| import OAI.Combinatorics.HadwigerCounterexample.PrimalChannelDecomposition |
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| namespace OAI |
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| /-! # Baseline Extension -/ |
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| noncomputable section |
| namespace HadwigerCounterexample |
| open Matrix Submodule |
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| variable {p : Shape} {n : ℕ} |
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| theorem project_table (a : Plan p n) (z : Place p) (Q : Space (Dir p n)) |
| (hq : Q ≤ primal p n) (hd : Disjoint (tabSpace a z) Q) : |
| ∃ B, ProjData (tabSpace a z) Q B ∧ ∀ x∈ primal p n, B*ᵥx∈prSmall a z := by |
| classical |
| let H := channels p n |
| let A := primal p n |
| let P := prSmall a z |
| have hp : P≤ A := prSmall_le _ _ |
| have he : tabSpace a z=P ⊔ H := prSmall_table a z |
| obtain ⟨X,hX⟩ := proj_exists H A channels_disjoint.symm |
| have hj : Disjoint P (Q⊔H) := by |
| rw [disjoint_iff_inf_le] |
| rintro v ⟨hv,hv'⟩ |
| obtain ⟨y,hy,w,hw,g⟩ := Submodule.mem_sup.mp hv' |
| have h := (show w=v-y by rw [←g]; abel) ▸ |
| A.sub_mem (hp hv) (hq hy) |
| have hz : w∈ A ⊓ H := ⟨h,hw⟩ |
| rw [channels_disjoint.eq_bot] at hz |
| have hi : w=0 := by simpa using hz |
| rw [hi,add_zero] at g |
| have ht : v∈tabSpace a z ⊓ Q := |
| ⟨he.symm ▸ Submodule.mem_sup_left hv,g ▸ hy⟩ |
| rwa [hd.eq_bot] at ht |
| obtain ⟨Y,hY⟩ := proj_exists P (Q⊔H) hj |
| refine ⟨X+Y,⟨fun x => ?_,fun x hx => ?_,fun x hx => ?_⟩, fun x hx => ?_⟩ |
| · rw [add_mulVec,he,add_comm]; exact add_mem_sup (hY.onto _) (hX.onto _) |
| · rw [he] at hx |
| obtain ⟨v,hv,w,hw,rfl⟩ := Submodule.mem_sup.mp hx |
| rw [add_mulVec,mulVec_add,mulVec_add,hX.kills v (hp hv),hX.fixes _ hw,hY.fixes _ hv, |
| hY.kills _ (Submodule.mem_sup_right hw),add_zero,zero_add,add_comm] |
| · rw [add_mulVec,hX.kills x (hq hx),hY.kills x (Submodule.mem_sup_left hx),zero_add] |
| · rw [add_mulVec,hX.kills x hx,zero_add]; exact hY.onto _ |
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| abbrev CrossM (p : Shape) (n : ℕ) := Mat (JointIdx p n) (JointIdx p n) |
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| set_option maxHeartbeats 800000 in |
| theorem base_pair (a b : Plan p n) (u v : Place p) |
| (K J : Space (Dir p n)) (hk : K≤primal p n) (hj : J≤primal p n) |
| (hK : Disjoint (tabSpace a u) K) (hJ : Disjoint (tabSpace b v) J) |
| (A B : CrossM p n) |
| (heq : ∀ x∈ tabSpace a u,∀ y∈tabSpace b v,x∈a u ∨ y∈b v → evmat A x y=evmat B x y) : |
| ∃ M : CrossM p n, |
| (∀ x∈tabSpace a u,∀ y∈tabSpace b v,evmat M x y=evmat B x y) ∧ |
| (∀ x y, x∈ (a u⊔K) ∨ y∈(b v⊔J) → evmat M x y=evmat A x y) ∧ |
| (∀ i,(Mᵀ*IP p n i).rank ≤ dim (a u⊔K : Submodule _ _) + dim (b v⊔J : Submodule _ _) + |
| 2*planRank a) ∧ |
| (∀ i,(M*IP p n i).rank ≤ dim (a u⊔K : Submodule _ _) + dim (b v⊔J : Submodule _ _) + |
| 2*planRank b) := by |
| let P := a u⊔K |
| let Q := b v⊔J |
| obtain ⟨X,hX⟩ := proj_exists P ⊥ disjoint_bot_right |
