Step
*
of Lemma
divide-and-conquer
∀[Q:a:ℤ ⟶ {a...} ⟶ ℙ]
∀s:{2...}
((∀a:ℤ. ∀b:{a..a + s-}. Q[a;b])
⇒ (∀a,b,c:ℤ. (Q[a;c]
⇒ Q[a;b]) ∨ (Q[c;b]
⇒ Q[a;b]) supposing a < c ∧ c < b)
⇒ (∀a:ℤ. ∀b:{a...}. Q[a;b]))
BY
{ RepeatFor 4 ((D 0 THENA Auto)) }
1
1. [Q] : a:ℤ ⟶ {a...} ⟶ ℙ
2. s : {2...}
3. ∀a:ℤ. ∀b:{a..a + s-}. Q[a;b]
4. ∀a,b,c:ℤ. (Q[a;c]
⇒ Q[a;b]) ∨ (Q[c;b]
⇒ Q[a;b]) supposing a < c ∧ c < b
⊢ ∀a:ℤ. ∀b:{a...}. Q[a;b]
Latex:
Latex:
\mforall{}[Q:a:\mBbbZ{} {}\mrightarrow{} \{a...\} {}\mrightarrow{} \mBbbP{}]
\mforall{}s:\{2...\}
((\mforall{}a:\mBbbZ{}. \mforall{}b:\{a..a + s\msupminus{}\}. Q[a;b])
{}\mRightarrow{} (\mforall{}a,b,c:\mBbbZ{}. (Q[a;c] {}\mRightarrow{} Q[a;b]) \mvee{} (Q[c;b] {}\mRightarrow{} Q[a;b]) supposing a < c \mwedge{} c < b)
{}\mRightarrow{} (\mforall{}a:\mBbbZ{}. \mforall{}b:\{a...\}. Q[a;b]))
By
Latex:
RepeatFor 4 ((D 0 THENA Auto))
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