| import OAI.Dynamics.StandardMap.ArrayTests |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open Set Filter |
| open scoped Topology |
| namespace StandardMapEntropy |
| lemma dyadic_zsmul_val (i : ℤ) (h : DyadicTime) : ((i • h : DyadicTime):ℝ)=(i:ℝ)*(h:ℝ) := by |
| simp only [AddSubgroup.coe_zsmul,zsmul_eq_mul] |
| lemma dyadic_mid_grid (h : DyadicTime) (i : ℤ) : dyadicMid (i • h) ((i+2) • h)=(i+1) • h := by |
| apply Subtype.ext |
| simp only [dyadicMid_val,dyadic_zsmul_val,Int.cast_add,Int.cast_one,Int.cast_ofNat] |
| ring |
| lemma int_constant_of_succ (f : ℤ → ℝ) (hf : ∀ i,f (i+1)=f i) (i : ℤ) : f i=f 0 := by |
| induction i using Int.induction_on with |
| | zero => rfl |
| | succ n ih => rw [hf,ih] |
| | pred n ih => rw [← hf (- (n:ℤ)-1),sub_add_cancel,ih] |
| lemma array_grid_edges (d : DistanceArray) |
| (hz : ∀ s t : DyadicTime, (s:ℝ)<(t:ℝ) → arrayJ s t d+arrayV s t d=0) |
| (h : DyadicTime) (hh : 0<(h:ℝ)) (i : ℤ) : |
| d.val (i • h) ((i+1) • h)=d.val 0 h := by |
| let f : ℤ → ℝ := fun i => d.val (i • h) ((i+1) • h) |
| have hf (i : ℤ) : f (i+1)=f i := by |
| have hi : ((i • h : DyadicTime):ℝ)<(((i+2) • h : DyadicTime):ℝ) := by |
| simp only [dyadic_zsmul_val,Int.cast_add,Int.cast_ofNat]; nlinarith |
| obtain ⟨hL,hR⟩ := array_zero_test_halves (i • h) ((i+2) • h) hi d (hz _ _ hi) |
| rw [dyadic_mid_grid] at hL hR |
| dsimp only [f] |
| simpa only [show i+1+1=i+2 by ring] using hR.trans hL.symm |
| have he := int_constant_of_succ f hf i |
| simpa only [f,zero_smul,zero_add,one_zsmul] using he |
| lemma array_grid_upper (d : DistanceArray) (h : DyadicTime) (c : ℝ) |
| (he : ∀ i : ℤ,d.val (i • h) ((i+1) • h)=c) (i : ℤ) (n : ℕ) : |
| d.val (i • h) ((i+(n:ℤ)) • h) ≤ (n:ℝ)*c := by |
| induction n with |
| | zero => simpa using (show d.val (i • h) (i • h) ≤ 0 by rw [d.property.2.1]) |
| | succ n ih => |
| have ht := d.property.2.2.2.1 (i • h) ((i+(n:ℤ)) • h) ((i+((n+1:ℕ):ℤ)) • h) |
| have hn : i+((n+1:ℕ):ℤ)=i+(n:ℤ)+1 := by omega |
| rw [hn,he] at ht |
| rw [hn] |
| push_cast |
| linarith |
| lemma array_grid_power (d : DistanceArray) |
| (hz : ∀ s t : DyadicTime, (s:ℝ)<(t:ℝ) → arrayJ s t d+arrayV s t d=0) |
| (h : DyadicTime) (hh : 0<(h:ℝ)) (i : ℤ) (m : ℕ) : |
| d.val (i • h) ((i+(2:ℤ)^m) • h)=(2:ℝ)^m*d.val 0 h := by |
| induction m with |
| | zero => simpa using array_grid_edges d hz h hh i |
| | succ m ih => |
| have hm : (0:ℝ)<(2:ℝ)^(m+1) := by positivity |
| have hi : ((i • h : DyadicTime):ℝ)<(((i+(2:ℤ)^(m+1)) • h : DyadicTime):ℝ) := by |
