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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | import Mathlib
namespace OAI
section
noncomputable section
open Function
namespace CriticalPercolation.LinearGrowth
variable {R : Type*} [NormedRing R] [NormedAlgebra ℝ R]
lemma eq_zero_of_bounded_nat_multiples (x : R) (C : ℝ)
(h : ∀ n : ℕ, ‖(n:ℝ) • x‖≤C) : x=0 := by
by_contra hx
have hxpos : 0<‖x‖ := norm_pos_iff.mpr hx
obtain ⟨n,hn⟩ := exists_nat_gt (C/‖x‖)
have hh := h n
rw [norm_smul,Real.norm_natCast] at hh
exact ((div_lt_iff₀ hxpos).mp hn).not_ge hh
lemma pow_mul_of_affine_step (a b c : R) (hab : a*b=b+c) (hac : a*c=c) :
∀ n : ℕ, a^n*b=b+(n:ℝ) • c := by
intro n
induction n with
| zero => simp
| succ n ih =>
rw [pow_succ',mul_assoc,ih,mul_add,mul_smul_comm,hab,hac]
simp only [Nat.cast_add,Nat.cast_one,add_smul,one_smul]
abel
lemma eq_one_of_power_bounded_unipotent (a : R) (hni : IsNilpotent (a-1))
(C : ℝ) (hb : ∀ n : ℕ, ‖a^n‖≤C) : a=1 := by
let N := a-1
have hdrop : ∀ k : ℕ, N^k=0 → N=0 := by
intro k
induction k with
| zero =>
intro hz
have ho : (1:R)=0 := by simpa only [pow_zero] using hz
exact (mul_one N).symm.trans ((congrArg (N*·) ho).trans (mul_zero N))
| succ k ih =>
intro hz
by_cases hk : k=0
· simpa only [hk,zero_add,pow_one] using hz
have hp : 1≤k := Nat.one_le_iff_ne_zero.mpr hk
let b := N^(k-1)
let c := N^k
have hab : a*b=b+c := by
have he : a=1+N := by dsimp [N]; noncomm_ring
rw [he,add_mul,one_mul]
congr 1
dsimp [b,c]
rw [←pow_succ',Nat.sub_add_cancel hp]
have hac : a*c=c := by
have he : a=1+N := by dsimp [N]; noncomm_ring
rw [he,add_mul,one_mul]
have hn : N*c=0 := by simpa only [c,←pow_succ'] using hz
rw [hn,add_zero]
have he : c=0 := by
apply eq_zero_of_bounded_nat_multiples c ((C+1)*‖b‖)
intro n
have heq : (n:ℝ) • c=a^n*b-b := by
rw [pow_mul_of_affine_step a b c hab hac n,add_sub_cancel_left]
rw [heq]
calc
‖a^n*b-b‖ ≤ ‖a^n*b‖+‖b‖ := norm_sub_le _ _
_ ≤ C*‖b‖+‖b‖ := add_le_add ((norm_mul_le _ _).trans
(mul_le_mul_of_nonneg_right (hb n) (norm_nonneg _))) le_rfl
_ = (C+1)*‖b‖ := by ring
exact ih he
obtain ⟨k,hk⟩ := hni
exact sub_eq_zero.mp (hdrop k hk)
end CriticalPercolation.LinearGrowth
end
end
end OAI
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