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e320e82 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 | import Mathlib
import OAI.Combinatorics.IndependentSets.Machines.ExprTables
import OAI.Combinatorics.IndependentSets.Encoding.GeometryExpr
namespace OAI
namespace LargeIndependentSets.GeometryNames
open IndependentSetsCut.CounterMachine UniformLC ShortestPaths
open scoped Classical BigOperators
noncomputable section
variable (L R : Type) [Fintype L] [Fintype R] (n D : ℕ) [NeZero D]
def pointCost (a b : StaticPoint L R n D) : ℕ :=
(Capped.clip D (copyCost (decode (fun _ => ()) (fun _ => ()) a)
(decode (fun _ => ()) (fun _ => ()) b))).val
omit [NeZero D] in
lemma copyCost_same {U V : Type} [DecidableEq U] [DecidableEq V]
(u u' : Fin n → U) (v v' : Fin n → V) (a b : StaticPoint L R n D) :
(Capped.clip D (copyCost (decode u v a) (decode u' v' b))).val =
if sameQuestion L R n D u u' v v' a b then pointCost L R n D a b else D+1 := by
by_cases hq : sameQuestion L R n D u u' v v' a b
· rw [ite_eq_left hq]
have hh := (sameQuestion_iff L R n D u u' v v' a b).mp hq
rcases a with ⟨i,a⟩
rcases b with ⟨j,b⟩
have hij : i=j := congrArg (fun w => w.1) hh
subst j
have hq' : mixed u v i.val=mixed u' v' i.val :=
eq_of_heq (Sigma.mk.inj_iff.mp hh).2
have hcopy (q q' : MixedTuple U V n i.val) (he : q=q') :
copyCost (M:=LayerAnswer L R) (⟨⟨i,q⟩,a⟩ : GridLocation _ D) ⟨⟨i,q'⟩,b⟩ =
(Finset.univ.sup (fun k => circleCost (a k) (b k)) : ℕ) := by
subst q'
simp [copyCost]
rw [show copyCost (decode u v ⟨i,a⟩) (decode u' v' ⟨i,b⟩)=
(Finset.univ.sup (fun k => circleCost (a k) (b k)) : ℕ) from hcopy _ _ hq']
simp [pointCost,copyCost,decode]
rfl
· have hh := (sameQuestion_iff L R n D u u' v v' a b).not.mp hq
simp [hq,copyCost,hh]
def questionExpr (x y c : Expr) (h : Fin n) : Expr :=
.mul (Expr.equal (uExpr L R n D x h) (fieldExpr c 0))
(.mul (Expr.equal (vExpr L R n D y h) (fieldExpr c 1))
(Expr.listAll (List.finRange n) (fun k => if k=h then .const 1 else
if k.val<h.val then Expr.equal (vExpr L R n D x k) (vExpr L R n D y k)
else Expr.equal (uExpr L R n D x k) (uExpr L R n D y k))))
lemma questionExpr_nonzero (t : Table L R) (x y c : Expr) (a : ℕ → ℕ)
(i j : Fin (size L R n D t)) (e : Fin t.ne)
(hi : x.eval t.bits a=i.val) (hj : y.eval t.bits a=j.val) (he : c.eval t.bits a=e.val) (h : Fin n) :
(questionExpr L R n D x y c h).eval t.bits a≠0 ↔
QuestionStep t.lc h e ((nodeEquiv L R n D t.nu t.nv).symm i).1.1
((nodeEquiv L R n D t.nu t.nv).symm j).1.1
((nodeEquiv L R n D t.nu t.nv).symm i).1.2
((nodeEquiv L R n D t.nu t.nv).symm j).1.2 := by
simp only [questionExpr,Expr.eval,Expr.equal_eval,Expr.listAll_eval,
uExpr_eval L R n D t x a i hi,vExpr_eval L R n D t y a j hj,
fieldExpr_eval t a c e he ⟨0,by omega⟩,fieldExpr_eval t a c e he ⟨1,by omega⟩]
have indicator (P : Prop) [Decidable P] : (if P then (1:ℕ) else 0)≠0 ↔ P := by
split_ifs <;> simp_all
simp only [↓reduceIte,Nat.one_ne_zero,mul_ne_zero_iff,indicator,Fin.val_inj]
unfold QuestionStep Table.lc
apply and_congr_right
