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you may not use this file except in compliance with the License.
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import FormalConjecturesUtilErdős Problem 1054
namespace Erdos1054
open Classical Filter AsymptoticsLet $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$.
noncomputable def f (n : ℕ) : ℕ :=
if h : ∃ᵉ (m) (k ≥ 1), n = ∑ i < k, Nat.nth (· ∈ m.divisors) i then
Nat.find h
else 0Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?
@[category research open, AMS 11]
theorem erdos_1054.parts.i : answer(sorry) ↔ (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := ⊢ True ↔ (fun n => ↑(f n)) =o[atTop] fun n => ↑n
All goals completed! 🐙Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$?
@[category research open, AMS 11]
theorem erdos_1054.parts.ii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧
(fun (n : A) ↦ (f ↑n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := ⊢ True ↔ ∃ A, A.HasDensity 1 ∧ (fun n => ↑(f ↑n)) =o[atTop] fun n => ↑↑n
All goals completed! 🐙Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$?
@[category research open, AMS 11]
theorem erdos_1054.parts.iii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧
atTop.limsup (fun n ↦ (f n : EReal) / n) = ⊤ := ⊢ True ↔ ∃ A, A.HasDensity 1 ∧ limsup (fun n => ↑(f n) / ↑n) atTop = ⊤
All goals completed! 🐙Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$.
@[category textbook, AMS 11]
theorem f_undefined_at_2 : f 2 = 0 := ⊢ f 2 = 0
⊢ ¬∃ m, ∃ k ≥ 1, 2 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) i
m:ℕk:ℕhk:k ≥ 1hsum:2 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) i⊢ False
k:ℕhk:k ≥ 1hsum:2 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ Nat.divisors 0) i⊢ Falsem:ℕk:ℕhk:k ≥ 1hsum:2 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) ihm:m ≠ 0⊢ False
k:ℕhk:k ≥ 1hsum:2 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ Nat.divisors 0) i⊢ False All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hsum:2 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) ihm:m ≠ 0⊢ False -- For `m ≠ 0` the smallest divisor is `1` and every later term is `≥ 2`, so the sum of the
-- `k` smallest divisors is `1` or `≥ 3`, never `2`.
have hk0 : (0 : ℕ) ∈ Finset.Iio k := Finset.mem_Iio.mpr (m:ℕk:ℕhk:k ≥ 1hsum:2 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) ihm:m ≠ 0⊢ 0 < k All goals completed! 🐙)
m:ℕk:ℕhk:k ≥ 1hsum:2 = 1 + ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x => x ∈ m.divisors) xhm:m ≠ 0hk0:0 ∈ Finset.Iio k⊢ False
obtain ⟨i, hi_mem, hi_ne⟩ :=
Finset.exists_ne_zero_of_sum_ne_zero (s := (Finset.Iio k).erase 0)
(f := Nat.nth (· ∈ m.divisors)) (m:ℕk:ℕhk:k ≥ 1hsum:2 = 1 + ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x => x ∈ m.divisors) xhm:m ≠ 0hk0:0 ∈ Finset.Iio k⊢ ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x => x ∈ m.divisors) x ≠ 0 All goals completed! 🐙)
m:ℕk:ℕhk:k ≥ 1hsum:2 = 1 + ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x => x ∈ m.divisors) xhm:m ≠ 0hk0:0 ∈ Finset.Iio ki:ℕhi_mem:i ∈ (Finset.Iio k).erase 0hi_ne:Nat.nth (fun x => x ∈ m.divisors) i ≠ 0h2:2 ≤ Nat.nth (fun x => x ∈ m.divisors) i := Nat.two_le_nth_divisors hm (Finset.ne_of_mem_erase hi_mem) hi_ne⊢ False
m:ℕk:ℕhk:k ≥ 1hsum:2 = 1 + ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x => x ∈ m.divisors) xhm:m ≠ 0hk0:0 ∈ Finset.Iio ki:ℕhi_mem:i ∈ (Finset.Iio k).erase 0hi_ne:Nat.nth (fun x => x ∈ m.divisors) i ≠ 0h2:2 ≤ Nat.nth (fun x => x ∈ m.divisors) i := Nat.two_le_nth_divisors hm (Finset.ne_of_mem_erase hi_mem) hi_nethis:2 ≤ ∑ x ∈ (Finset.Iio k).erase 0, Nat.nth (fun x => x ∈ m.divisors) x := LE.le.trans h2 (Finset.single_le_sum (fun j x => Nat.zero_le (Nat.nth (fun x => x ∈ m.divisors) j)) hi_mem)⊢ False
All goals completed! 🐙Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$.
