Step
*
2
2
1
3
1
1
of Lemma
range_sublist
1. T : Type
2. u : T
3. v : T List
4. ∀n:ℕ. ∀f:ℕn ⟶ ℕ||v||.
∃L1:T List. ((||L1|| = n ∈ ℤ) c∧ (increasing(f;||L1||) ∧ (∀j:ℕ||L1||. (L1[j] = v[f j] ∈ T))))
supposing increasing(f;n)
5. n : ℕ
6. f : ℕn ⟶ ℕ||[u / v]||
7. increasing(f;n)
8. ¬(n = 0 ∈ ℤ)
9. (f 0) = 0 ∈ ℤ
10. L1 : T List
11. ||L1|| = (n - 1) ∈ ℤ
12. increasing(λi.((f (i + 1)) - 1);||L1||)
13. ∀j:ℕ||L1||. (L1[j] = v[(λi.((f (i + 1)) - 1)) j] ∈ T)
14. (||L1|| + 1) = n ∈ ℤ
15. increasing(f;||L1|| + 1)
16. j : ℕ||L1|| + 1
17. j = 0 ∈ ℤ
⊢ [u / L1][j] = [u / v][f j] ∈ T
BY
{ (HypSubstSq (-1) 0 THEN Reduce 0) }
1
1. T : Type
2. u : T
3. v : T List
4. ∀n:ℕ. ∀f:ℕn ⟶ ℕ||v||.
∃L1:T List. ((||L1|| = n ∈ ℤ) c∧ (increasing(f;||L1||) ∧ (∀j:ℕ||L1||. (L1[j] = v[f j] ∈ T))))
supposing increasing(f;n)
5. n : ℕ
6. f : ℕn ⟶ ℕ||[u / v]||
7. increasing(f;n)
8. ¬(n = 0 ∈ ℤ)
9. (f 0) = 0 ∈ ℤ
10. L1 : T List
11. ||L1|| = (n - 1) ∈ ℤ
12. increasing(λi.((f (i + 1)) - 1);||L1||)
13. ∀j:ℕ||L1||. (L1[j] = v[(λi.((f (i + 1)) - 1)) j] ∈ T)
14. (||L1|| + 1) = n ∈ ℤ
15. increasing(f;||L1|| + 1)
16. j : ℕ||L1|| + 1
17. j = 0 ∈ ℤ
⊢ u = [u / v][f 0] ∈ T
Latex:
Latex:
1. T : Type
2. u : T
3. v : T List
4. \mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}||v||.
\mexists{}L1:T List. ((||L1|| = n) c\mwedge{} (increasing(f;||L1||) \mwedge{} (\mforall{}j:\mBbbN{}||L1||. (L1[j] = v[f j]))))
supposing increasing(f;n)
5. n : \mBbbN{}
6. f : \mBbbN{}n {}\mrightarrow{} \mBbbN{}||[u / v]||
7. increasing(f;n)
8. \mneg{}(n = 0)
9. (f 0) = 0
10. L1 : T List
11. ||L1|| = (n - 1)
12. increasing(\mlambda{}i.((f (i + 1)) - 1);||L1||)
13. \mforall{}j:\mBbbN{}||L1||. (L1[j] = v[(\mlambda{}i.((f (i + 1)) - 1)) j])
14. (||L1|| + 1) = n
15. increasing(f;||L1|| + 1)
16. j : \mBbbN{}||L1|| + 1
17. j = 0
\mvdash{} [u / L1][j] = [u / v][f j]
By
Latex:
(HypSubstSq (-1) 0 THEN Reduce 0)
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