intro _
apply and_congr_right
intro _
simp only [List.mem_finRange,forall_true_left]
apply forall_congr'
intro k
by_cases hk : k=h
· simp [hk,Expr.eval]
· by_cases hl : k.val<h.val <;> simp [hk,hl,Expr.equal_eval,
uExpr_eval L R n D t x a i hi,uExpr_eval L R n D t y a j hj,
vExpr_eval L R n D t x a i hi,vExpr_eval L R n D t y a j hj,Fin.val_inj]
def pointStepExpr (x y c : Expr) (h : Fin n) : Expr :=
Expr.static3 (fun (a b : StaticPoint L R n D) (f : L → R) => if PointStep h f a b then 1 else 0)
(pointExpr L R n D x) (pointExpr L R n D y) (fieldExpr c 2)
lemma pointStepExpr_eval (t : Table L R) (x y c : Expr) (a : ℕ → ℕ)
(i j : Fin (size L R n D t)) (e : Fin t.ne)
(hi : x.eval t.bits a=i.val) (hj : y.eval t.bits a=j.val) (he : c.eval t.bits a=e.val) (h : Fin n) :
(pointStepExpr L R n D x y c h).eval t.bits a =
if PointStep h (t.project e) ((nodeEquiv L R n D t.nu t.nv).symm i).2
((nodeEquiv L R n D t.nu t.nv).symm j).2 then 1 else 0 := by
apply Expr.static3_eval
· exact pointExpr_eval L R n D t x a i hi
· exact pointExpr_eval L R n D t y a j hj
· exact fieldExpr_eval t a c e he ⟨2,by omega⟩
def forwardExpr (x y : Expr) : Expr :=
.sum neExpr (Expr.listSum (List.finRange n) (fun h =>
.mul (pointStepExpr L R n D (x.rename Nat.succ) (y.rename Nat.succ) (.arg 0) h)
(questionExpr L R n D (x.rename Nat.succ) (y.rename Nat.succ) (.arg 0) h)))
lemma forwardExpr_nonzero (t : Table L R) (x y : Expr) (a : ℕ → ℕ)
(i j : Fin (size L R n D t)) (hi : x.eval t.bits a=i.val) (hj : y.eval t.bits a=j.val) :
(forwardExpr L R n D x y).eval t.bits a≠0 ↔
(layeredSystem t.lc n).ForwardZero D (names L R n D t.nu t.nv i) (names L R n D t.nu t.nv j) := by
dsimp only [names,nodeDecode]
rw [forward_iff]
change _ ↔ ∃ h e, PointStep h (t.project e) ((nodeEquiv L R n D t.nu t.nv).symm i).2
((nodeEquiv L R n D t.nu t.nv).symm j).2 ∧ QuestionStep t.lc h e _ _ _ _
simp only [forwardExpr,Expr.sum_ne_zero,neExpr_eval,Expr.listSum_ne_zero,List.mem_finRange,true_and]
have each (c : ℕ) (hc : c<t.ne) (h : Fin n) :
(Expr.mul (pointStepExpr L R n D (x.rename Nat.succ) (y.rename Nat.succ) (.arg 0) h)
(questionExpr L R n D (x.rename Nat.succ) (y.rename Nat.succ) (.arg 0) h)).eval
t.bits (Expr.bind a c)≠0 ↔
PointStep h (t.project ⟨c,hc⟩) ((nodeEquiv L R n D t.nu t.nv).symm i).2
((nodeEquiv L R n D t.nu t.nv).symm j).2 ∧ QuestionStep t.lc h ⟨c,hc⟩
((nodeEquiv L R n D t.nu t.nv).symm i).1.1
((nodeEquiv L R n D t.nu t.nv).symm j).1.1
((nodeEquiv L R n D t.nu t.nv).symm i).1.2
((nodeEquiv L R n D t.nu t.nv).symm j).1.2 := by
have hx : (x.rename Nat.succ).eval t.bits (Expr.bind a c)=i.val := by simpa [Expr.bind] using hi
have hy : (y.rename Nat.succ).eval t.bits (Expr.bind a c)=j.val := by simpa [Expr.bind] using hj
rw [Expr.eval,mul_ne_zero_iff,pointStepExpr_eval L R n D t _ _ _ _ i j ⟨c,hc⟩ hx hy rfl h,
questionExpr_nonzero L R n D t _ _ _ _ i j ⟨c,hc⟩ hx hy rfl h]
split_ifs <;> simp_all
constructor
· rintro ⟨c,hc,h,hh⟩; exact ⟨h,⟨c,hc⟩,(each c hc h).mp hh⟩
· rintro ⟨h,e,hh⟩; exact ⟨e.val,e.isLt,h,(each e.val e.isLt h).mpr hh⟩
