openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Rooted /ForestAffineCompletion.lean
| import OAI.Combinatorics.ContingencyTables.Rooted.ForestAffineCoefficients | |
| namespace OAI | |
| /-! # The original integral forest solution is the cut-defined affine completion -/ | |
| namespace ContingencyTables.ForestAffineCoefficients | |
| open scoped BigOperators Classical | |
| open CompanionFactorialApplication GraphIntegralCompletion GraphCompletionCounts | |
| open ForestCapacity ForestCut ForestPhysicalCompletion MaximumForest | |
| variable {I J : Type*} [Fintype I] [Fintype J] | |
| lemma selected_nonroot (A : Instance I J) (t : TreeCell A.T) (v : I⊕J) | |
| (hv : cutSide A t v=true) : v≠componentRoot A.T v := by | |
| intro he | |
| let root := componentRoot A.T (.inl t.val.1) | |
| have hrootRoot : componentRoot A.T root=root := componentRoot_idempotent A.T (.inl t.val.1) | |
| have hreach : (A.T.deleteEdges {edge t.val}).Reachable (cutVertex A t) v := by | |
| simpa only [cutSide,side,decide_eq_true_eq] using hv | |
| have hcomp := (cut_properties A t).1.trans (hreach.mono (A.T.deleteEdges_le {edge t.val})) | |
| have hsame : componentRoot A.T v=root := by | |
| rw [← componentRoot_constant A.T hcomp,hrootRoot] | |
| have hvr : v=root := he.trans hsame | |
| have hbad : cutSide A t root=true := by simpa only [hvr] using hv | |
| exact Bool.false_ne_true ((cut_properties A t).2.1.symm.trans hbad) | |
| lemma other_tree_sign_zero (A : Instance I J) (t : TreeCell A.T) (a : I×J) | |
| (ha : A.T.Adj (.inl a.1) (.inr a.2)) (hne : a≠t.val) : sign A t a=0 := by | |
| have hh := side_eq_of_tree_edge A.T (edge t.val) (cutVertex A t) ha | |
| (by intro he; exact hne (cell_edge_injective he)) | |
| exact sub_eq_zero.mpr (congrArg indicator hh) | |
| noncomputable def smallContribution (A : Instance I J) (t : TreeCell A.T) | |
| (x q : I → J → ℝ) : ℝ := | |
| ∑ a : I×J,if A.small a.1 a.2 then | |
| indicator (cutSide A t (.inl a.1))*(x a.1 a.2-A.X a.1 a.2)- | |
| indicator (cutSide A t (.inr a.2))*(q a.1 a.2-A.X a.1 a.2) else 0 | |
| noncomputable def offset (A : Instance I J) (t : TreeCell A.T) (x q : I → J → ℝ) : ℝ := | |
| A.X t.val.1 t.val.2-smallContribution A t x q/sign A t t.val- | |
| ∑ a : I×J,coefficient A t a*(A.X a.1 a.2:ℝ) | |
| noncomputable def affineTree (A : Instance I J) (t : TreeCell A.T) | |
| (x q : I → J → ℝ) (z : I×J → ℝ) : ℝ := | |
| AffinePenalty.affineValue (offset A t x q) (coefficient A t) z | |
| /-- Isolating the actual selected tree edge in the root-away cut gives | |
| its affine value for every real solution of the nonroot equations. -/ | |
| theorem affine_of_nonroot_equations (A : Instance I J) (t : TreeCell A.T) | |
| (x q z : I → J → ℝ) | |
| (hr : ∀ i,Sum.inl i≠componentRoot A.T (.inl i) → | |
| rowDifference A.small x z (fun i j => (A.X i j:ℝ)) i=0) | |
| (hc : ∀ j,Sum.inr j≠componentRoot A.T (.inr j) → | |
| columnDifference A.small q z (fun i j => (A.X i j:ℝ)) j=0) : | |
| z t.val.1 t.val.2=affineTree A t x q (fun a => z a.1 a.2) := by | |
| let f : I×J → ℝ := fun a => cutTerm A.small | |
| (fun i => cutSide A t (.inl i)) (fun j => cutSide A t (.inr j)) | |
| x q z (fun i j => (A.X i j:ℝ)) a.1 a.2 | |
| have hsum : (∑ a,f a)=0 := by | |
| rw [Fintype.sum_prod_type] | |
| exact cut_sum_zero A.small _ _ x q z _ | |
| (fun i hi => hr i (selected_nonroot A t (.inl i) hi)) | |
| (fun j hj => hc j (selected_nonroot A t (.inr j) hj)) | |
| have hterm (a : I×J) : f a= | |
| (if a=t.val then sign A t t.val*(z a.1 a.2-A.X a.1 a.2) else 0)+ | |
| (if A.small a.1 a.2 then | |
| indicator (cutSide A t (.inl a.1))*(x a.1 a.2-A.X a.1 a.2)- | |
| indicator (cutSide A t (.inr a.2))*(q a.1 a.2-A.X a.1 a.2) else 0)- | |
| sign A t t.val*coefficient A t a*(z a.1 a.2-A.X a.1 a.2) := by | |
| by_cases he : a=t.val | |
| · subst a | |
| have hs : A.small t.val.1 t.val.2=false := A.hTG t.property | |
| simp only [f,cutTerm,hs,Bool.false_eq_true,ite_false,ite_true,coefficient,t.property, | |
| not_true_eq_false,and_false,sign,mul_zero,zero_mul,add_zero,sub_zero] | |
