| import OAI.Combinatorics.ContingencyTables.Dense.GridAdjacency |
|
|
| namespace OAI |
|
|
| /-! # Integer signed-coordinate proposals for the dense grid -/ |
|
|
| namespace ContingencyTables.GridBoundary |
|
|
| open scoped Classical |
|
|
| variable [DecidableEq I] |
|
|
| /-- A signed step is rejected at the boundary of the declared finite grid. -/ |
| def axisStep (a : Grid (I := I) K) (i : I) (sign : Bool) : Option (Grid (I := I) K) := |
| if sign then |
| if h : corner a i + 1 ≤ (K : ℤ) - 1 then |
| some (Function.update a i ⟨corner a i + 1, by |
| constructor |
| · have ha : 0 ≤ corner a i := (a i).property.1; omega |
| · exact h⟩) |
| else none |
| else |
| if h : 0 ≤ corner a i - 1 then |
| some (Function.update a i ⟨corner a i - 1, by |
| constructor |
| · exact h |
| · have ha : corner a i ≤ (K : ℤ) - 1 := (a i).property.2; omega⟩) |
| else none |
|
|
| theorem axisStep_sound (a b : Grid (I := I) K) (i : I) (sign : Bool) |
| (h : axisStep a i sign = some b) : OrientedNeighbor a b i sign := by |
| cases sign |
| · unfold axisStep at h |
| simp only [Bool.false_eq_true, ite_false] at h |
| split_ifs at h with hc |
| · have hb := Option.some.inj h |
| rw [← hb] |
| change IsGridEdge _ |
| constructor |
| · simp only [corner, Function.update_self] |
| omega |
| · intro j hj |
| change j ≠ i at hj |
| simp [Function.update_of_ne hj] |
| · unfold axisStep at h |
| simp only [ite_true] at h |
| split_ifs at h with hc |
| · have hb := Option.some.inj h |
| rw [← hb] |
| change IsGridEdge _ |
| constructor |
| · simp [corner] |
| · intro j hj |
| change j ≠ i at hj |
| simp [Function.update_of_ne hj] |
|
|
| theorem axisStep_complete (a b : Grid (I := I) K) (i : I) (sign : Bool) |
| (h : OrientedNeighbor a b i sign) : axisStep a i sign = some b := by |
| cases sign |
| · change IsGridEdge (b, a, i) at h |
| have hc : 0 ≤ corner a i - 1 := by |
| have := (b i).property.1 |
| have he := h.1 |
| change corner a i = corner b i + 1 at he |
| change 0 ≤ corner b i at this |
| omega |
| unfold axisStep |
| simp only [Bool.false_eq_true, ite_false, dite_eq_left hc, Option.some.injEq] |
| funext j |
| apply Subtype.ext |
| by_cases hj : j = i |
| · subst j |
| simp only [Function.update_self] |
| have he := h.1 |
| change corner a i = corner b i + 1 at he |
| change corner a i - 1 = corner b i |
| omega |
| · simp only [Function.update_of_ne hj] |
| exact congrArg Subtype.val (h.2 j hj) |
| · change IsGridEdge (a, b, i) at h |
| have hc : corner a i + 1 ≤ (K : ℤ) - 1 := by |
| have := (b i).property.2 |
| have he := h.1 |
| change corner b i = corner a i + 1 at he |
| change corner b i ≤ (K : ℤ) - 1 at this |
| omega |
| unfold axisStep |
| simp only [ite_true, dite_eq_left hc, Option.some.injEq] |
| funext j |
| apply Subtype.ext |
| by_cases hj : j = i |
| · subst j |
| simp only [Function.update_self] |
| exact h.1.symm |
| · simp only [Function.update_of_ne hj] |
| exact congrArg Subtype.val (h.2 j hj).symm |
|
|
| theorem axisStep_some_iff (a b : Grid (I := I) K) (i : I) (sign : Bool) : |
| axisStep a i sign = some b ↔ OrientedNeighbor a b i sign := |
| ⟨axisStep_sound a b i sign, axisStep_complete a b i sign⟩ |
|
|
| /-- Each neighbouring bin has exactly one signed-coordinate proposal. -/ |
| theorem oriented_direction_unique (a b : Grid (I := I) K) (i j : I) (s t : Bool) |
| (hi : OrientedNeighbor a b i s) (hj : OrientedNeighbor a b j t) : (i, s) = (j, t) := by |
| have hij : i = j := by |
| by_contra hne |
| have he : corner a i = corner b i := by |
| cases t |
| · exact congrArg Subtype.val (hj.2 i hne) |
| · exact (congrArg Subtype.val (hj.2 i hne)).symm |
| cases s |
| · have hp := hi.1 |
| change corner a i = corner b i + 1 at hp |
| omega |
| · have hp := hi.1 |
| change corner b i = corner a i + 1 at hp |
| omega |
| subst j |
| have hst : s = t := by |
| cases s <;> cases t |
| · rfl |
| · have h₁ := hi.1 |
| have h₂ := hj.1 |
| change corner a i = corner b i + 1 at h₁ |
| change corner b i = corner a i + 1 at h₂ |
| omega |
| · have h₁ := hi.1 |
| have h₂ := hj.1 |
| change corner b i = corner a i + 1 at h₁ |
| change corner a i = corner b i + 1 at h₂ |
| omega |
| · rfl |
| subst t |
| rfl |
|
|
| end ContingencyTables.GridBoundary |
|
|
| end OAI |
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|
