openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionHistoryReadback.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionStoredRepresentation | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionOuterPhaseSemantics | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionCanonicalHistoryLaw | |
| namespace OAI | |
| /-! # Reading a represented physical history as source matrix data -/ | |
| namespace ContingencyTables.CompanionHistoryReadback | |
| open MatchingFPRAS CompanionMultiplierMatrixProgram CompanionStoredRepresentation | |
| open CompanionFactorialApplication CompanionComputedInitial CompanionEvaluatedWeightProgram | |
| open GridBoundary | |
| open scoped Classical | |
| noncomputable section | |
| abbrev Stored := CompanionPhaseFinishProgram.Stored | |
| def lookup (initial : Stored) (history : List Stored) : ℕ→Stored | |
| | 0 => initial | |
| | i+1 => history.getD i initial | |
| def matrixAt (p : ℕ) (initial : Stored) (history : List Stored) (i : ℕ) : | |
| Fin p→Fin p→TreeFraction.Fraction := entries p (lookup initial history i).1 | |
| def estimateAt (initial : Stored) (history : List Stored) (i : ℕ) : TreeFraction.Fraction := | |
| (lookup initial history i).2 | |
| lemma getD_rep {p : ℕ} (history : List Stored) (initial : Stored) | |
| (hi : StoredRep p initial) (hh : ∀ s∈history,StoredRep p s) (i : ℕ) : | |
| StoredRep p (history.getD i initial) := by | |
| by_cases h : i < history.length | |
| · rw [List.getD_eq_getElem _ _ h] | |
| exact hh _ (List.getElem_mem h) | |
| · rw [List.getD_eq_default _ _ (Nat.le_of_not_gt h)] | |
| exact hi | |
| lemma lookup_rep {p : ℕ} (initial : Stored) (history : List Stored) | |
| (hi : StoredRep p initial) (hh : ∀ s∈history,StoredRep p s) (i : ℕ) : | |
| StoredRep p (lookup initial history i) := by | |
| cases i with | |
| | zero => exact hi | |
| | succ i => exact getD_rep history initial hi hh i | |
| theorem lookup_readback {p : ℕ} (initial : Stored) (history : List Stored) | |
| (hi : StoredRep p initial) (hh : ∀ s∈history,StoredRep p s) (i : ℕ) : | |
| (matrixCode (matrixAt p initial history i),estimateAt initial history i)= | |
| lookup initial history i := | |
| stored_readback (lookup_rep initial history hi hh i) | |
| theorem initial_readback {p : ℕ} (initial : Stored) (history : List Stored) | |
| (hi : StoredRep p initial) : | |
| (matrixCode (matrixAt p initial history 0),estimateAt initial history 0)=initial := | |
| stored_readback hi | |
| theorem history_readback {p : ℕ} (initial : Stored) (history : List Stored) | |
| (hi : StoredRep p initial) (hh : ∀ s∈history,StoredRep p s) : | |
| List.ofFn (fun i : Fin history.length => | |
| (matrixCode (matrixAt p initial history (i.val+1)),estimateAt initial history (i.val+1)))= | |
| history := by | |
| apply List.ext_getElem | |
| · exact List.length_ofFn | |
| · intro i _ hj | |
| simp only [List.getElem_ofFn] | |
| have h := lookup_readback initial history hi hh (i+1) | |
| change (matrixCode (matrixAt p initial history (i+1)),estimateAt initial history (i+1))= | |
| history.getD i initial at h | |
| rw [List.getD_eq_getElem _ _ hj] at h | |
| exact h | |
| def viewLookup {Z : Type} (initial : Z) (history : List Z) : ℕ→Z | |
| | 0 => initial | |
| | i+1 => history.getD i initial | |
| theorem view_lookup (p : ℕ) (initial : Stored) (history : List Stored) (i : ℕ) : | |
| CompanionPhaseFinishProgram.view (p := p) (lookup initial history i)= | |
| viewLookup (CompanionPhaseFinishProgram.view (p := p) initial) | |
| (history.map (CompanionPhaseFinishProgram.view (p := p))) i := by | |
| cases i with | |
| | zero => rfl | |
| | succ i => exact (List.getD_map history initial | |
| (CompanionPhaseFinishProgram.view (p := p))).symm | |
| theorem old_eq_lookup (fixed : CompanionOuterHistoryProgram.Fixed) (history : List Stored) | |
| (bits : List Bool) : | |
| CompanionOuterPhaseProgram.old ((fixed,history),bits)=lookup fixed.2 history history.length := by | |
| cases history with | |
| | nil => rfl | |
| | cons s history => | |
| change (s::history).getD history.length default=(s::history).getD history.length fixed.2 | |
| rw [List.getD_eq_getElem _ _ (by simp : history.length < (s::history).length), | |
| List.getD_eq_getElem _ _ (by simp : history.length < (s::history).length)] | |
| theorem view_old (p : ℕ) (fixed : CompanionOuterHistoryProgram.Fixed) (history : List Stored) | |
| (bits : List Bool) : | |
| CompanionPhaseFinishProgram.view (p := p) (CompanionOuterPhaseProgram.old ((fixed,history),bits))= | |
| viewLookup (CompanionPhaseFinishProgram.view (p := p) fixed.2) | |
| (history.map (CompanionPhaseFinishProgram.view (p := p))) history.length := by | |
| rw [old_eq_lookup,view_lookup] | |
| theorem view_old_canonical {I J : Type} [Fintype I] [Fintype J] | |
| (A : Instance I J) {p : ℕ} (g : Fin p ≃ Labels A) | |
| (fixed : CompanionOuterHistoryProgram.Fixed) (history : List Stored) (bits : List Bool) | |
| (hi : CompanionPhaseFinishProgram.view (p := p) fixed.2= | |
| CompanionCanonicalHistory.project A g (RootedStoredEstimate.initialStored A)) : | |
| CompanionPhaseFinishProgram.view (p := p) (CompanionOuterPhaseProgram.old ((fixed,history),bits))= | |
| CompanionCanonicalHistory.canonicalLookup A g | |
| (history.map (CompanionPhaseFinishProgram.view (p := p))) history.length := by | |
| rw [view_old,hi] | |
| rfl | |
| variable {m n : ℕ} | |
| theorem outer_data_readback (A : Instance (Fin m) (Fin n)) (C K H T N R L : ℕ) | |
| (epsilon : TreeFraction.Fraction) (z : View A→ℕ) | |
| (v : Grid (I := Fin (RootedCatalogGrid.freeCount A)) K) | |
| (initial : Stored) (history : List Stored) | |
| (hi : StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length initial) | |
| (hh : ∀ s∈history,StoredRep (CanonicalInitialEncoding.copyOrder (capacities A)).length s) | |
| (bits : List Bool) : | |
| CompanionOuterPhaseProgram.data A C K H T N R L history.length epsilon z v | |
| (matrixAt (CanonicalInitialEncoding.copyOrder (capacities A)).length initial history) | |
| (estimateAt initial history) bits= | |
| (((CompanionRestartInitializer.data A C K H T N R L epsilon z v,initial),history),bits) := by | |
| unfold CompanionOuterPhaseProgram.data | |
| rw [initial_readback initial history hi,history_readback initial history hi hh] | |
| end | |
| end ContingencyTables.CompanionHistoryReadback | |
| end OAI | |