Step
*
1
2
2
2
1
of Lemma
div-search-lemma
1. aa : ℤ
2. bb : {aa + 1...}
3. f : ℤ ⟶ 𝔹
4. ∃x:{aa..bb-} [((∀y:{aa..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..bb + 1-}. (↑(f z))))]
5. a : ℤ
6. b : ℤ
7. c : ℤ
8. a < c
9. c < b
10. ¬↑(f c)
11. ∃x:{a..b-} [((∀y:{a..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..b + 1-}. (↑(f z))))]
12. x : {c..b-}
13. ∀y:{c..x + 1-}. (¬↑(f y))
14. ∀z:{x + 1..b + 1-}. (↑(f z))
15. y : {a..x + 1-}
16. ¬(c ≤ y)
⊢ ¬↑(f y)
BY
{ TACTIC:(ExRepD THEN Try (Complete (Auto))) }
1
1. aa : ℤ
2. bb : {aa + 1...}
3. f : ℤ ⟶ 𝔹
4. x2 : {aa..bb-}
5. ∀y:{aa..x2 + 1-}. (¬↑(f y))
6. ∀z:{x2 + 1..bb + 1-}. (↑(f z))
7. a : ℤ
8. b : ℤ
9. c : ℤ
10. a < c
11. c < b
12. ¬↑(f c)
13. x1 : {a..b-}
14. ∀y:{a..x1 + 1-}. (¬↑(f y))
15. ∀z:{x1 + 1..b + 1-}. (↑(f z))
16. x : {c..b-}
17. ∀y:{c..x + 1-}. (¬↑(f y))
18. ∀z:{x + 1..b + 1-}. (↑(f z))
19. y : {a..x + 1-}
20. ¬(c ≤ y)
⊢ ¬↑(f y)
Latex:
Latex:
1. aa : \mBbbZ{}
2. bb : \{aa + 1...\}
3. f : \mBbbZ{} {}\mrightarrow{} \mBbbB{}
4. \mexists{}x:\{aa..bb\msupminus{}\} [((\mforall{}y:\{aa..x + 1\msupminus{}\}. (\mneg{}\muparrow{}(f y))) \mwedge{} (\mforall{}z:\{x + 1..bb + 1\msupminus{}\}. (\muparrow{}(f z))))]
5. a : \mBbbZ{}
6. b : \mBbbZ{}
7. c : \mBbbZ{}
8. a < c
9. c < b
10. \mneg{}\muparrow{}(f c)
11. \mexists{}x:\{a..b\msupminus{}\} [((\mforall{}y:\{a..x + 1\msupminus{}\}. (\mneg{}\muparrow{}(f y))) \mwedge{} (\mforall{}z:\{x + 1..b + 1\msupminus{}\}. (\muparrow{}(f z))))]
12. x : \{c..b\msupminus{}\}
13. \mforall{}y:\{c..x + 1\msupminus{}\}. (\mneg{}\muparrow{}(f y))
14. \mforall{}z:\{x + 1..b + 1\msupminus{}\}. (\muparrow{}(f z))
15. y : \{a..x + 1\msupminus{}\}
16. \mneg{}(c \mleq{} y)
\mvdash{} \mneg{}\muparrow{}(f y)
By
Latex:
TACTIC:(ExRepD THEN Try (Complete (Auto)))
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