| obtain ⟨Y,hY⟩ := proj_exists Q ⊥ disjoint_bot_right |
| obtain ⟨L,hL,gL⟩ := project_table a u K hk hK |
| obtain ⟨R,hR,gR⟩ := project_table b v J hj hJ |
| let C := pullM X (1:CrossM p n) A + pullM (1-X) Y A |
| let M := C + pullM L R (B-C) |
| have h1 (x y) (hx : x∈P ∨ y∈Q) : evmat C x y=evmat A x y := by |
| rw [show C=_ from rfl,evmat_addM,evmat_pull,evmat_pull,sub_mulVec,Matrix.one_mulVec, |
| Matrix.one_mulVec] |
| rcases hx with h|h |
| · rw [hX.fixes _ h,sub_self]; simp [evmat] |
| rw [hY.fixes _ h] |
| simp [evmat,sub_dotProduct] |
| have h2 (x y) : evmat M x y=evmat C x y+ |
| (evmat B (L *ᵥ x) (R *ᵥ y)-evmat C (L*ᵥ x) (R*ᵥ y)) := by |
| rw [show M=_ from rfl,evmat_addM,evmat_pull,evmat_subM] |
| have hRank : C.rank≤dim P+dim Q := by |
| apply (mrk_add_le ..).trans |
| have hx : X.rank≤dim P := rank_maps _ hX.onto |
| have hy : Y.rank≤dim Q := rank_maps _ hY.onto |
| apply Nat.add_le_add |
| · rw [pullM_def] |
| refine ((Matrix.rank_mul_le_left ..).trans (Matrix.rank_mul_le_left ..)).trans ?_ |
| rwa [Matrix.rank_transpose] |
| · rw [pullM_def]; exact (Matrix.rank_mul_le_right ..).trans hy |
| have hr (i) {a : Plan p n} (u) |
| (R : CrossM p n) (h : ∀ x∈primal p n,R*ᵥ x∈prSmall a u) : |
| (R*IP p n i).rank≤ 2*planRank a := by |
| apply le_trans (rank_maps (prSmall a u) _) |
| (prSmall_dim a u) |
| intro x |
| rw [← Matrix.mulVec_mulVec] |
| exact h _ (ip_le_primal i (show IP p n i*ᵥx∈colSp (IP p n i) from ⟨x,rfl⟩)) |
| refine ⟨M, fun x hx y hy => ?_, fun x y h => ?_, fun i => ?_, fun i => ?_⟩ |
| · rw [h2,hL.fixes x hx,hR.fixes y hy,add_sub_cancel] |
| · have hi : L*ᵥ x ∈ a u ∨ R *ᵥ y ∈ b v := by |
| rcases h with h|h |
| · left; obtain ⟨x,hx,y,hy,rfl⟩ := Submodule.mem_sup.mp h |
| rwa [Matrix.mulVec_add,hL.kills _ hy,hL.fixes _ (pin_le_table _ _ hx),add_zero] |
| right; obtain ⟨x,hx,y,hy,rfl⟩ := Submodule.mem_sup.mp h |
| rwa [Matrix.mulVec_add,hR.kills _ hy,hR.fixes _ (pin_le_table _ _ hx),add_zero] |
| have he : L*ᵥx ∈ P ∨ R*ᵥy∈Q := hi.imp Submodule.mem_sup_left Submodule.mem_sup_left |
| rw [h2,h1 _ _ h,h1 _ _ he,heq _ (hL.onto _) _ (hR.onto _) hi,sub_self,add_zero] |
| · unfold M |
| rw [transpose_add,Matrix.add_mul] |
| refine (mrk_add_le ..).trans (Nat.add_le_add ?_ ?_) |
| · exact (Matrix.rank_mul_le_left ..).trans ((Matrix.rank_transpose _).trans_le hRank) |
| rw [pullM_def,transpose_mul,transpose_mul,transpose_transpose,Matrix.mul_assoc, |
| Matrix.mul_assoc] |
| exact ((Matrix.rank_mul_le_right ..).trans (Matrix.rank_mul_le_right ..)).trans (hr i u L gL) |
| · unfold M |
| rw [Matrix.add_mul] |
| refine (mrk_add_le ..).trans (Nat.add_le_add ?_ ?_) |
| · exact (Matrix.rank_mul_le_left ..).trans hRank |
| rw [pullM_def,Matrix.mul_assoc] |
| exact (Matrix.rank_mul_le_right ..).trans (hr i v R gR) |
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| end HadwigerCounterexample |
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| end |
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| end OAI |
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