@[category textbook, AMS 11]
theorem f_undefined_at_3 : f 5 = 0 := ⊢ f 5 = 0
⊢ ¬∃ m, ∃ k ≥ 1, 5 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) i
m:ℕk:ℕhk:k ≥ 1hsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) i⊢ False
k:ℕhk:k ≥ 1hsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ Nat.divisors 0) i⊢ Falsem:ℕk:ℕhk:k ≥ 1hsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) ihm:m ≠ 0⊢ False
k:ℕhk:k ≥ 1hsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ Nat.divisors 0) i⊢ False All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth (fun x => x ∈ m.divisors) ihm:m ≠ 0⊢ False m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rfl⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisors⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hm⊢ False
-- The `j`-th smallest divisor is at least `j + 1` (for `j` below the number of divisors).
have hlb : ∀ j, j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j := ⊢ f 5 = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmj:ℕ⊢ j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j
induction j with
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hm⊢ 0 < hfin.toFinset.card → 0 + 1 ≤ Nat.nth p 0 m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hma✝:0 < hfin.toFinset.card⊢ 0 + 1 ≤ Nat.nth p 0; All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmn:ℕih:n < hfin.toFinset.card → n + 1 ≤ Nat.nth p n⊢ n + 1 < hfin.toFinset.card → n + 1 + 1 ≤ Nat.nth p (n + 1)
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmn:ℕih:n < hfin.toFinset.card → n + 1 ≤ Nat.nth p nhj:n + 1 < hfin.toFinset.card⊢ n + 1 + 1 ≤ Nat.nth p (n + 1)
have h1 := Nat.nth_lt_nth_of_lt_card hfin (show n < n + 1 ⊢ f 5 = 0 All goals completed! 🐙)
(show n + 1 < hfin.toFinset.card ⊢ f 5 = 0 All goals completed! 🐙)
have h2 := ih (m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmn:ℕih:n < hfin.toFinset.card → n + 1 ≤ Nat.nth p nhj:n + 1 < hfin.toFinset.cardh1:Nat.nth p n < Nat.nth p (n + 1) :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this)⊢ n < hfin.toFinset.card All goals completed! 🐙)
All goals completed! 🐙
-- The second smallest divisor of `m` is never `4`: if `4 ∣ m` then `2 ∣ m`, so `2` would be
-- the second smallest divisor.
have refute4 : Nat.nth p 1 ≠ 4 := ⊢ f 5 = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4⊢ False
have hne : Nat.nth p 1 ≠ 0 := ⊢ f 5 = 0 m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4⊢ 4 ≠ 0; All goals completed! 🐙
have hcard1 : 1 < hfin.toFinset.card := ⊢ f 5 = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4hne:Nat.nth p 1 ≠ 0 :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false))hcon:¬1 < hfin.toFinset.card⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4hne:Nat.nth p 1 ≠ 0 :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false))hcon:hfin.toFinset.card ≤ 1⊢ False
All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4hne:Nat.nth p 1 ≠ 0 :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false))hcard1:1 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hmem:p (Nat.nth p 1) := Nat.nth_mem_of_lt_card hfin hcard1⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4hne:Nat.nth p 1 ≠ 0 :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false))hcard1:1 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hmem:p 4⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4hne:Nat.nth p 1 ≠ 0 :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false))hcard1:1 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hmem:p 4h4:4 ∣ m := (Nat.mem_divisors.mp hmem).left⊢ False
have h2d : (2 : ℕ) ∣ m := dvd_trans (m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4hne:Nat.nth p 1 ≠ 0 :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false))hcard1:1 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hmem:p 4h4:4 ∣ m := (Nat.mem_divisors.mp hmem).left⊢ 2 ∣ 4 All goals completed! 🐙) h4
have h2mem : p 2 := ⊢ f 5 = 0 All goals completed! 🐙
have hcount : Nat.count p 2 = 1 := ⊢ f 5 = 0
All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4hne:Nat.nth p 1 ≠ 0 :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false))hcard1:1 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hmem:p 4h4:4 ∣ m := (Nat.mem_divisors.mp hmem).lefth2d:2 ∣ m :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4h2mem:p 2 :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d))hcount:Nat.count p 2 = 1 :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0 (IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1))hnc:Nat.nth p (Nat.count p 2) = 2 := Nat.nth_count h2mem⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jh:Nat.nth p 1 = 4hne:Nat.nth p 1 ≠ 0 :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false))hcard1:1 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hmem:p 4h4:4 ∣ m := (Nat.mem_divisors.mp hmem).lefth2d:2 ∣ m :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4h2mem:p 2 :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d))hcount:Nat.count p 2 = 1 :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0 (IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1))hnc:4 = 2⊢ False
All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:k < 3⊢ Falsem:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ k⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:k < 3⊢ False -- `k = 1` or `k = 2`.
