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fcab081 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import OAI.Combinatorics.HadwigerCounterexample.BaselineExtension
namespace OAI
/-! # Baseline Forms -/
noncomputable section
namespace HadwigerCounterexample
open scoped BigOperators
open Matrix
variable {p : Shape} {n N : ℕ}
def lowCross (K : ℕ) (B : Cross p n) :=
∀ z i,( (B z)ᵀ * IP p n i).rank ≤ K
def fitsTable (a b : Plan p n) (B T : Cross p n) :=
∀ z, ∀ x∈tabSpace a z,∀ y∈tabSpace b (flipZ p z), evmat (B z) x y=evmat (T z) x y
theorem fits_recip {B T : Cross p n} {a b : Plan p n} (h : fitsTable a b B T) :
fitsTable b a (reciprocal p B) (reciprocal p T) := by
intro z x hx y hy
have hx' : x∈tabSpace b (flipZ p (flipZ p z)) := by rwa [flipZ_flip]
change evmat _ _ _=evmat _ _ _
unfold reciprocal
rw [evmat_transpose,evmat_transpose,h _ y hy x hx']
def blin (K : ℕ) := 4*K+56
set_option maxHeartbeats 800000 in
theorem start_form (a b : Pin p n N) (x y : UnitType p n N) (T : Cross p n)
(ka kb : Plan p n)
(hA : ∀ z,ka z≤primal p n) (hB : ∀ z,kb z≤primal p n)
(hd : ∀ z, Disjoint (tabSpace a.sp z) (ka z))
(hd' : ∀ z, Disjoint (tabSpace b.sp z) (kb z))
(ha : a.agree x) (hb : b.agree y) (h1 : tableFlag a b T x)
(h2 : tableFlag b a (reciprocal p T) y)
(K : ℕ) (hk : planRank a.sp ≤ K) (hk' : planRank b.sp ≤ K)
(hv : ∀ z, dim (ka z) ≤ 28) (hv' : ∀ z, dim (kb z) ≤ 28) :
∃ M : Cross p n, fitsTable a.sp b.sp M T ∧
rowAgree (a.sp ⊔ ka) M (actual p x y) ∧
rowAgree (b.sp ⊔ kb) (reciprocal p M) (actual p y x) ∧
lowCross (blin K) M ∧ lowCross (blin K) (reciprocal p M) := by
let A := actual p x y
have hh (z : Place p) (u v) (hu : u∈tabSpace a.sp z) (hv : v∈tabSpace b.sp (flipZ p z))
(h : u∈a.sp z ∨ v∈b.sp (flipZ p z)) : evmat (A z) u v=evmat (T z) u v := by
rcases h with h|h
· have h' : u ∈ a.sp (flipZ p (flipZ p z)) := by rwa [flipZ_flip]
have he := h2 _ v hv u h'
have hf := flag_actual (a:=b) y x ha (flipZ p z) v hv u h'
have hg := hf.trans he.symm
rw [← reciprocal_actual x y] at hg
simp only [reciprocal,flipZ_flip,evmat_transpose] at hg
exact hg
exact (flag_actual (a:=a) x y hb _ u hu v h).trans (h1 _ u hu v h).symm
choose M hC hD hE hF using fun z =>
base_pair a.sp b.sp z (flipZ p z) (ka z) (kb (flipZ p z)) (hA _) (hB _) (hd _) (hd' _)
(A z) (T z) (fun u hu v hv g => hh z u v hu hv g)
have hr (z) {a s : Plan p n} (h : planRank a≤K) (hg : ∀ z,dim (s z)≤28) :
dim (a z ⊔ s z : Submodule _ _) ≤ K+28 :=
(Submodule.finrank_add_le_finrank_add_finrank ..).trans
(Nat.add_le_add ((dim_le_plan a _).trans h) (hg _))
have hm z {L} (hl : L≤K) : dim (a.sp z ⊔ ka z : Submodule _ _)+
dim (b.sp (flipZ p z) ⊔ kb (flipZ p z):Submodule _ _)+2*L ≤ blin K := by
apply le_trans (Nat.add_le_add (Nat.add_le_add (hr z hk hv) (hr _ hk' hv'))
(Nat.mul_le_mul_left 2 hl))
unfold blin; omega
refine ⟨M,hC,(fun z u hu v => hD z u v (Or.inl hu)),?_,
fun z i => (hE z i).trans (hm z hk),?_⟩
· rw [← reciprocal_actual x y]
intro z v hv u
unfold reciprocal; rw [evmat_transpose,evmat_transpose]