| simp only [dyadic_zsmul_val,Int.cast_add,Int.cast_pow,Int.cast_ofNat] |
| nlinarith |
| have hmid : dyadicMid (i • h) ((i+(2:ℤ)^(m+1)) • h)=(i+(2:ℤ)^m) • h := by |
| apply Subtype.ext |
| simp only [dyadicMid_val,dyadic_zsmul_val,Int.cast_add,Int.cast_pow,Int.cast_ofNat,pow_succ] |
| push_cast |
| ring |
| have hl := (array_zero_test_halves _ _ hi d (hz _ _ hi)).1 |
| rw [hmid,ih] at hl |
| calc |
| _ = 2*((2:ℝ)^m*d.val 0 h) := by linarith |
| _ = _ := by rw [pow_succ]; ring |
| lemma nat_le_two_power (n : ℕ) : n ≤ 2^n := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| have hp : 1 ≤ (2:ℕ)^n := one_le_pow₀ (by norm_num) |
| rw [pow_succ] |
| omega |
| lemma array_grid_distance_nat (d : DistanceArray) |
| (hz : ∀ s t : DyadicTime, (s:ℝ)<(t:ℝ) → arrayJ s t d+arrayV s t d=0) |
| (h : DyadicTime) (hh : 0<(h:ℝ)) (i : ℤ) (n : ℕ) : |
| d.val (i • h) ((i+(n:ℤ)) • h)=(n:ℝ)*d.val 0 h := by |
| have he := array_grid_edges d hz h hh |
| apply le_antisymm (array_grid_upper d h (d.val 0 h) he i n) |
| let N : ℕ := 2^n |
| have hn : n ≤ N := nat_le_two_power n |
| have hu := array_grid_upper d h (d.val 0 h) he (i+(n:ℤ)) (N-n) |
| have hidx : i+(n:ℤ)+((N-n:ℕ):ℤ)=i+(2:ℤ)^n := by |
| rw [Nat.cast_sub hn] |
| dsimp only [N] |
| push_cast; ring |
| rw [hidx] at hu |
| have ht := d.property.2.2.2.1 (i • h) ((i+(n:ℤ)) • h) ((i+(2:ℤ)^n) • h) |
| rw [array_grid_power d hz h hh i n] at ht |
| have hcast : ((N-n:ℕ):ℝ)=(2:ℝ)^n-(n:ℝ) := by rw [Nat.cast_sub hn]; simp [N] |
| rw [hcast] at hu |
| nlinarith |
| lemma array_grid_distance (d : DistanceArray) |
| (hz : ∀ s t : DyadicTime, (s:ℝ)<(t:ℝ) → arrayJ s t d+arrayV s t d=0) |
| (h : DyadicTime) (hh : 0<(h:ℝ)) (i j : ℤ) : |
| d.val (i • h) (j • h)=|(j:ℝ)-(i:ℝ)| *d.val 0 h := by |
| wlog hij : i ≤ j generalizing i j |
| · rw [d.property.2.2.1,abs_sub_comm] |
| exact this j i (le_of_not_ge hij) |
| have hn : i+((j-i).toNat:ℤ)=j := by rw [Int.toNat_of_nonneg (sub_nonneg.mpr hij)]; ring |
| have he := array_grid_distance_nat d hz h hh i (j-i).toNat |
| rw [hn] at he |
| have hcast : ((j-i).toNat:ℝ)=(j:ℝ)-(i:ℝ) := by |
| have hh := congrArg (fun a : ℤ => (a:ℝ)) (Int.toNat_of_nonneg (sub_nonneg.mpr hij)) |
| simpa only [Int.cast_natCast,Int.cast_sub] using hh |
| rw [hcast] at he |
| rw [abs_of_nonneg (sub_nonneg.mpr (by exact_mod_cast hij))] |
| exact he |
| end StandardMapEntropy |
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| end OAI |
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