m:ℕk:ℕhm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk:1 ≥ 1hsum:5 = ∑ i ∈ Finset.Iio 1, Nat.nth p ihk3:1 < 3⊢ Falsem:ℕk:ℕhm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk:2 ≥ 1hsum:5 = ∑ i ∈ Finset.Iio 2, Nat.nth p ihk3:2 < 3⊢ False
m:ℕk:ℕhm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk:1 ≥ 1hsum:5 = ∑ i ∈ Finset.Iio 1, Nat.nth p ihk3:1 < 3⊢ False m:ℕk:ℕhm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk:1 ≥ 1hsum:5 = 1hk3:1 < 3⊢ False
All goals completed! 🐙
m:ℕk:ℕhm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk:2 ≥ 1hsum:5 = ∑ i ∈ Finset.Iio 2, Nat.nth p ihk3:2 < 3⊢ False m:ℕk:ℕhm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk:2 ≥ 1hsum:5 = 1 + Nat.nth p 1hk3:2 < 3⊢ False
exact refute4 (m:ℕk:ℕhm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk:2 ≥ 1hsum:5 = 1 + Nat.nth p 1hk3:2 < 3⊢ Nat.nth p 1 = 4 All goals completed! 🐙)
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ k⊢ False -- `k ≥ 3`.
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0⊢ Falsem:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0⊢ False -- At most two divisors are involved, so the sum equals `1 + Nat.nth p 1`.
have hz : ∀ i, 2 ≤ i → Nat.nth p i = 0 := ⊢ f 5 = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hp0':p 0right✝:2 = 0⊢ ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hf:(setOf p).Finitehcle:hf.toFinset.card ≤ 2⊢ ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hp0':p 0right✝:2 = 0⊢ ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 exact absurd hp0' (m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hp0':p 0right✝:2 = 0⊢ ¬p 0 All goals completed! 🐙)
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hf:(setOf p).Finitehcle:hf.toFinset.card ≤ 2⊢ ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 intro i m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hf:(setOf p).Finitehcle:hf.toFinset.card ≤ 2i:ℕhi:2 ≤ i⊢ Nat.nth p i = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hf:(setOf p).Finitehcle:hf.toFinset.card ≤ 2i:ℕhi:2 ≤ i⊢ hfin.toFinset.card ≤ i
have heq : hf.toFinset.card = hfin.toFinset.card := ⊢ f 5 = 0 All goals completed! 🐙
All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio 2, Nat.nth p ihpdef:p = fun x => x ∈ m.divisorshfin:(setOf p).Finitehg0:Nat.nth p 0 = 1hlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p jrefute4:Nat.nth p 1 ≠ 4hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0⊢ Falsem:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))⊢ Finset.Iio 2 ⊆ Finset.Iio km:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))⊢ ∀ x ∈ Finset.Iio k, x ∉ Finset.Iio 2 → Nat.nth p x = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio 2, Nat.nth p ihpdef:p = fun x => x ∈ m.divisorshfin:(setOf p).Finitehg0:Nat.nth p 0 = 1hlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p jrefute4:Nat.nth p 1 ≠ 4hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0⊢ False m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = 1 + Nat.nth p 1hpdef:p = fun x => x ∈ m.divisorshfin:(setOf p).Finitehg0:Nat.nth p 0 = 1hlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p jrefute4:Nat.nth p 1 ≠ 4hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0⊢ False
exact refute4 (m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = 1 + Nat.nth p 1hpdef:p = fun x => x ∈ m.divisorshfin:(setOf p).Finitehg0:Nat.nth p 0 = 1hlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p jrefute4:Nat.nth p 1 ≠ 4hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0⊢ Nat.nth p 1 = 4 All goals completed! 🐙)
