Step
*
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. alpha : ℕ ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
14. ∀n:ℕ. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) n alpha (alpha n))
⊢ ↓∃n:ℕ. Barred(<n, alpha>)
BY
{ ((((RepUR ``param-W`` 10⋅ THEN D 10⋅) THEN Unhide) THEN Auto) THEN Fold `pcw-partial` 0 THEN BackThruSomeHyp) }
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. w : pco-W par
11. ∀path:Path. (StepAgree(path 0;par;w) ⇒ (↓∃n:ℕ. Barred(pcw-partial(path;n))))
12. 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
13. ∀[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)
14. alpha : ℕ ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
15. ∀n:ℕ. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) n alpha (alpha n))
⊢ alpha ∈ Path
2
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 : pco-W par
11. ∀path:Path. (StepAgree(path 0;par;w) ⇒ (↓∃n:ℕ. Barred(pcw-partial(path;n))))
12. 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
13. ∀[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)
14. alpha : ℕ ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
15. ∀n:ℕ. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) n alpha (alpha n))
⊢ StepAgree(alpha 0;par;w)
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. alpha : \mBbbN{} {}\mrightarrow{} pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])
14. \mforall{}n:\mBbbN{}. (param-W-rel(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];par;w) n alpha (alpha n))
\mvdash{} \mdownarrow{}\mexists{}n:\mBbbN{}. Barred(<n, alpha>)
By
Latex:
((((RepUR ``param-W`` 10\mcdot{} THEN D 10\mcdot{}) THEN Unhide) THEN Auto)
THEN Fold `pcw-partial` 0
THEN BackThruSomeHyp)
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