File size: 3,274 Bytes
f1f7953 | 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 80 81 82 83 84 85 86 87 88 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
namespace OAI
section
namespace Pureness
namespace OrderedSemigroup
variable {S : Type*} [AddCommMonoid S] [PartialOrder S] [IsOrderedAddMonoid S]
theorem iterate_with_buffer {a b c : S} (h : a + c ≤ b + c) (n : ℕ) :
n • a + c ≤ n • b + c := by
induction n with
| zero => simp
| succ n ih =>
calc
(n+1) • a + c = a + (n • a + c) := by simp [succ_nsmul, add_comm, add_left_comm]
_ ≤ a + (n • b + c) := add_le_add le_rfl ih
_ = n • b + (a + c) := by ac_rfl
_ ≤ n • b + (b + c) := add_le_add le_rfl h
_ = (n+1) • b + c := by simp [succ_nsmul, add_assoc]
theorem strict_multiples_le
(hzero : ∀ a : S, 0 ≤ a)
(hAU : ∀ a b : S, ∀ k : ℕ, 1 ≤ k → (k+1) • a ≤ k • b → a ≤ b)
{a b : S} {m n : ℕ} (hmn : m < n) (h : n • a ≤ m • b) : a ≤ b := by
by_cases hm : m = 0
· subst m
have ha : a ≤ n • a := by
simpa only [one_nsmul] using nsmul_le_nsmul_left (hzero a) (show 1 ≤ n by omega)
exact ha.trans (by simpa using h) |>.trans (hzero b)
· apply hAU a b m (by omega)
exact (nsmul_le_nsmul_left (hzero a) (show m+1 ≤ n by omega)).trans h
theorem strict_multiples_le_of_buffer
(hzero : ∀ a : S, 0 ≤ a)
(hAU : ∀ a b : S, ∀ k : ℕ, 1 ≤ k → (k+1) • a ≤ k • b → a ≤ b)
{a b c : S} {m n k : ℕ} (hmn : m < n)
(h : n • a + c ≤ m • b + c) (hc : c ≤ k • b) : a ≤ b := by
have hi := iterate_with_buffer h (k+1)
have hcomp : ((k+1)*n) • a ≤ ((k+1)*m+k) • b := by
calc
((k+1)*n) • a ≤ ((k+1)*n) • a + c := le_add_of_nonneg_right (hzero c)
_ ≤ ((k+1)*m) • b + c := by simpa only [← mul_nsmul, Nat.mul_comm] using hi
_ ≤ ((k+1)*m) • b + k • b := add_le_add le_rfl hc
_ = ((k+1)*m+k) • b := (add_nsmul _ _ _).symm
apply strict_multiples_le hzero hAU (m := (k+1)*m+k) (n := (k+1)*n) _ hcomp
nlinarith
theorem properlyInfinite_of_multiple
(hzero : ∀ a : S, 0 ≤ a)
(hAU : ∀ a b : S, ∀ k : ℕ, 1 ≤ k → (k+1) • a ≤ k • b → a ≤ b)
{u : S} {k : ℕ} (hk : 0 < k) (h : (2*k) • u ≤ k • u) : 2 • u ≤ u := by
have hstep : (k+1) • u ≤ k • u :=
(nsmul_le_nsmul_left (hzero u) (by omega : k+1 ≤ 2*k)).trans h
have hi : ∀ j : ℕ, (k+j) • u ≤ k • u := by
intro j
induction j with
| zero => simp
| succ j ih =>
calc
(k+(j+1)) • u = (k+j) • u + u := by simp [add_nsmul, add_assoc]
_ ≤ k • u + u := add_le_add ih le_rfl
_ = (k+1) • u := by simp [add_nsmul]
_ ≤ k • u := hstep
apply hAU (2 • u) u k (by omega)
simpa only [← mul_nsmul, show 2*(k+1) = k+(k+2) by omega] using hi (k+2)
theorem le_of_properlyInfinite {u a : S} (hu : 2 • u ≤ u)
(ha : ∃ n : ℕ, a ≤ n • u) (hzero : 0 ≤ u) : a ≤ u := by
obtain ⟨n, hn⟩ := ha
apply hn.trans
clear hn
induction n with
| zero => simpa using hzero
| succ n ih =>
calc
(n+1) • u = n • u + u := by simp [add_nsmul]
_ ≤ u + u := add_le_add ih le_rfl
_ ≤ u := by simpa [two_nsmul] using hu
end OrderedSemigroup
end Pureness
end
end OAI
|