openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /ComputedCompanionInstance.lean
| import OAI.Combinatorics.ContingencyTables.Rooted.ComputedLargeForest | |
| import OAI.Combinatorics.ContingencyTables.Machines.TightenedInputProgram | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionTerminalMass | |
| namespace OAI | |
| /-! # A literal companion instance from binary bounded margins | |
| All operational data are read from the executed endpoint, flow, and | |
| maximum-forest programs. The existential endpoint witnesses are used | |
| only in the terminal-mass proof, never to define an executed output. | |
| -/ | |
| namespace ContingencyTables.ComputedCompanionInstance | |
| open CompanionFactorialApplication ComputedTableTightening | |
| open scoped BigOperators Classical | |
| variable {m n : ℕ} | |
| def baseline (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (i : Fin m) (j : Fin n) : ℕ := | |
| ((TightenedInputProgram.baseline (NetworkVertexCode.data r c b)).getD i.val []).getD j.val 0 | |
| lemma baseline_properties (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1) | |
| (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (hne : ∃ X,TableTightening.Feasible r c b X) : | |
| HasMargins (baseline r c b) (newRows r c b) (newColumns r c b) ∧ | |
| ∀ i j,baseline r c b i j≤newBounds r c b i j := by | |
| obtain ⟨Y,hcode,hm,hb⟩ := TightenedInputProgram.baseline_correct hEK r c b hne | |
| have he : baseline r c b=Y := by | |
| funext i j | |
| simp only [baseline,hcode,ResidualGraphProgram.natCode, | |
| List.getD_eq_getElem?_getD,List.getElem?_ofFn,i.isLt,j.isLt, | |
| dite_true,Option.getD_some,Fin.eta] | |
| simpa only [he] using And.intro hm hb | |
| /-- The package consumed by the proved factorial, terminal, and oracle | |
| lemmas. Its forest and baseline are decoded executed values. -/ | |
| noncomputable def instanceData (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1) | |
| (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (hne : ∃ X,TableTightening.Feasible r c b X) (W : ℕ) (hW : 2≤W) : | |
| Instance (Fin m) (Fin n) where | |
| small:=ComputedLargeForest.small (newBounds r c b) W | |
| b:=newBounds r c b | |
| T:=ComputedLargeForest.computed (newRows r c b) (newColumns r c b) (newBounds r c b) W | |
| hTG:=(ComputedLargeForest.computed_spec _ _ _ W (by omega)).1 | |
| hT:=(ComputedLargeForest.computed_spec _ _ _ W (by omega)).2.1 | |
| hmax:=(ComputedLargeForest.computed_spec _ _ _ W (by omega)).2.2 | |
| r:=fun i=>(newRows r c b i:ℤ) | |
| c:=fun j=>(newColumns r c b j:ℤ) | |
| X:=fun i j=>(baseline r c b i j:ℤ) | |
| d:=1+m*n | |
| W:=W | |
| hd:=by omega | |
| hcard:=by simp only [Fintype.card_fin]; omega | |
| hW:=hW | |
| hlarge:=by | |
| intro i j hs | |
| have hb : W≤newBounds r c b i j := by | |
| simpa only [ComputedLargeForest.small,decide_eq_false_iff_not,not_lt] using hs | |
| exact ⟨hb,(show 1≤W by omega).trans hb⟩ | |
| hsmall:=by | |
| intro i j hs | |
| simpa only [ComputedLargeForest.small,decide_eq_true_eq] using hs | |
| hX:=by | |
| intro i j | |
