Merge pull request #193 from trishullab/george

optParam and notational improvements.

Author: GeorgeTsoukalas

lean4/src/putnam_1962_a2.lean

@@ -6,8 +6,10 @@ open MeasureTheory
 abbrev putnam_1962_a2_solution : Set (ℝ → ℝ) := sorry
 -- {f : ℝ → ℝ | ∃ a c : ℝ, a > 0 ∧ f = fun x => a / (1 - c * x)^2}
 theorem putnam_1962_a2
-(hf : ℝ → (ℝ → ℝ) → Prop := fun (e : ℝ) (f : ℝ → ℝ) => ∀ x ∈ Set.Ioo 0 e, (⨍ v in Set.Icc 0 x, f v) = Real.sqrt ((f 0) * (f x)))
-(hfinf : (ℝ → ℝ) → Prop := fun (f : ℝ → ℝ) => ∀ x > 0, (⨍ v in Set.Icc 0 x, f v) = Real.sqrt ((f 0) * (f x)))
+(hf : ℝ → (ℝ → ℝ) → Prop)
+(hf_def : hf = fun (e : ℝ) (f : ℝ → ℝ) => ∀ x ∈ Set.Ioo 0 e, (⨍ v in Set.Icc 0 x, f v) = Real.sqrt ((f 0) * (f x)))
+(hfinf : (ℝ → ℝ) → Prop)
+(hfinf_def : hfinf = fun (f : ℝ → ℝ) => ∀ x > 0, (⨍ v in Set.Icc 0 x, f v) = Real.sqrt ((f 0) * (f x)))
 : (∀ f : ℝ → ℝ, (hfinf f → ∃ g ∈ putnam_1962_a2_solution, ∀ x ≥ 0, g x = f x) ∧
 ∀ e > 0, hf e f → ∃ g ∈ putnam_1962_a2_solution, ∀ x ∈ Set.Ico 0 e, g x = f x) ∧
 ∀ f ∈ putnam_1962_a2_solution, hfinf f ∨ (∃ e > 0, hf e f) :=

lean4/src/putnam_1962_a4.lean

@@ -1,8 +1,6 @@
 import Mathlib
 open BigOperators
 
-open MeasureTheory
-
 theorem putnam_1962_a4
 (f : ℝ → ℝ)
 (a b : ℝ)

lean4/src/putnam_1962_a5.lean

@@ -1,8 +1,6 @@
 import Mathlib
 open BigOperators
 
-open MeasureTheory
-
 abbrev putnam_1962_a5_solution : ℕ → ℕ := sorry
 -- fun n : ℕ => n * (n + 1) * 2^(n - 2)
 theorem putnam_1962_a5

lean4/src/putnam_1962_a6.lean

@@ -1,8 +1,6 @@
 import Mathlib
 open BigOperators
 
-open MeasureTheory
-
 theorem putnam_1962_a6
 (S : Set ℚ)
 (hSadd : ∀ a ∈ S, ∀ b ∈ S, a + b ∈ S)

lean4/src/putnam_1962_b1.lean

@@ -1,8 +1,6 @@
 import Mathlib
 open BigOperators
 
-open MeasureTheory
-
 theorem putnam_1962_b1
 (p : ℕ → ℝ → ℝ)
 (x y : ℝ)

lean4/src/putnam_1962_b6.lean

@@ -1,14 +1,14 @@
 import Mathlib
 open BigOperators
 
-open MeasureTheory
+open MeasureTheory Real
 
 theorem putnam_1962_b6
-(π : ℝ := Real.pi)
 (n : ℕ)
 (a b : ℕ → ℝ)
 (xs : Set ℝ)
-(f : ℝ → ℝ := fun x : ℝ => ∑ k in Finset.Icc 0 n, ((a k) * Real.sin (k * x) + (b k) * Real.cos (k * x)))
+(f : ℝ → ℝ)
+(hf : f = fun x : ℝ => ∑ k in Finset.Icc 0 n, ((a k) * Real.sin (k * x) + (b k) * Real.cos (k * x)))
 (hf1 : ∀ x ∈ Set.Icc 0 (2 * π), |f x| ≤ 1)
 (hxs : xs.ncard = 2 * n ∧ xs ⊆ Set.Ico 0 (2 * π))
 (hfxs : ∀ x ∈ xs, |f x| = 1)

