| import OAI.Combinatorics.MatrixRemoval.Model |
|
|
| namespace OAI |
|
|
| /-! |
| # Exact body signatures and extraction of body roles |
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|
| All signatures refer to `fixedH`, including its zero entries. |
| -/ |
|
|
| namespace Problem348.BodySignatures52 |
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|
| def anchorIndex (u : Fin 64) : Fin 66 := ⟨u.val, by omega⟩ |
|
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| def bodyIndex (b : Fin 2) : Fin 66 := ⟨64 + b.val, by omega⟩ |
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|
| theorem fixedH_anchor_anchor (u v : Fin 64) : |
| fixedH (anchorIndex u) (anchorIndex v) = anchor64 u v := by |
| simp [fixedH, anchorIndex, u.isLt, v.isLt] |
|
|
| theorem fixedH_body_anchor (b : Fin 2) (v : Fin 64) : |
| fixedH (bodyIndex b) (anchorIndex v) = decide (v.val = b.val) := by |
| fin_cases b <;> simp [fixedH, bodyIndex, anchorIndex, v.isLt] |
|
|
| theorem fixedH_anchor_body (u : Fin 64) (b : Fin 2) : |
| fixedH (anchorIndex u) (bodyIndex b) = decide (u.val = b.val) := by |
| fin_cases b <;> simp [fixedH, bodyIndex, anchorIndex, u.isLt] |
|
|
| theorem fixedH_body_body (b c : Fin 2) : |
| fixedH (bodyIndex b) (bodyIndex c) = decide (b.val = 1 ∨ c.val = 0) := by |
| fin_cases b <;> fin_cases c <;> rfl |
|
|
| / |
| theorem anchor64_row_ne_body_signature (u : Fin 64) (b : Fin 2) : |
| (fun v => anchor64 u v) ≠ (fun v : Fin 64 => decide (v.val = b.val)) := by |
| intro h |
| have h2 := congrFun h (2 : Fin 64) |
| have h3 := congrFun h (3 : Fin 64) |
| fin_cases b <;> simp [anchor64] at h2 h3 <;> omega |
|
|
| / |
| theorem anchor64_col_ne_body_signature (v : Fin 64) (b : Fin 2) : |
| (fun u => anchor64 u v) ≠ (fun u : Fin 64 => decide (u.val = b.val)) := by |
| intro h |
| have h2 := congrFun h (2 : Fin 64) |
| have h3 := congrFun h (3 : Fin 64) |
| fin_cases b <;> simp [anchor64] at h2 h3 <;> omega |
|
|
| theorem zero_ne_body_signature (b : Fin 2) : |
| (fun _ : Fin 64 => false) ≠ (fun v : Fin 64 => decide (v.val = b.val)) := by |
| intro h |
| have hb := congrFun h (⟨b.val, by omega⟩ : Fin 64) |
| simp at hb |
|
|
| theorem copy_body_row_signature {n : ℕ} {A : BinaryMatrix n} |
| {rc : IncreasingMap 66 n × IncreasingMap 66 n} |
| (hcopy : rc ∈ orderedCopies fixedH A) (b : Fin 2) (v : Fin 64) : |
| A (rc.1.val (bodyIndex b)) (rc.2.val (anchorIndex v)) = |
| decide (v.val = b.val) := by |
| classical |
| have hm := (Finset.mem_filter.mp hcopy).2 (bodyIndex b) (anchorIndex v) |
| exact hm.trans (fixedH_body_anchor b v) |
|
|
| theorem copy_body_col_signature {n : ℕ} {A : BinaryMatrix n} |
| {rc : IncreasingMap 66 n × IncreasingMap 66 n} |
| (hcopy : rc ∈ orderedCopies fixedH A) (b : Fin 2) (u : Fin 64) : |
| A (rc.1.val (anchorIndex u)) (rc.2.val (bodyIndex b)) = |
| decide (u.val = b.val) := by |
| classical |
| have hm := (Finset.mem_filter.mp hcopy).2 (anchorIndex u) (bodyIndex b) |
| exact hm.trans (fixedH_anchor_body u b) |
|
|
| end Problem348.BodySignatures52 |
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| end OAI |
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|
