reduce 0
THEN (InstLemma `param-co-W\_wf` [\mkleeneopen{}P\mkleeneclose{};\mkleeneopen{}A\mkleeneclose{};\mkleeneopen{}B\mkleeneclose{};\mkleeneopen{}C\mkleeneclose{}]\mcdot{} THENA Auto)
THEN (GenConclTerm \mkleeneopen{}C[p;a;x]\mkleeneclose{}\mcdot{} THENA Auto)
THEN (GenConcl \mkleeneopen{}(w2 x) = z\mkleeneclose{}\mcdot{} THENA Auto)
THEN pcoWD (-2)
THEN D -2
THEN RecUnfold `pW-rec` 0
THEN Reduce 0
THEN Fold `pW-sup` 0
THEN RepUR ``so\_apply`` 0)
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