| import OAI.Combinatorics.ContingencyTables.Model |
| import Mathlib.Tactic.SplitIfs |
|
|
| namespace OAI |
|
|
| /-! # A literal one-cell lower-bound reduction |
|
|
| Requiring the selected entry to be at least t is equivalent to subtracting |
| t from that entry, its incident margins and its capacity. The three guards |
| are retained explicitly, including for zero margins and structural zeros. |
| -/ |
|
|
| namespace ContingencyTables.BoundedCellShift |
|
|
| open scoped BigOperators |
|
|
| variable {m n : ℕ} |
|
|
| def margin (r : Fin m→ℕ) (i₀ : Fin m) (t : ℕ) (i : Fin m) : ℕ := |
| if i=i₀ then r i-t else r i |
|
|
| def lower (X : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) (t : ℕ) |
| (i : Fin m) (j : Fin n) : ℕ := |
| if i=i₀ ∧ j=j₀ then X i j-t else X i j |
|
|
| def raise (X : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) (t : ℕ) |
| (i : Fin m) (j : Fin n) : ℕ := |
| X i j+if i=i₀ ∧ j=j₀ then t else 0 |
|
|
| lemma lower_raise (X : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) (t : ℕ) : |
| lower (raise X i₀ j₀ t) i₀ j₀ t=X := by |
| funext i j |
| by_cases hi : i=i₀ <;> by_cases hj : j=j₀ <;> simp [lower,raise,hi,hj] |
|
|
| lemma raise_lower (X : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) (t : ℕ) |
| (ht : t≤X i₀ j₀) : raise (lower X i₀ j₀ t) i₀ j₀ t=X := by |
| funext i j |
| by_cases hi : i=i₀ <;> by_cases hj : j=j₀ |
| · subst i; subst j |
| simpa only [raise,lower,ite_true,true_and,add_zero] using Nat.sub_add_cancel ht |
| all_goals simp [lower,raise,hi,hj] |
|
|
| lemma raise_row_sum (X : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) |
| (t : ℕ) (i : Fin m) : |
| (∑ j,raise X i₀ j₀ t i j)=(∑ j,X i j)+if i=i₀ then t else 0 := by |
| by_cases hi : i=i₀ |
| · subst i |
| simp [raise,Finset.sum_add_distrib] |
| · simp [raise,hi] |
|
|
| lemma raise_column_sum (X : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) |
| (t : ℕ) (j : Fin n) : |
| (∑ i,raise X i₀ j₀ t i j)=(∑ i,X i j)+if j=j₀ then t else 0 := by |
| by_cases hj : j=j₀ |
| · subst j |
| simp [raise,Finset.sum_add_distrib] |
| · simp [raise,hj] |
|
|
| lemma lower_margins (X : Fin m→Fin n→ℕ) (r : Fin m→ℕ) (c : Fin n→ℕ) |
| (i₀ : Fin m) (j₀ : Fin n) (t : ℕ) (hm : HasMargins X r c) |
| (ht : t≤X i₀ j₀) : |
| HasMargins (lower X i₀ j₀ t) (margin r i₀ t) (margin c j₀ t) := by |
| have he := raise_lower X i₀ j₀ t ht |
| constructor |
| · intro i |
| by_cases hi : i=i₀ |
| · subst i |
| have hs := raise_row_sum (lower X i₀ j₀ t) i₀ j₀ t i₀ |
| rw [he,hm.1 i₀] at hs |
| simp only [ite_true] at hs |
| simp only [margin,ite_true] |
| omega |
| · have hs := raise_row_sum (lower X i₀ j₀ t) i₀ j₀ t i |
