| import OAI.Combinatorics.BinaryEdit.FiniteTreeInstantiation |
| import OAI.Combinatorics.BinaryEdit.FiniteTreeConcreteCosts |
|
|
| namespace OAI |
|
|
| /-! |
| # The finite tree’s pointwise translation estimates |
|
|
| The concrete `finiteWord` and cyclic-state vectors satisfy `lem:finite-cost`. |
| Coordinate support follows from the vector definitions. |
| -/ |
|
|
| namespace TreeEdit.BinaryLower.FiniteTree |
|
|
| theorem wordU_off_prefix {m k : ℕ} (primes : Fin k → Fin m → ℕ) |
| (i₀ : Fin m) (q : List (Fin m)) (hq : q.length ≤ k) |
| (l : Leaf m k) (hl : ¬ q.IsPrefix l.val) : |
| wordU primes i₀ q l = 0 := by |
| change FiniteVectors.u primes q.length (extendPrefix i₀ q) (leafEquiv m k l) = 0 |
| apply FiniteVectors.u_off_subtree |
| intro ha |
| have hp := (agrees_extendPrefix_iff i₀ q hq (leafEquiv m k l)).1 ha |
| exact hl (by simpa only [ofFn_leafEquiv] using hp) |
|
|
| theorem wordW_off_prefix {m k : ℕ} (primes : Fin k → Fin m → ℕ) |
| (i₀ : Fin m) (q : List (Fin m)) (hq : q.length ≤ k) |
| (l : Leaf m k) (hl : ¬ q.IsPrefix l.val) : |
| wordW primes i₀ q l = 0 := by |
| change FiniteVectors.w primes q.length (extendPrefix i₀ q) (leafEquiv m k l) = 0 |
| apply FiniteVectors.w_off_subtree |
| intro ha |
| have hp := (agrees_extendPrefix_iff i₀ q hq (leafEquiv m k l)).1 ha |
| exact hl (by simpa only [ofFn_leafEquiv] using hp) |
|
|
| theorem wordW_leaf_other {m k : ℕ} (primes : Fin k → Fin m → ℕ) |
| (i₀ : Fin m) (ell l : Leaf m k) (hne : l ≠ ell) : |
| wordW primes i₀ ell.val l = 0 := by |
| change FiniteVectors.w primes ell.val.length (extendPrefix i₀ ell.val) |
| (leafEquiv m k l) = 0 |
| rw [ell.property, extendPrefix_leaf i₀ ell.val ell.property] |
| have hn : leafEquiv m k l ≠ leafEquiv m k ell := by |
| intro he |
| exact hne ((leafEquiv m k).injective he) |
| simp only [FiniteVectors.w_leaf, ite_eq_right hn] |
|
|
| / |
| cyclic leaf coordinate. The distinguished vector at a leaf is its unit vector. -/ |
| theorem finiteWord_leaf_translation_script {m k : ℕ} |
| (primes : Fin k → Fin m → ℕ) (i₀ : Fin m) (T : ℕ) |
| (ell : Leaf m k) (t : ℤ) (x : State (wordPeriods primes)) : |
| EditScript (finiteWord primes i₀ T k [] x) |
| (finiteWord primes i₀ T k [] (x + t • wordW primes i₀ ell.val)) (T ^ k) := by |
| apply root_leaf_supported_script T (wordPeriods primes) |
| ⟨List.replicate k i₀, by simp⟩ ell (wordU primes i₀) |
| intro l hne |
| simp only [Pi.smul_apply, wordW_leaf_other primes i₀ ell l hne, smul_zero] |
|
|
| / |
| multiple of a leaf unit vector has edit cost at most `M/m^k`. -/ |
| theorem finiteWord_leaf_translation_cost {m k : ℕ} |
| (primes : Fin k → Fin m → ℕ) (i₀ : Fin m) (T : ℕ) |
| (ell : Leaf m k) (t : ℤ) (x : State (wordPeriods primes)) : |
| (ed (finiteWord primes i₀ T k [] x) |
| (finiteWord primes i₀ T k [] (x + t • wordW primes i₀ ell.val)) : ℝ) ≤ |
| ((m * T : ℕ) : ℝ) ^ k / (m : ℝ) ^ k := by |
| have hm : 0 < m := by have hi := i₀.isLt; omega |
| apply root_leaf_supported_normalized_cost T hm (wordPeriods primes) |
| ⟨List.replicate k i₀, by simp⟩ ell (wordU primes i₀) |
| intro l hne |
| simp only [Pi.smul_apply, wordW_leaf_other primes i₀ ell l hne, smul_zero] |
|
|
| / |
| theorem finiteWord_node_translation_script {m k : ℕ} |
| (primes : Fin k → Fin m → ℕ) (i₀ : Fin m) (T : ℕ) |
| (q : List (Fin m)) (hq : q.length < k) |
| (s : ℕ) (hs : s ≤ T) (x : State (wordPeriods primes)) : |
| EditScript (finiteWord primes i₀ T k [] x) |
| (finiteWord primes i₀ T k [] (x + s • wordU primes i₀ q)) |
| (T ^ q.length * (2 * s * m * (m * T) ^ (k - (q.length + 1)))) := by |
| apply root_internal_supported_script T (wordPeriods primes) |
| ⟨List.replicate k i₀, by simp⟩ (wordU primes i₀) (k - (q.length + 1)) q |
| (by omega) s hs |
| exact wordU_off_prefix primes i₀ q hq.le |
|
|
| / |
| vector, and exact source constants. It holds uniformly at every state and |
| also includes the harmless zero shift. -/ |
| theorem finiteWord_node_translation_cost {m k : ℕ} |
| (primes : Fin k → Fin m → ℕ) (i₀ : Fin m) (T : ℕ) (hT : 0 < T) |
| (q : List (Fin m)) (hq : q.length < k) |
| (s : ℕ) (hs : s ≤ T) (x : State (wordPeriods primes)) : |
| (ed (finiteWord primes i₀ T k [] x) |
| (finiteWord primes i₀ T k [] (x + s • wordU primes i₀ q)) : ℝ) ≤ |
| (2 * (s : ℝ) / T) * (((m * T : ℕ) : ℝ) ^ k / (m : ℝ) ^ q.length) := by |
| have hm : 0 < m := by have hi := i₀.isLt; omega |
| apply root_internal_supported_normalized_cost T hm hT (wordPeriods primes) |
| ⟨List.replicate k i₀, by simp⟩ (wordU primes i₀) (k - (q.length + 1)) q |
| (by omega) s hs |
| exact wordU_off_prefix primes i₀ q hq.le |
|
|
| end TreeEdit.BinaryLower.FiniteTree |
|
|
| end OAI |
|
|
