| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
| |
| namespace GridClosedLabels |
|
|
| def potential {A : Type*} [AddCommGroup A] (H V : ℕ → ℕ → A) (i j : ℕ) : A := |
| (∑ k ∈ Finset.range i, H k 0) + ∑ k ∈ Finset.range j, V i k |
|
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| lemma potential_vertical {A : Type*} [AddCommGroup A] (H V : ℕ → ℕ → A) (i j : ℕ) : |
| potential H V i (j+1)-potential H V i j = V i j := by |
| simp only [potential,Finset.sum_range_succ] |
| abel |
|
|
| lemma telescope_faces {A : Type*} [AddCommGroup A] (H V : ℕ → ℕ → A) |
| {N i j : ℕ} (hi : i < N) (hj : j ≤ N) |
| (hface : ∀ a < N, ∀ b < N, H a (b+1)-H a b = V (a+1) b-V a b) : |
| H i j-H i 0 = (∑ k ∈ Finset.range j, V (i+1) k) - ∑ k ∈ Finset.range j, V i k := by |
| rw [← Finset.sum_sub_distrib,← Finset.sum_range_sub (H i)] |
| apply Finset.sum_congr rfl |
| intro k hk |
| exact hface i hi k ((Finset.mem_range.mp hk).trans_le hj) |
|
|
| lemma potential_horizontal {A : Type*} [AddCommGroup A] (H V : ℕ → ℕ → A) |
| {N i j : ℕ} (hi : i < N) (hj : j ≤ N) |
| (hface : ∀ a < N, ∀ b < N, H a (b+1)-H a b = V (a+1) b-V a b) : |
| potential H V (i+1) j-potential H V i j = H i j := by |
| have h := telescope_faces H V hi hj hface |
| calc |
| potential H V (i+1) j-potential H V i j = |
| H i 0 + ((∑ k ∈ Finset.range j, V (i+1) k) - ∑ k ∈ Finset.range j, V i k) := by |
| simp only [potential, Finset.sum_range_succ] |
| abel |
| _ = H i 0 + (H i j - H i 0) := by rw [← h] |
| _ = H i j := by abel |
|
|
| theorem exact_of_closed {A : Type*} [AddCommGroup A] (N : ℕ) (H V : ℕ → ℕ → A) |
| (hface : ∀ a < N, ∀ b < N, H a (b+1)-H a b = V (a+1) b-V a b) : |
| ∃ f : ℕ → ℕ → A, |
| (∀ i < N, ∀ j ≤ N, f (i+1) j-f i j=H i j) ∧ |
| (∀ i ≤ N, ∀ j < N, f i (j+1)-f i j=V i j) := by |
| refine ⟨potential H V,?_,?_⟩ |
| · intro i hi j hj |
| · intro i hi j hj |
|
|
| end GridClosedLabels |
|
|
| namespace GridClosedLabels |
|
|
| def rayPotential {A : Type*} [AddCommGroup A] (H : ℤ → ℤ → A) (a x y : ℤ) : A := |
| ∑ k ∈ Finset.range (x-a).toNat, H (a+k) y |
|
|
| lemma ray_horizontal {A : Type*} [AddCommGroup A] (H : ℤ → ℤ → A) (a x y : ℤ) |
| (hleft : ∀ x < a, ∀ y, H x y=0) : |
| rayPotential H a (x+1) y-rayPotential H a x y=H x y := by |
| by_cases hx : a ≤ x |
| · have hn : (x+1-a).toNat=(x-a).toNat+1 := by omega |
| have he : a+((x-a).toNat:ℤ)=x := by omega |
| simp only [rayPotential,hn,Finset.sum_range_succ,he] |
| abel |
| · have hn : (x-a).toNat=0 := by omega |
| have hn' : (x+1-a).toNat=0 := by omega |
| simp [rayPotential,hn,hn',hleft x (by omega) y] |
|
|
| lemma ray_vertical {A : Type*} [AddCommGroup A] (H V : ℤ → ℤ → A) (a x y : ℤ) |
| (hleft : ∀ x ≤ a, ∀ y, V x y=0) |
| (hface : ∀ x y, H x (y+1)-H x y=V (x+1) y-V x y) : |
| rayPotential H a x (y+1)-rayPotential H a x y=V x y := by |
| by_cases hx : a ≤ x |
| · have he : a+((x-a).toNat:ℤ)=x := by omega |
| change (∑ k ∈ Finset.range (x-a).toNat, H (a+k) (y+1)) - |
| (∑ k ∈ Finset.range (x-a).toNat, H (a+k) y) = V x y |
| rw [← Finset.sum_sub_distrib] |
| calc |
| _ = ∑ k ∈ Finset.range (x-a).toNat, |
| (V (a+((k+1:ℕ):ℤ)) y-V (a+k) y) := by |
| apply Finset.sum_congr rfl |
| intro k hk |
| simpa [Nat.cast_add,Nat.cast_one,add_assoc] using hface (a+k) y |
| _ = V (a+((x-a).toNat:ℤ)) y-V (a+0) y := by |
| simpa only [Nat.cast_zero] using |
| Finset.sum_range_sub (fun k : ℕ => V (a+(k:ℤ)) y) (x-a).toNat |
| _ = V x y := by rw [he,add_zero,hleft a le_rfl y,sub_zero] |
| · have hn : (x-a).toNat=0 := by omega |
| simp [rayPotential,hn,hleft x (by omega) y] |
|
|
| lemma square_binary (b0 b1 b2 b3 : Bool) (s0 s1 s2 s3 : ZMod 2) |
| (h0 : b0=true → s0=0) (h1 : b1=true → s1=0) |
| (h2 : b2=true → s2=0) (h3 : b3=true → s3=0) |
| (h01 : b0=false → b1=false → s0=s1) |
| (h12 : b1=false → b2=false → s1=s2) |
| (h23 : b2=false → b3=false → s2=s3) |
| (h30 : b3=false → b0=false → s3=s0) : |
| ((if b3 then s2 else 0)+(if b2 then s3 else 0)) - |
| ((if b0 then s1 else 0)+(if b1 then s0 else 0)) = |
| ((if b1 then s2 else 0)+(if b2 then s1 else 0)) - |
| ((if b0 then s3 else 0)+(if b3 then s0 else 0)) := by |
| have hneg : ∀ z : ZMod 2, -z=z := by decide |
| cases b0 < |
| ring |
|
|
| end GridClosedLabels |
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| end |
|
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| end OAI |
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|