| · by_cases hs : A.small a.1 a.2=true | |
| · simp [f,cutTerm,hs,he,coefficient] | |
| · have hs' : A.small a.1 a.2=false := Bool.eq_false_iff.mpr hs | |
| by_cases ht : A.T.Adj (.inl a.1) (.inr a.2) | |
| · have hz := other_tree_sign_zero A t a ht he | |
| simp only [f,cutTerm,hs',Bool.false_eq_true,ite_false,he,coefficient,ht, | |
| not_true_eq_false,and_false,zero_add,mul_zero,zero_mul,sub_zero] | |
| change sign A t a*(z a.1 a.2-(A.X a.1 a.2:ℝ))=0 | |
| rw [hz,zero_mul] | |
| · simp only [f,cutTerm,hs',Bool.false_eq_true,ite_false,he,coefficient,ht, | |
| not_false_eq_true,and_self,ite_true,zero_add,zero_sub] | |
| change sign A t a*(z a.1 a.2-(A.X a.1 a.2:ℝ))= | |
| -(sign A t t.val*(-sign A t a/sign A t t.val)*(z a.1 a.2-A.X a.1 a.2)) | |
| field_simp [sign_self_ne_zero A t] | |
| simp_rw [hterm] at hsum | |
| rw [Finset.sum_sub_distrib,Finset.sum_add_distrib] at hsum | |
| have hsingle : (∑ a : I×J,if a=t.val then sign A t t.val*(z a.1 a.2-A.X a.1 a.2) else 0)= | |
| sign A t t.val*(z t.val.1 t.val.2-A.X t.val.1 t.val.2) := by simp | |
| rw [hsingle] at hsum | |
| change sign A t t.val*(z t.val.1 t.val.2-A.X t.val.1 t.val.2)+smallContribution A t x q- | |
| (∑ a : I×J,sign A t t.val*coefficient A t a*(z a.1 a.2-A.X a.1 a.2))=0 at hsum | |
| have hprod : (∑ a : I×J,sign A t t.val*coefficient A t a*(z a.1 a.2-A.X a.1 a.2))= | |
| sign A t t.val*((∑ a : I×J,coefficient A t a*z a.1 a.2)- | |
| ∑ a : I×J,coefficient A t a*(A.X a.1 a.2:ℝ)) := by | |
| rw [mul_sub,Finset.mul_sum,Finset.mul_sum,← Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| ring | |
| rw [hprod] at hsum | |
| unfold affineTree AffinePenalty.affineValue offset | |
| apply (mul_left_cancel₀ (sign_self_ne_zero A t)) | |
| rw [mul_add,mul_sub,mul_sub,← mul_div_assoc,mul_div_cancel_left₀ _ (sign_self_ne_zero A t)] | |
| nlinarith [hsum] | |
| /-- In particular this affine display is the original integral solution | |
| used inside h_j, rather than an unrelated continuous extension. -/ | |
| theorem solved_eq_affine (A : Instance I J) (t : TreeCell A.T) | |
| (x q free : I → J → ℤ) : | |
| (solved A.small A.T A.r A.c x q free t.val.1 t.val.2:ℝ)= | |
| affineTree A t (fun i j => (x i j:ℝ)) (fun i j => (q i j:ℝ)) | |
| (fun a => (free a.1 a.2:ℝ)) := by | |
| let z : I → J → ℝ := fun i j => (completed A.small A.T A.r A.c x q free i j:ℝ) | |
| have hh := affine_of_nonroot_equations A t (fun i j => (x i j:ℝ)) | |
| (fun i j => (q i j:ℝ)) z (by | |
| intro i hi | |
| have hr := row_completed_sum A.small A.T A.hTG A.r A.c x q free i hi | |
| have hX := A.hXr i | |
| unfold rowDifference | |
| have he : (∑ j,if A.small i j then (x i j:ℝ) else z i j)=(A.r i:ℝ) := by | |
| dsimp only [z] | |
| exact_mod_cast hr | |
| have hX' : (∑ j,(A.X i j:ℝ))=(A.r i:ℝ) := by exact_mod_cast hX | |
| calc | |
| _=(∑ j,if A.small i j then (x i j:ℝ) else z i j)-(∑ j,(A.X i j:ℝ)) := by | |
| rw [← Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| split_ifs <;> rfl | |
| _=0 := by rw [he,hX',sub_self]) (by | |
| intro j hj | |
| have hc := column_completed_sum A.small A.T A.hTG A.r A.c x q free j hj | |
| have hX := A.hXc j | |
| unfold columnDifference | |
| have he : (∑ i,if A.small i j then (q i j:ℝ) else z i j)=(A.c j:ℝ) := by | |
| dsimp only [z] | |
| exact_mod_cast hc | |
| have hX' : (∑ i,(A.X i j:ℝ))=(A.c j:ℝ) := by exact_mod_cast hX | |
| calc | |
| _=(∑ i,if A.small i j then (q i j:ℝ) else z i j)-(∑ i,(A.X i j:ℝ)) := by | |
| rw [← Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| split_ifs <;> rfl | |
| _=0 := by rw [he,hX',sub_self]) | |
| have hz : z t.val.1 t.val.2=(solved A.small A.T A.r A.c x q free t.val.1 t.val.2:ℝ) := by | |
| simp only [z,completed,t.property,ite_true] | |
| rw [hz] at hh | |
| rw [hh] | |
| unfold affineTree AffinePenalty.affineValue | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| by_cases ht : A.T.Adj (.inl a.1) (.inr a.2) | |
| · simp only [coefficient,ht,not_true_eq_false,and_false,ite_false,zero_mul] | |
| · simp only [z,completed,ht,ite_false] | |
| end ContingencyTables.ForestAffineCoefficients | |
| end OAI | |