lean4/src/putnam_1963_a3.lean

@@ -8,10 +8,12 @@ noncomputable abbrev putnam_1963_a3_solution : (ℝ → ℝ) → ℕ → ℝ →
 theorem putnam_1963_a3
 (n : ℕ)
 (f : ℝ → ℝ)
-(P : ℕ → (ℝ → ℝ) → (ℝ → ℝ))
-(δ : (ℝ → ℝ) → (ℝ → ℝ) := fun g : ℝ → ℝ ↦ (fun x : ℝ ↦ x) * deriv g)
-(D : ℕ → (ℝ → ℝ) → (ℝ → ℝ) := fun (m : ℕ) (g : ℝ → ℝ) ↦ δ g - (fun x : ℝ ↦ (m : ℝ)) * g)
-(y : ℝ → ℝ := fun x : ℝ ↦ ∫ t in Set.Ioo 1 x, putnam_1963_a3_solution f n x t)
+(P D : ℕ → (ℝ → ℝ) → (ℝ → ℝ))
+(δ : (ℝ → ℝ) → (ℝ → ℝ))
+(hδ : δ = fun g : ℝ → ℝ ↦ (fun x : ℝ ↦ x) * deriv g)
+(hD : D = fun (m : ℕ) (g : ℝ → ℝ) ↦ δ g - (fun x : ℝ ↦ (m : ℝ)) * g)
+(y : ℝ → ℝ)
+(hy : y = fun x : ℝ ↦ ∫ t in Set.Ioo 1 x, putnam_1963_a3_solution f n x t)
 (hn : n ≥ 1)
 (hf : Continuous f)
 (hP : P 0 y = y ∧ ∀ m ∈ Finset.range n, P (m + 1) y = D (n - 1 - m) (P m y))

lean4/src/putnam_1963_a4.lean

@@ -4,7 +4,9 @@ open BigOperators
 open Topology Filter
 
 theorem putnam_1963_a4
-(apos : (ℕ → ℝ) → Prop := fun a => ∀ n, a n > 0)
-(f : (ℕ → ℝ) → ℕ → ℝ := fun a n => n * (((1 + a (n+1)) / (a n)) - 1))
+(apos : (ℕ → ℝ) → Prop)
+(hapos : apos = fun a => ∀ n, a n > 0)
+(f : (ℕ → ℝ) → ℕ → ℝ)
+(hf : f = fun (a : ℕ → ℝ) n => n * (((1 + a (n+1)) / (a n)) - 1))
 : (∀ a, apos a → limsup (f a) atTop ≥ 1) ∧ (¬∃ c > 1, ∀ a, apos a → limsup (f a) atTop ≥ c) :=
 sorry

lean4/src/putnam_1963_a6.lean

@@ -6,8 +6,10 @@ open Topology Filter
 theorem putnam_1963_a6
 (F1 F2 U V A B C D P Q : EuclideanSpace ℝ (Fin 2))
 (r : ℝ)
-(E : Set (Fin 2 → ℝ) := {H : EuclideanSpace ℝ (Fin 2) | dist F1 H + dist F2 H = r})
-(M : EuclideanSpace ℝ (Fin 2) := midpoint ℝ U V)
+(E : Set (EuclideanSpace ℝ (Fin 2)))
+(hE : E = {H : EuclideanSpace ℝ (Fin 2) | dist F1 H + dist F2 H = r})
+(M : EuclideanSpace ℝ (Fin 2))
+(hM : M = midpoint ℝ U V)
 (hr : r > dist F1 F2)
 (hUV : U ∈ E ∧ V ∈ E ∧ U ≠ V)
 (hAB : A ∈ E ∧ B ∈ E ∧ A ≠ B)

lean4/src/putnam_1963_b2.lean

@@ -6,7 +6,7 @@ open Topology Filter Polynomial
 abbrev putnam_1963_b2_solution : Prop := sorry
 -- True
 theorem putnam_1963_b2
-(S : Set ℝ := {2 ^ m * 3 ^ n | (m : ℤ) (n : ℤ)})
-(P : Set ℝ := Set.Ioi 0)
-: closure S ⊇ P ↔ putnam_1963_b2_solution :=
+(S : Set ℝ)
+(hS : S = {2 ^ m * 3 ^ n | (m : ℤ) (n : ℤ)})
+: closure S ⊇ Set.Ioi (0 : ℝ) ↔ putnam_1963_b2_solution :=
 sorry