`Compfiles/Bulgaria1998P8.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

Bulgarian Mathematical Olympiad 1998, Problem 8

The polynomials Pₙ(x,y) for n = 1, 2, ... are defined by P₁(x,y) = 1 and Pₙ₊₁(x,y) = (x + y - 1)(y + 1)Pₙ(x,y+2) + (y - y²)Pₙ(x,y) Prove that Pₙ(x,y) = Pₙ(y,x) for all x,y,n.

Imports (3)

Imported by 1 file

Compfiles.lean

def Bulgaria1998P8.P line 29 source
standard axioms: propext what this means
Statement uses: CommRing, Nat
2 instances, coercions and auxiliary constants left out.
def Bulgaria1998P8.S helper function Sₙ₋₁(x,y) = (x + y)² - 1(x + 1)Pₙ₋₁(y+2, x+2) line 38 source
Source comment
helper function Sₙ₋₁(x,y) = [(x + y)² - 1](y + 1)(x + 1)Pₙ₋₁(y+2, x+2).

L4DEL explanation
helper function Sₙ₋₁(x,y) = (x + y)² - 1(x + 1)Pₙ₋₁(y+2, x+2).
Sources and attribution: Pinned Lean source (47b19bbfe5f3)

standard axioms: propext what this means
Statement uses: CommRing, Nat
Proof uses: Bulgaria1998P8.S.match_1, HMul.hMul, HSub.hSub, HPow.hPow, HAdd.hAdd, OfNat.ofNat, Bulgaria1998P8.P
24 instances, coercions and auxiliary constants left out.
```
def S : ℕ → R → R → R
| n, x, y => ((x + y)^2 - 1) * (y + 1) * (x + 1) * P n (y + 2) (x + 2)

/- helper function
   Tₙ₋₁(x,y) = (y - y²)(x - x²)Pₙ₋₁(y, x).
-/
```
Mentions: Bulgaria1998P8.P
def Bulgaria1998P8.T helper function Tₙ₋₁(x,y) = (y - y²)(x - x²)Pₙ₋₁(y, x) line 44 source
Source comment
helper function Tₙ₋₁(x,y) = (y - y²)(x - x²)Pₙ₋₁(y, x).

standard axioms: propext what this means
Statement uses: CommRing, Nat
Proof uses: Bulgaria1998P8.S.match_1, HMul.hMul, HSub.hSub, HPow.hPow, OfNat.ofNat, Bulgaria1998P8.P
16 instances, coercions and auxiliary constants left out.
```
def T : ℕ → R → R → R
| n, x, y => (y - y^2) * (x - x^2) * P n y x

/- helper function
   Uₙ₋₁(x,y) = (x + y - 1) [(y + 1)(x - x²)Pₙ₋₁(y + 2, x)
                           + (x + 1)(y - y²) Pₙ₋₁(y, x + 2)]
-/
```
Mentions: Bulgaria1998P8.P
def Bulgaria1998P8.U helper function Uₙ₋₁(x,y) = (x + y - 1) [(y + 1)(x - x²)Pₙ₋₁(y + 2, x) + (x + 1)(y - y²) Pₙ₋₁(y, x + 2)] line 51 source
Source comment
helper function Uₙ₋₁(x,y) = (x + y - 1) [(y + 1)(x - x²)Pₙ₋₁(y + 2, x) + (x + 1)(y - y²) Pₙ₋₁(y, x + 2)]

standard axioms: propext what this means
Statement uses: CommRing, Nat
Proof uses: Bulgaria1998P8.S.match_1, HMul.hMul, HSub.hSub, HAdd.hAdd, OfNat.ofNat, HPow.hPow, Bulgaria1998P8.P
24 instances, coercions and auxiliary constants left out.
```
def U : ℕ → R → R → R
| n, x, y => (x + y - 1) *((y + 1)*(x - x^2) * P n (y + 2) x
                           + (x + 1) * (y - y^2) * P n y (x + 2))

snip end
```
Mentions: Bulgaria1998P8.P
problem Bulgaria1998P8.bulgaria1998_p8 line 57 source
standard axioms: Quot.sound, propext what this means
Statement uses: CommRing, Nat, Eq, Bulgaria1998P8.P
Proof uses: Nat.strong_induction_on, LT.lt, Nat.casesAuxOn, OfNat.ofNat, Eq.refl, Eq.symm, HAdd.hAdd, id, Mathlib.Tactic.Ring.of_eq, HMul.hMul, HSub.hSub, HPow.hPow, Int.rawCast, inferInstance, Ring, Int.negOfNat, Nat.rawCast, Mathlib.Tactic.Ring.Common.add_congr, Mathlib.Tactic.Ring.Common.mul_congr, Mathlib.Tactic.Ring.Common.sub_congr and 61 more
60 instances, coercions and auxiliary constants left out.

Tactic trace: 49 steps
Open to load the checked before-and-after goal states.