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))⊢ Finset.Iio 2 ⊆ Finset.Iio k intro x m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))x:ℕhx:x ∈ Finset.Iio 2⊢ x ∈ Finset.Iio k; m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))x:ℕhx:x < 2⊢ x < k; All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))⊢ ∀ x ∈ Finset.Iio k, x ∉ Finset.Iio 2 → Nat.nth p x = 0 intro x m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))x:ℕhx:x ∈ Finset.Iio k⊢ x ∉ Finset.Iio 2 → Nat.nth p x = 0 m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))x:ℕhx:x ∈ Finset.Iio khx2:x ∉ Finset.Iio 2⊢ Nat.nth p x = 0; m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))x:ℕhx:x < khx2:¬x < 2⊢ Nat.nth p x = 0; exact hz x (m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 = 0hz:∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0 :=
Or.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) (Nat.nth_eq_zero.mp hg2)
(fun h =>
And.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hp0' right =>
absurd hp0'
(of_eq_true
(Eq.trans
(congrArg Not
(Eq.trans (congrFun (Eq.trans hpdef (funext fun x => Nat.mem_divisors._simp_1)) 0)
(Eq.trans (congrArg (fun x => x ∧ ¬m = 0) zero_dvd_iff._simp_1) and_not_self._simp_1)))
not_false_eq_true)))
fun h =>
Exists.casesOn (motive := fun x => ∀ (i : ℕ), 2 ≤ i → Nat.nth p i = 0) h fun hf hcle i hi =>
Nat.nth_eq_zero.mpr
(Or.inr
(Exists.intro hfin
(have heq := Eq.refl hf.toFinset.card;
Decidable.byContradiction fun a => f_undefined_at_3._proof_7 m k hfin hf hcle i hi a)))x:ℕhx:x < khx2:¬x < 2⊢ 2 ≤ x All goals completed! 🐙)
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0⊢ False -- Three distinct divisors `1 < d₁ < d₂` force the sum to be at least `6`.
have hc3 : 2 < hfin.toFinset.card := ⊢ f 5 = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hcon:¬2 < hfin.toFinset.card⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hcon:hfin.toFinset.card ≤ 2⊢ False
All goals completed! 🐙
have hg1 := hlb 1 (m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))⊢ 1 < hfin.toFinset.card All goals completed! 🐙)
have hg2' := hlb 2 (m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)⊢ 2 < hfin.toFinset.card All goals completed! 🐙)
have hsub : ∑ i ∈ Finset.Iio 3, Nat.nth p i ≤ ∑ i ∈ Finset.Iio k, Nat.nth p i := ⊢ f 5 = 0
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)hg2':2 + 1 ≤ Nat.nth p 2 := hlb 2 hc3⊢ Finset.Iio 3 ⊆ Finset.Iio km:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)hg2':2 + 1 ≤ Nat.nth p 2 := hlb 2 hc3⊢ ∀ i ∈ Finset.Iio k, i ∉ Finset.Iio 3 → 0 ≤ Nat.nth p i
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)hg2':2 + 1 ≤ Nat.nth p 2 := hlb 2 hc3⊢ Finset.Iio 3 ⊆ Finset.Iio k intro x m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)hg2':2 + 1 ≤ Nat.nth p 2 := hlb 2 hc3x:ℕhx:x ∈ Finset.Iio 3⊢ x ∈ Finset.Iio k; m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)hg2':2 + 1 ≤ Nat.nth p 2 := hlb 2 hc3x:ℕhx:x < 3⊢ x < k; All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)hg2':2 + 1 ≤ Nat.nth p 2 := hlb 2 hc3⊢ ∀ i ∈ Finset.Iio k, i ∉ Finset.Iio 3 → 0 ≤ Nat.nth p i m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)hg2':2 + 1 ≤ Nat.nth p 2 := hlb 2 hc3i✝:ℕa✝¹:i✝ ∈ Finset.Iio ka✝:i✝ ∉ Finset.Iio 3⊢ 0 ≤ Nat.nth p i✝; All goals completed! 🐙
