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f0f05de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | 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]
/-- The exact `T^k`-cost script for arbitrary integer changes to a single
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]
/-- First assertion of `lem:finite-cost`: every state and every integer
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]
/-- An ordinary edit script advances the rows at every occurrence of a node. -/
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
/-- Equation `eq:finite-cost` with the concrete root word, actual cheap
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
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