`Compfiles/Usa1987P3.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1987, Problem 3

X is the smallest set of polynomials p(x) such that: (1) p(x) = x belongs to X; and (2) if r(x) belongs to X, then x·r(x) and (x + (1 - x)·r(x)) both belong to X.

Show that if r(x) and s(x) are distinct elements of X, then r(x) ≠ s(x) for any 0 < x < 1.

Imports (3)

Imported by 1 file

Compfiles.lean

inductive Usa1987P3.InX Documented declaration: InX line 37 source
Source documentation
The set X of the problem, as an inductive predicate: it is the smallest set of polynomials that contains the identity polynomial X and is closed under the two operations r ↦ X * r and r ↦ X + (1 - X) * r.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real
1 instances, coercions and auxiliary constants left out.
```
inductive InX : Polynomial ℝ → Prop
  | base : InX X
  | mul {r : Polynomial ℝ} (hr : InX r) : InX (X * r)
  | comb {r : Polynomial ℝ} (hr : InX r) : InX (X + (1 - X) * r)

/-- Every polynomial of `X` maps the open interval `(0, 1)` into itself. -/
```
Mentions: base
lemma Usa1987P3.InX.eval_mem_Ioo Every polynomial of X maps the open interval (0, 1) into itself line 43 source
Source documentation
Every polynomial of X maps the open interval (0, 1) into itself.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real, Usa1987P3.InX, LT.lt, OfNat.ofNat, And, Polynomial.eval
Proof uses: Usa1987P3.InX.rec, Eq.mpr, Polynomial.X, id, Eq, congrArg, Polynomial.eval_X, And.intro, HMul.hMul, congr, Eq.trans, Polynomial.eval_mul, congrFun', mul_pos, And.left, LT.lt.trans, LT.lt.trans_eq, mul_lt_mul_of_pos_left, And.right, mul_one and 11 more
79 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma InX.eval_mem_Ioo {r : Polynomial ℝ} (hr : InX r)
    {x : ℝ} (hx0 : 0 < x) (hx1 : x < 1) :
    0 < r.eval x ∧ r.eval x < 1 := by
…
```
Mentions: Usa1987P3.InX
lemma Usa1987P3.InX.trichotomy The key trichotomy: two elements of X are either equal as polynomials, or one of them is strictly smaller than the other throughout the whole open interval (0, 1) line 67 source
Source documentation
The key trichotomy: two elements of X are either equal as polynomials, or one of them is strictly smaller than the other throughout the whole open interval (0, 1). The proof goes by induction on the construction of r and case analysis on the construction of s; it is a form of lexicographic comparison of the operation sequences (cf. the informal solution, where + counts as larger than "no further operation", which in turn counts as larger than -).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real, Usa1987P3.InX, Or, Eq, LT.lt, OfNat.ofNat, Polynomial.eval
Proof uses: Usa1987P3.InX.rec, HEq, Polynomial.X, Usa1987P3.InX.base, Or.inl, rfl, Eq.symm, HMul.hMul, Usa1987P3.InX.mul, Or.inr, Eq.mpr, id, congr, congrArg, Eq.trans, Polynomial.eval_mul, congrFun', Polynomial.eval_X, LT.lt.trans_eq, mul_lt_mul_of_pos_left and 19 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 54 steps
Open to load the checked before-and-after goal states.
```
lemma InX.trichotomy {r : Polynomial ℝ} (hr : InX r) :
    ∀ {s : Polynomial ℝ}, InX s →
      r = s ∨ (∀ x : ℝ, 0 < x → x < 1 → r.eval x < s.eval x) ∨
              (∀ x : ℝ, 0 < x → x < 1 → s.eval x < r.eval x) := by
…
```
Mentions: Usa1987P3.InX
problem Usa1987P3.usa1987_p3 line 135 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Polynomial, Real, Membership.mem, Polynomial.X, HMul.hMul, HAdd.hAdd, HSub.hSub, OfNat.ofNat, LE.le, Ne, LT.lt, Polynomial.eval
Proof uses: Set.ofPred, Usa1987P3.InX, Usa1987P3.InX.base, Usa1987P3.InX.mul, Usa1987P3.InX.comb, Eq, Or, Usa1987P3.InX.trichotomy, absurd, ne_of_lt, ne_of_gt
34 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.