Step
*
2
2
2
2
1
of Lemma
vdf-wf+
.....subterm..... T:t
1:n
1. A : Type
2. B : Type
3. C : A ⟶ B ⟶ Type
4. n : ℤ
5. 0 < n
6. vdf(A;B;a,b.C[a;b];n - 1) ∈ Type
7. ∀f:vdf(A;B;a,b.C[a;b];n - 1). ∀L:(a:A × b:B × C[a;b]) List.
((||L|| ≤ ((n - 1) + 1))
⇒ ((vdf-eq(A;f;L) ∈ ℙ) ∧ (vdf-eq(A;f;L) ⊆r (∀[i:ℕ||L||]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))) ∈ A)))))
8. vdf(A;B;a,b.C[a;b];n) ∈ Type
9. f : f:vdf(A;B;a,b.C[a;b];n - 1) ⋂ {L:(a:A × b:B × C[a;b]) List| (||L|| ≤ n) ∧ vdf-eq(A;f;L)} ⟶ B ⟶ A
10. f ∈ vdf(A;B;a,b.C[a;b];n - 1)
11. f ∈ {L:(a:A × b:B × C[a;b]) List| (||L|| ≤ n) ∧ vdf-eq(A;f;L)} ⟶ B ⟶ A
12. L : (a:A × b:B × C[a;b]) List
13. ||L|| ≤ (n + 1)
14. ∀m:ℕ
(m < ||L||
⇒ (dep-all(m;i.let a,b,c = firstn(||L|| - 1;L)[i] in
a = (f firstn(i;firstn(||L|| - 1;L)) b) ∈ A) ~ dep-all(m;i.let a,b,c = L[i] in
a = (f firstn(i;L) b) ∈ A)))
15. ¬(||L|| ≤ n)
16. ¬||L|| < 1
17. vdf-eq(A;f;L) ∈ ℙ
18. x : vdf-eq(A;f;L)
⊢ x ∈ ∀[i:ℕ||L||]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))) ∈ A)
BY
{ ((InstHyp [⌜f⌝;⌜firstn(||L|| - 1;L)⌝] 7⋅ THENA Auto) THEN D -1 THEN (D -1 THENA Auto)) }
1
1. A : Type
2. B : Type
3. C : A ⟶ B ⟶ Type
4. n : ℤ
5. 0 < n
6. vdf(A;B;a,b.C[a;b];n - 1) ∈ Type
7. ∀f:vdf(A;B;a,b.C[a;b];n - 1). ∀L:(a:A × b:B × C[a;b]) List.
((||L|| ≤ ((n - 1) + 1))
⇒ ((vdf-eq(A;f;L) ∈ ℙ) ∧ (vdf-eq(A;f;L) ⊆r (∀[i:ℕ||L||]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))) ∈ A)))))
8. vdf(A;B;a,b.C[a;b];n) ∈ Type
9. f : f:vdf(A;B;a,b.C[a;b];n - 1) ⋂ {L:(a:A × b:B × C[a;b]) List| (||L|| ≤ n) ∧ vdf-eq(A;f;L)} ⟶ B ⟶ A
10. f ∈ vdf(A;B;a,b.C[a;b];n - 1)
11. f ∈ {L:(a:A × b:B × C[a;b]) List| (||L|| ≤ n) ∧ vdf-eq(A;f;L)} ⟶ B ⟶ A
12. L : (a:A × b:B × C[a;b]) List
13. ||L|| ≤ (n + 1)
14. ∀m:ℕ
(m < ||L||
⇒ (dep-all(m;i.let a,b,c = firstn(||L|| - 1;L)[i] in
a = (f firstn(i;firstn(||L|| - 1;L)) b) ∈ A) ~ dep-all(m;i.let a,b,c = L[i] in
a = (f firstn(i;L) b) ∈ A)))
15. ¬(||L|| ≤ n)
16. ¬||L|| < 1
17. vdf-eq(A;f;L) ∈ ℙ
18. x : vdf-eq(A;f;L)
19. vdf-eq(A;f;firstn(||L|| - 1;L)) ∈ ℙ
20. vdf-eq(A;f;firstn(||L|| - 1;L)) ⊆r (∀[i:ℕ||firstn(||L|| - 1;L)||]
((fst(firstn(||L|| - 1;L)[i]))
= (f firstn(i;firstn(||L|| - 1;L)) (fst(snd(firstn(||L|| - 1;L)[i]))))
∈ A))
⊢ x ∈ ∀[i:ℕ||L||]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))) ∈ A)
Latex:
Latex:
.....subterm..... T:t
1:n
1. A : Type
2. B : Type
3. C : A {}\mrightarrow{} B {}\mrightarrow{} Type
4. n : \mBbbZ{}
5. 0 < n
6. vdf(A;B;a,b.C[a;b];n - 1) \mmember{} Type
7. \mforall{}f:vdf(A;B;a,b.C[a;b];n - 1). \mforall{}L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List.
((||L|| \mleq{} ((n - 1) + 1))
{}\mRightarrow{} ((vdf-eq(A;f;L) \mmember{} \mBbbP{})
\mwedge{} (vdf-eq(A;f;L) \msubseteq{}r (\mforall{}[i:\mBbbN{}||L||]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))))))))
8. vdf(A;B;a,b.C[a;b];n) \mmember{} Type
9. f : f:vdf(A;B;a,b.C[a;b];n - 1) \mcap{} \{L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List| (||L|| \mleq{} n) \mwedge{} vdf-eq(A;f;L)\}
{}\mrightarrow{} B
{}\mrightarrow{} A
10. f \mmember{} vdf(A;B;a,b.C[a;b];n - 1)
11. f \mmember{} \{L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List| (||L|| \mleq{} n) \mwedge{} vdf-eq(A;f;L)\} {}\mrightarrow{} B {}\mrightarrow{} A
12. L : (a:A \mtimes{} b:B \mtimes{} C[a;b]) List
13. ||L|| \mleq{} (n + 1)
14. \mforall{}m:\mBbbN{}
(m < ||L||
{}\mRightarrow{} (dep-all(m;i.let a,b,c = firstn(||L|| - 1;L)[i] in
a = (f firstn(i;firstn(||L|| - 1;L)) b)) \msim{} dep-all(m;i.let a,b,c = L[i] in
a = (f firstn(i;L) b))))
15. \mneg{}(||L|| \mleq{} n)
16. \mneg{}||L|| < 1
17. vdf-eq(A;f;L) \mmember{} \mBbbP{}
18. x : vdf-eq(A;f;L)
\mvdash{} x \mmember{} \mforall{}[i:\mBbbN{}||L||]. ((fst(L[i])) = (f firstn(i;L) (fst(snd(L[i])))))
By
Latex:
((InstHyp [\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}firstn(||L|| - 1;L)\mkleeneclose{}] 7\mcdot{} THENA Auto) THEN D -1 THEN (D -1 THENA Auto))
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