Fix 2012 A4 #255
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Fix 2012 A4 #255
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This was disprovable because of the `n > 2` condition in the goal.
For example if we take:
* $q = 101$
* $r = 1$
* $A = [0, 100]$
* $B = [1, 2]$
Then $S = \{1, 2\}$ and there is no three-or-more term arithmetic
progression whose intersection with $A$ is $S$.
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| (hT : T = {x : ℤ | ∃ b m : ℤ, (b : ℝ) ∈ Set.Icc (B 0) (B 1) ∧ x = b + m * q}) | ||
| (hS : S = {a : ℤ | (a : ℝ) ∈ Set.Icc (A 0) (A 1) ∧ r * a ∈ T}) | ||
| (prod_lt : (A 1 - A 0) * (B 1 - B 0) < q) : | ||
| IsFiniteAP {x | x ∈ S ∧ (x : ℝ) ∈ Set.Icc (A 0) (A 1)} := |
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This final goal line states that the A \cap S is an arithmetic progression, but the problem statement says S is A \cap X where X is a set of points of an (finite) AP. The latter implies the former but my feeling is that the goal as stated does not reflect the informal statement, what do you think?
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This was disprovable because of the
n > 2condition in the goal.For example if we take:
Then$S = \{1, 2\}$ and there is no three-or-more term arithmetic progression whose intersection with $A$ is $S$ .