apply hD
right
rw [flipZ_flip]; exact hv
intro z i
unfold reciprocal; rw [transpose_transpose]
exact (hF (flipZ p z) i).trans (hm _ hk')
theorem channel_eval (p : Shape) {n} (M : Cross p n) (j : Bool) (z : Place p)
(x : Dir p n) (y : Vec (Chan p)) :
dotProduct y (channelOut p M j z *ᵥ x)=evmat (M z) x (IH p n j *ᵥ y) := by
unfold channelOut evmat
rw [dotProduct_transpose_mulVec, Matrix.mulVec_mulVec]
theorem channel_same {M T : Cross p n} (i z u)
(h : ∀ v : Vec (Chan p), evmat (M z) u (IH p n i *ᵥ v)=evmat (T z) u (IH p n i *ᵥ v)) :
channelOut p M i z *ᵥ u = channelOut p T i z *ᵥ u := by
classical
funext j
have he v : dotProduct v (channelOut p M i z *ᵥ u)=dotProduct v (channelOut p T i z*ᵥ u) := by
rw [channel_eval, channel_eval, h]
simpa using he (Pi.single j 1)
theorem fits_channel {M T : Cross p n} {a b : Plan p n} (h : fitsTable a b M T)
(z : Place p) (x : Dir p n) (hx : x∈tabSpace a z) (j : Bool) :
channelOut p M j z *ᵥ x = channelOut p T j z *ᵥ x := by
apply channel_same j
intro v
exact h z x hx _ (channel_le_table b _ j ⟨_,rfl⟩)
theorem fits_inj {M T : Cross p n} {a b : Plan p n} (h : fitsTable a b M T)
(ht : tableInj a b T) : tableInj a b M := by
intro z i j v hv hh
apply ht z i j v hv
rwa [fits_channel h z _ (pin_le_table a z hv) j] at hh
private theorem cout_ev (A B : Mat (Chan p) (JointIdx p n)) (x y) :
evmat (Aᵀ*B) x y=dotProduct (A *ᵥ x) (B *ᵥ y) := by
rw [evmat,← mulVec_mulVec,dotProduct_transpose_mulVec,dotProduct_comm]
theorem fits_contract {M T : Cross p n} {a b : Plan p n} (h : fitsTable a b M T)
(i j : Bool) (x : Profile p n) (hx : x ∈ effective a i) :
contract p M i j x=contract p T i j x := by
change ∑ e : Comp p.g, _ = _
refine Finset.sum_congr rfl fun e _ => ?_
apply (box_eval_iff (tabSpace a (e,true)) (tabSpace a (e,false)) _ _).mp ?_ _
· have hp := lift_mem_box (IP p n i) (IP p n i) ((effective_mem.mp hx).2 e)
exact box_le (le_trans le_sup_left (half_le_table a _ i))
(le_trans le_sup_left (half_le_table a _ i)) hp
intro x hx y hy; rw [cout_ev, cout_ev,fits_channel h _ _ hx,fits_channel h _ _ hy]
variable {I : Type} [Fintype I]
theorem frozen_contract_key {M T : Cross p n} {a} (v : Volley p n I) (t : Tagging p I)
(h : rowAgree (a⊔v.keyPlan t) M T) (i : I) (j : Bool) :
contract p M (v.obj i) j (v.atomAt t i)=contract p T (v.obj i) j (v.atomAt t i) := by
classical
change ∑ e : Comp p.g, _ = _
refine Finset.sum_congr rfl fun e _ => ?_
by_cases he : inc e (t i)
· have hx b :
v.rayLoc i ∈ (v.keyPlan t (e,b)) := colSp_mem (P:=v.keyM t e)
(⟨i,he⟩ : Volley.visits p t e)
have hE (b : Bool) : channelOut p M j (e,b) *ᵥ v.rayLoc i =
channelOut p T j (e,b) *ᵥ v.rayLoc i :=
channel_same j _ _ (fun y => h _ _ (Submodule.mem_sup_right (hx b)) _)
simp only [Volley.atomAt,atom_def,he, ite_true,mom,liftM_outer,mpair_outer]
rw [cout_ev,cout_ev]
exact congrArg₂ dotProduct (hE true) (hE false)
simp only [Volley.atomAt,atom_def,he,ite_false,map_zero]
end HadwigerCounterexample
end
end OAI
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