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Fix 2012 A4 #255

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24 changes: 14 additions & 10 deletions lean4/src/putnam_2012_a4.lean
Original file line number Diff line number Diff line change
Expand Up @@ -6,13 +6,17 @@ open Matrix Function
Let $q$ and $r$ be integers with $q>0$, and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B$, and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T$. Show that if the product of the lengths of $A$ and $B$ is less than $q$, then $S$ is the intersection of $A$ with some arithmetic progression.
-/
theorem putnam_2012_a4
(q r : ℤ)
(A B : Fin 2 → ℝ)
(T : Set ℝ)
(S : Set ℤ)
(qpos : q > 0)
(ABlt : A 0 < A 1 ∧ B 0 < B 1)
(hT : T = {x : ℝ | ∃ 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)) ∧ (∃ t ∈ T, r * a = t)})
: ((A 1 - A 0) * (B 1 - B 0) < q) → (∃ n : ℕ, ∃ a1 d : ℝ, n > 2 ∧ {s : ℝ | s = round s ∧ round s ∈ S} = (Set.Icc (A 0) (A 1)) ∩ {x : ℝ | ∃ i : Fin n, x = a1 + i * d}) :=
sorry
(IsFiniteAP : Set ℤ → Prop)
(IsFiniteAP_def : ∀ s,
IsFiniteAP s ↔ ∃ n : ℕ, ∃ a d : ℤ, 0 < d ∧ s = {a + i * d | i : Fin n})
(q r : ℤ)
(A B : Fin 2 → ℝ)
(T : Set ℤ)
(S : Set ℤ)
(qpos : q > 0)
(ABlt : A 0 < A 1 ∧ B 0 < B 1)
(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?

sorry
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