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.Iio k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisors := rflhfin:(setOf p).Finite := Set.finite_mem_finset m.divisorshg0:Nat.nth p 0 = 1 := Nat.nth_divisors_zero hmhlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p j :=
fun j =>
Nat.recAux (motive := fun j => j < hfin.toFinset.card → j + 1 ≤ Nat.nth p j)
(fun a => Decidable.byContradiction fun a => f_undefined_at_3._proof_1 m k hfin hg0 a)
(fun n ih hj =>
have h1 :=
Nat.nth_lt_nth_of_lt_card hfin
(have this := Decidable.byContradiction fun a => f_undefined_at_3._proof_2 m k hfin n a;
this)
(have this := hj;
this);
have h2 := ih (Decidable.byContradiction fun a => f_undefined_at_3._proof_3 m k hfin n hj a);
Decidable.byContradiction fun a => f_undefined_at_3._proof_4 m k hfin n h1 h2 a)
jrefute4:Nat.nth p 1 ≠ 4 :=
fun h =>
have hne :=
Eq.mpr (id (congrArg (fun _a => _a ≠ 0) h))
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 0)) (Eq.refl false));
have hcard1 :=
Decidable.byContradiction fun hcon =>
hne (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))));
have hmem := Nat.nth_mem_of_lt_card hfin hcard1;
have h4 := (Nat.mem_divisors.mp (Eq.mp (congrArg (fun _a => p _a) h) hmem)).left;
have h2d :=
dvd_trans
(Mathlib.Meta.NormNum.isNat_dvd_true (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4)) (Eq.refl 0))
h4;
have h2mem :=
of_eq_true
(Eq.trans
(congrFun
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2)
(eq_true h2d));
have hcount :=
of_eq_true
(Eq.trans
(congrArg (fun x => x = 1)
(Eq.trans
(Eq.trans
(Nat.count.congr_simp p (fun x => x ∣ m)
(Eq.trans hpdef
(funext fun x =>
Eq.trans Nat.mem_divisors._simp_1
(Eq.trans (congrArg (And (x ∣ m)) (Eq.trans (congrArg Not (eq_false hm)) not_false_eq_true))
(and_true (x ∣ m)))))
2 2 (Eq.refl 2))
(Nat.count_succ (fun x => x ∣ m) 1))
(Eq.trans
(congr
(congrArg HAdd.hAdd
(Eq.trans (Nat.count_succ (fun x => x ∣ m) 0)
(Eq.trans
(congr (congrArg HAdd.hAdd (Nat.count_zero fun x => x ∣ m))
(ite_cond_eq_false 1 0 (Eq.trans zero_dvd_iff._simp_1 (eq_false hm))))
(add_zero 0))))
(ite_cond_eq_true 1 0
(IsUnit.dvd._simp_1 (of_eq_true (Eq.trans isUnit_iff_eq_one._simp_1 (eq_self 1))))))
(zero_add 1))))
(eq_self 1));
have hnc := Nat.nth_count h2mem;
False.elim
(Eq.mp
(eq_false
(Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 4))
(Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 2)) (Eq.refl false)))
(Eq.mp (congrArg (fun _a => _a = 2) h) (Eq.mp (congrArg (fun _a => Nat.nth p _a = 2) hcount) hnc)))hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.card := Decidable.byContradiction fun hcon => hg2 (Nat.nth_eq_zero.mpr (Or.inr (Exists.intro hfin (Eq.mp not_lt._simp_1 hcon))))hg1:1 + 1 ≤ Nat.nth p 1 := hlb 1 (Decidable.byContradiction fun a => f_undefined_at_3._proof_11 m k hfin hc3 a)hg2':2 + 1 ≤ Nat.nth p 2 := hlb 2 hc3hsub:1 + Nat.nth p 1 + Nat.nth p 2 ≤ ∑ i ∈ Finset.range k, Nat.nth p i⊢ False
m:ℕk:ℕhk:k ≥ 1hm:m ≠ 0p:ℕ → Prop := fun x => x ∈ m.divisorshsum:5 = ∑ i ∈ Finset.range k, Nat.nth p ihpdef:p = fun x => x ∈ m.divisorshfin:(setOf p).Finitehg0:Nat.nth p 0 = 1hlb:∀ j < hfin.toFinset.card, j + 1 ≤ Nat.nth p jrefute4:Nat.nth p 1 ≠ 4hk3:3 ≤ khg2:Nat.nth p 2 ≠ 0hc3:2 < hfin.toFinset.cardhg1:1 + 1 ≤ Nat.nth p 1hg2':2 + 1 ≤ Nat.nth p 2hsub:1 + Nat.nth p 1 + Nat.nth p 2 ≤ ∑ i ∈ Finset.range k, Nat.nth p i⊢ False
All goals completed! 🐙
end Erdos1054