| exact ⟨Nat.cast_nonneg _,by exact_mod_cast (baseline_properties hEK r c b hne).2 i j⟩ | |
| hXr:=by | |
| intro i | |
| exact_mod_cast (baseline_properties hEK r c b hne).1.1 i | |
| hXc:=by | |
| intro j | |
| exact_mod_cast (baseline_properties hEK r c b hne).1.2 j | |
| /-- Tightening preserves the count consumed by the terminal estimator. -/ | |
| theorem original_count (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1) | |
| (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (hne : ∃ X,TableTightening.Feasible r c b X) (W : ℕ) (hW : 2≤W) : | |
| CompanionTerminalMass.tableCount (instanceData hEK r c b hne W hW)= | |
| (Fintype.card (BoundedTable r c b):ℝ) := by | |
| unfold CompanionTerminalMass.tableCount | |
| have he := (ComputedTableTightening.count_eq hEK r c b hne).symm | |
| calc | |
| _ = (Fintype.card (BoundedTable (newRows r c b) (newColumns r c b) | |
| (newBounds r c b)):ℝ) := by | |
| apply congrArg (fun k : ℕ=>(k:ℝ)) | |
| exact Fintype.card_congr (Equiv.refl _) | |
| _ = _ := congrArg (fun k : ℕ=>(k:ℝ)) he | |
| /-- The endpoint assumption in the terminal proof follows from the | |
| executed searches. The witnesses here are mathematical proof data. -/ | |
| noncomputable def endpoints (hEK : PublishedEdmondsKarp.EdmondsKarp1972Theorem1) | |
| (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) | |
| (hne : ∃ X,TableTightening.Feasible r c b X) (W : ℕ) (hW : 2≤W) : | |
| CompanionTerminalMass.Endpoints (instanceData hEK r c b hne W hW) := by | |
| have hends:=ComputedTableTightening.attained_endpoints hEK r c b hne | |
| let lo : ∀ i j,Fin m→Fin n→ℕ:=fun i j=>Classical.choose (hends.1 i j) | |
| let hi : ∀ i j,Fin m→Fin n→ℕ:=fun i j=>Classical.choose (hends.2 i j) | |
| have hlo (i : Fin m) (j : Fin n) := Classical.choose_spec (hends.1 i j) | |
| have hhi (i : Fin m) (j : Fin n) := Classical.choose_spec (hends.2 i j) | |
| refine { lowerTable:=fun t i j=>(lo t.val.1 t.val.2 i j:ℤ) | |
| upperTable:=fun t i j=>(hi t.val.1 t.val.2 i j:ℤ) | |
| lowerBounds:=?_,upperBounds:=?_,lowerRows:=?_,lowerColumns:=?_, | |
| upperRows:=?_,upperColumns:=?_,lowerAttains:=?_,upperAttains:=?_ } | |
| · intro t i j | |
| exact ⟨Nat.cast_nonneg _,by exact_mod_cast (hlo t.val.1 t.val.2).2.1 i j⟩ | |
| · intro t i j | |
| exact ⟨Nat.cast_nonneg _,by exact_mod_cast (hhi t.val.1 t.val.2).2.1 i j⟩ | |
| · intro t i | |
| change (∑ j,(lo t.val.1 t.val.2 i j:ℤ))=(newRows r c b i:ℤ) | |
| exact_mod_cast (hlo t.val.1 t.val.2).1.1 i | |
| · intro t j | |
| change (∑ i,(lo t.val.1 t.val.2 i j:ℤ))=(newColumns r c b j:ℤ) | |
| exact_mod_cast (hlo t.val.1 t.val.2).1.2 j | |
| · intro t i | |
| change (∑ j,(hi t.val.1 t.val.2 i j:ℤ))=(newRows r c b i:ℤ) | |
| exact_mod_cast (hhi t.val.1 t.val.2).1.1 i | |
| · intro t j | |
| change (∑ i,(hi t.val.1 t.val.2 i j:ℤ))=(newColumns r c b j:ℤ) | |
| exact_mod_cast (hhi t.val.1 t.val.2).1.2 j | |
| · intro t | |
| exact_mod_cast (hlo t.val.1 t.val.2).2.2 | |
| · intro t | |
| exact_mod_cast (hhi t.val.1 t.val.2).2.2 | |
| end ContingencyTables.ComputedCompanionInstance | |
| end OAI | |