openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionCodeBounds.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionOuterPhaseProgram | |
| namespace OAI | |
| /-! # Encoded profile bounds independent of learned multiplier sizes -/ | |
| namespace ContingencyTables.CompanionCodeBounds | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped | |
| lemma restart_next (i : ℕ) (s : CompanionRestartProgram.State) : | |
| weight (CompanionRestartProgram.next (i,s)) ≤ 6*s.1.1.2.length+1 := by | |
| simpa only [CompanionRestartProgram.next,CompanionRestartProgram.bound,Nat.size_one, Nat.reduceMul,Nat.reduceAdd,Nat.mul_comm] using | |
| ProfilePrograms.capView_weight s.1.1.2.length 1 (CompanionRestartProgram.candidate (i,s)) | |
| lemma restart_fold (is : List ℕ) (s : CompanionRestartProgram.State) : | |
| weight ((is.foldl (fun s i => CompanionRestartProgram.advance (i,s)) s).2) ≤ | |
| max (weight s.2) (6*s.1.1.2.length+1) := by | |
| induction is using List.reverseRecOn with | |
| | nil => exact le_max_left _ _ | |
| | append_singleton is i _ => | |
| rw [List.foldl_append] | |
| simp only [List.foldl_cons,List.foldl_nil] | |
| apply (restart_next _ _).trans | |
| rw [CompanionRestartProgram.fold_fixed] | |
| exact le_max_right _ _ | |
| lemma restart_output (z : CompanionRestartProgram.Input) : | |
| weight (CompanionRestartProgram.output z) ≤ 7*weight z.1.2+1 := by | |
| have h := restart_fold (CompanionRestartProgram.indices z) (z,z.1.2) | |
| change weight (CompanionRestartProgram.output z) ≤ _ at h | |
| dsimp only at h | |
| have hl := list_length_le_weight z.1.2 | |
| exact h.trans (max_le (by omega) (by omega)) | |
| lemma initialized_output (z : CompanionInitializedRestartProgram.Input) : | |
| weight (CompanionInitializedRestartProgram.output z) ≤ | |
| 7*weight (CompanionInitializedRestartProgram.initialCode z)+1 := | |
| restart_output (CompanionInitializedRestartProgram.prepare z) | |
| lemma walk_next (bits : List Bool) (s : CompanionBoundedWalkProgram.State) : | |
| weight (CompanionBoundedWalkProgram.next (bits,s)) ≤ 6*s.1.2.length+1 := by | |
| simpa only [CompanionBoundedWalkProgram.next,CompanionBoundedWalkProgram.bound,Nat.size_one,Nat.reduceMul,Nat.reduceAdd,Nat.mul_comm] using | |
| ProfilePrograms.capView_weight s.1.2.length 1 (CompanionBoundedWalkProgram.candidate (bits,s)) | |
| lemma record_bound (words : List (List Bool)) (s : CompanionBoundedWalkProgram.RecordState) | |
| (B : ℕ) (hc : weight s.1.2 ≤ B) (hf : 6*s.1.1.2.length+1 ≤ B) | |
| (hh : ∀ code∈s.2,weight code ≤ B) : | |
| let out := words.foldl (fun s bits => CompanionBoundedWalkProgram.recordAdvance (bits,s)) s | |
| (∀ code∈out.2,weight code ≤ B) ∧ out.2.length=s.2.length+words.length := by | |
| induction words generalizing s with | |
| | nil => exact ⟨hh,by simp⟩ | |
| | cons word words ih => | |
| have hn := (walk_next word s.1).trans hf | |
| have h := ih (CompanionBoundedWalkProgram.recordAdvance (word,s)) hn hf (by | |
| intro code hcode | |
| rcases List.mem_cons.mp hcode with he | he | |
| · subst code; exact hn | |
| · exact hh code he) | |
| exact ⟨h.1,by simpa only [List.foldl_cons,CompanionBoundedWalkProgram.recordAdvance, | |
| List.length_cons,Nat.add_assoc,Nat.add_comm,Nat.add_left_comm] using h.2⟩ | |
| lemma history_bound (s : CompanionBoundedWalkProgram.State) (words : List (List Bool)) | |
| (B : ℕ) (hc : weight s.2 ≤ B) (hf : 6*s.1.2.length+1 ≤ B) : | |
| (∀ code∈CompanionBoundedWalkProgram.history (s,words),weight code ≤ B) ∧ | |
| (CompanionBoundedWalkProgram.history (s,words)).length=words.length+1 := by | |
| have h := record_bound words (s,[s.2]) B hc hf (by simpa only [List.mem_singleton] using | |
| fun code he => he ▸ hc) | |
| constructor | |
| · intro code hcode | |
| exact h.1 code (List.mem_reverse.mp hcode) | |
| · simpa only [CompanionBoundedWalkProgram.history,CompanionBoundedWalkProgram.recordWalk, | |
| CompanionBoundedWalkProgram.initialRecord,List.length_reverse,List.length_cons,List.length_nil, | |
| Nat.add_comm] using h.2 | |
| lemma trajectory_length (z : CompanionTrajectoryProgram.Input) : | |
| (CompanionTrajectoryProgram.output z).length ≤ z.2.length+1 := by | |
| let s := z.1.1 | |
| let words := CompanionFlatReturnProgram.words z | |
| have h := history_bound s words (weight s.2+6*s.1.2.length+1) (by omega) (by omega) | |
| have hlen : (CompanionTrajectoryProgram.output z).length=words.length+1 := h.2 | |
| have hl : words.length ≤ z.2.length := by | |
| unfold words CompanionFlatReturnProgram.words DensePrograms.wordChunks DensePrograms.completeWordRows | |
| simp only [List.length_take,List.length_map,List.length_range] | |
| exact (min_le_right _ _).trans (Nat.div_le_self _ _) | |
| omega | |
| lemma trajectory_codes (z : CompanionTrajectoryProgram.Input) (B : ℕ) | |
| (hc : weight z.1.1.2 ≤ B) (hf : 6*z.1.1.1.2.length+1 ≤ B) : | |
| ∀ code∈CompanionTrajectoryProgram.output z,weight code ≤ B := | |
| (history_bound z.1.1 (CompanionFlatReturnProgram.words z) B hc hf).1 | |
| end ContingencyTables.CompanionCodeBounds | |
| end OAI | |