Step
*
2
2
2
1
2
1
2
1
1
of Lemma
iroot-lemma
1. a : ℕ
2. n : ℕ+
3. b : ℕ+
4. k : ℕ+
5. a + k ∈ ℕ+
6. M : ℕ+
7. a + k < M^n
8. y : ℕ+
9. (n * b * M^(n - 1)) ≤ (k * y)
10. (a + k) * y^n ∈ ℕ+
11. iroot(n;(a + k) * y^n) < M * y
12. iroot(n;(a + k) * y^n)^n ≤ ((a + k) * y^n)
13. (a + k) * y^n < (iroot(n;(a + k) * y^n) + 1)^n
14. (iroot(n;(a + k) * y^n) + 1)^n ≤ (((iroot(n;(a + k) * y^n) ÷ b) * b)^n + (n * b * (M * y)^(n - 1)))
⊢ (n * b * M^(n - 1) * y^(n - 1)) ≤ (k * y^n)
BY
{ (InstLemma `mul_preserves_le` [⌜n * b * M^(n - 1)⌝;⌜k * y⌝;⌜y^(n - 1)⌝]⋅ THEN Auto) }
1
1. a : ℕ
2. n : ℕ+
3. b : ℕ+
4. k : ℕ+
5. a + k ∈ ℕ+
6. M : ℕ+
7. a + k < M^n
8. y : ℕ+
9. (n * b * M^(n - 1)) ≤ (k * y)
10. (a + k) * y^n ∈ ℕ+
11. iroot(n;(a + k) * y^n) < M * y
12. iroot(n;(a + k) * y^n)^n ≤ ((a + k) * y^n)
13. (a + k) * y^n < (iroot(n;(a + k) * y^n) + 1)^n
14. (iroot(n;(a + k) * y^n) + 1)^n ≤ (((iroot(n;(a + k) * y^n) ÷ b) * b)^n + (n * b * (M * y)^(n - 1)))
15. (y^(n - 1) * n * b * M^(n - 1)) ≤ (y^(n - 1) * k * y)
⊢ (n * b * M^(n - 1) * y^(n - 1)) ≤ (k * y^n)
Latex:
Latex:
1. a : \mBbbN{}
2. n : \mBbbN{}\msupplus{}
3. b : \mBbbN{}\msupplus{}
4. k : \mBbbN{}\msupplus{}
5. a + k \mmember{} \mBbbN{}\msupplus{}
6. M : \mBbbN{}\msupplus{}
7. a + k < M\^{}n
8. y : \mBbbN{}\msupplus{}
9. (n * b * M\^{}(n - 1)) \mleq{} (k * y)
10. (a + k) * y\^{}n \mmember{} \mBbbN{}\msupplus{}
11. iroot(n;(a + k) * y\^{}n) < M * y
12. iroot(n;(a + k) * y\^{}n)\^{}n \mleq{} ((a + k) * y\^{}n)
13. (a + k) * y\^{}n < (iroot(n;(a + k) * y\^{}n) + 1)\^{}n
14. (iroot(n;(a + k) * y\^{}n) + 1)\^{}n \mleq{} (((iroot(n;(a + k) * y\^{}n) \mdiv{} b) * b)\^{}n
+ (n * b * (M * y)\^{}(n - 1)))
\mvdash{} (n * b * M\^{}(n - 1) * y\^{}(n - 1)) \mleq{} (k * y\^{}n)
By
Latex:
(InstLemma `mul\_preserves\_le` [\mkleeneopen{}n * b * M\^{}(n - 1)\mkleeneclose{};\mkleeneopen{}k * y\mkleeneclose{};\mkleeneopen{}y\^{}(n - 1)\mkleeneclose{}]\mcdot{} THEN Auto)
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