| rw [he,hm.1 i] at hs |
| simpa only [margin,hi,ite_false,Nat.add_zero] using hs.symm |
| · intro j |
| by_cases hj : j=j₀ |
| · subst j |
| have hs := raise_column_sum (lower X i₀ j₀ t) i₀ j₀ t j₀ |
| rw [he,hm.2 j₀] at hs |
| simp only [ite_true] at hs |
| simp only [margin,ite_true] |
| omega |
| · have hs := raise_column_sum (lower X i₀ j₀ t) i₀ j₀ t j |
| rw [he,hm.2 j] at hs |
| simpa only [margin,hj,ite_false,Nat.add_zero] using hs.symm |
|
|
| lemma raise_margins (Y : Fin m→Fin n→ℕ) (r : Fin m→ℕ) (c : Fin n→ℕ) |
| (i₀ : Fin m) (j₀ : Fin n) (t : ℕ) |
| (hr : t≤r i₀) (hc : t≤c j₀) |
| (hm : HasMargins Y (margin r i₀ t) (margin c j₀ t)) : |
| HasMargins (raise Y i₀ j₀ t) r c := by |
| constructor |
| · intro i |
| rw [raise_row_sum,hm.1 i] |
| by_cases hi : i=i₀ |
| · subst i |
| simpa [margin] using Nat.sub_add_cancel hr |
| · simp [margin,hi] |
| · intro j |
| rw [raise_column_sum,hm.2 j] |
| by_cases hj : j=j₀ |
| · subst j |
| simpa [margin] using Nat.sub_add_cancel hc |
| · simp [margin,hj] |
|
|
| lemma lower_bounds (X b : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) |
| (t : ℕ) (hb : ∀ i j,X i j≤b i j) : |
| ∀ i j,lower X i₀ j₀ t i j≤lower b i₀ j₀ t i j := by |
| intro i j |
| unfold lower |
| split_ifs |
| · exact Nat.sub_le_sub_right (hb i j) t |
| · exact hb i j |
|
|
| lemma raise_bounds (Y b : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) |
| (t : ℕ) (ht : t≤b i₀ j₀) |
| (hb : ∀ i j,Y i j≤lower b i₀ j₀ t i j) : |
| ∀ i j,raise Y i₀ j₀ t i j≤b i j := by |
| intro i j |
| have h := hb i j |
| by_cases hi : i=i₀ <;> by_cases hj : j=j₀ |
| · subst i; subst j |
| simp only [lower,ite_true,true_and] at h |
| simp only [raise,ite_true,true_and] |
| omega |
| all_goals simpa [raise,lower,hi,hj] using h |
|
|
| theorem feasible_lower_iff (r : Fin m→ℕ) (c : Fin n→ℕ) |
| (b : Fin m→Fin n→ℕ) (i₀ : Fin m) (j₀ : Fin n) (t : ℕ) : |
| (∃ X,HasMargins X r c ∧ (∀ i j,X i j≤b i j) ∧ t≤X i₀ j₀) ↔ |
| t≤r i₀ ∧ t≤c j₀ ∧ t≤b i₀ j₀ ∧ |
| ∃ Y,HasMargins Y (margin r i₀ t) (margin c j₀ t) ∧ |
| ∀ i j,Y i j≤lower b i₀ j₀ t i j := by |
| constructor |
| · rintro ⟨X,hm,hb,ht⟩ |
| have hc : X i₀ j₀≤c j₀ := by |
| rw [←hm.2 j₀] |
| exact Finset.single_le_sum (f := fun i=>X i j₀) (fun _ _=>Nat.zero_le _) |
| (Finset.mem_univ i₀) |
| exact ⟨ht.trans (entry_le_row hm i₀ j₀),ht.trans hc,ht.trans (hb i₀ j₀), |
| lower X i₀ j₀ t,lower_margins X r c i₀ j₀ t hm ht,lower_bounds X b i₀ j₀ t hb⟩ |
| · rintro ⟨hr,hc,hb,Y,hm,hy⟩ |
| refine ⟨raise Y i₀ j₀ t,raise_margins Y r c i₀ j₀ t hr hc hm, |
| raise_bounds Y b i₀ j₀ t hb hy,?_⟩ |
| simp only [raise,ite_true,true_and] |
| omega |
|
|
| end ContingencyTables.BoundedCellShift |
|
|
| end OAI |
|
|