def levelExpr (x : Expr) : Expr :=
Expr.staticLookup (fun a : StaticPoint L R n D => a.1.val) (pointExpr L R n D x)
lemma levelExpr_eval (t : Table L R) (x : Expr) (a : ℕ → ℕ) (i : Fin (size L R n D t))
(hi : x.eval t.bits a=i.val) :
(levelExpr L R n D x).eval t.bits a=((nodeEquiv L R n D t.nu t.nv).symm i).2.1.val := by
exact Expr.staticLookup_eval _ _ _ _ _ (pointExpr_eval L R n D t x a i hi)
def sameQuestionExpr (x y : Expr) : Expr :=
.mul (Expr.equal (levelExpr L R n D x) (levelExpr L R n D y))
(Expr.listAll (List.finRange n) (fun k =>
Expr.cond (Expr.lt (.const k.val) (levelExpr L R n D x))
(Expr.equal (vExpr L R n D x k) (vExpr L R n D y k))
(Expr.equal (uExpr L R n D x k) (uExpr L R n D y k))))
lemma sameQuestionExpr_nonzero (t : Table L R) (x y : Expr) (a : ℕ → ℕ)
(i j : Fin (size L R n D t)) (hi : x.eval t.bits a=i.val) (hj : y.eval t.bits a=j.val) :
(sameQuestionExpr L R n D x y).eval t.bits a≠0 ↔
sameQuestion L R n D ((nodeEquiv L R n D t.nu t.nv).symm i).1.1
((nodeEquiv L R n D t.nu t.nv).symm j).1.1
((nodeEquiv L R n D t.nu t.nv).symm i).1.2
((nodeEquiv L R n D t.nu t.nv).symm j).1.2
((nodeEquiv L R n D t.nu t.nv).symm i).2
((nodeEquiv L R n D t.nu t.nv).symm j).2 := by
have indicator (P : Prop) [Decidable P] : (if P then (1:ℕ) else 0)≠0 ↔ P := by
split_ifs <;> simp_all
simp only [sameQuestionExpr,Expr.eval,mul_ne_zero_iff,Expr.equal_eval,Expr.listAll_eval,
levelExpr_eval L R n D t x a i hi,levelExpr_eval L R n D t y a j hj,
indicator,Fin.val_inj,List.mem_finRange,forall_true_left,sameQuestion]
apply and_congr_right
intro _
apply forall_congr'
intro k
simp only [Expr.cond_eval,Expr.lt_eval,Expr.eval,levelExpr_eval L R n D t x a i hi,
indicator,Expr.equal_eval,vExpr_eval L R n D t x a i hi,vExpr_eval L R n D t y a j hj,
uExpr_eval L R n D t x a i hi,uExpr_eval L R n D t y a j hj]
split_ifs <;> simp_all [Fin.val_inj]
def copyCostExpr (x y : Expr) : Expr :=
Expr.cond (sameQuestionExpr L R n D x y)
(Expr.static2 (pointCost L R n D) (pointExpr L R n D x) (pointExpr L R n D y))
(.const (D+1))
lemma copyCostExpr_eval (t : Table L R) (x y : Expr) (a : ℕ → ℕ)
(i j : Fin (size L R n D t)) (hi : x.eval t.bits a=i.val) (hj : y.eval t.bits a=j.val) :
(copyCostExpr L R n D x y).eval t.bits a=
(Capped.clip D (copyCost (names L R n D t.nu t.nv i) (names L R n D t.nu t.nv j))).val := by
dsimp only [names,nodeDecode]
rw [copyCost_same]
simp only [copyCostExpr,Expr.cond_eval,sameQuestionExpr_nonzero L R n D t x y a i j hi hj,
Expr.static2_eval _ _ _ _ _ _ _ (pointExpr_eval L R n D t x a i hi)
(pointExpr_eval L R n D t y a j hj),Expr.eval]
def costExpr (x y : Expr) : Expr :=
Expr.cond (.add (forwardExpr L R n D x y) (forwardExpr L R n D y x))
(.const 0) (copyCostExpr L R n D x y)
lemma costExpr_eval (t : Table L R) (x y : Expr) (a : ℕ → ℕ)
(i j : Fin (size L R n D t)) (hi : x.eval t.bits a=i.val) (hj : y.eval t.bits a=j.val) :
(costExpr L R n D x y).eval t.bits a=
(Capped.clip D (linkCost (layeredSystem t.lc n)
(names L R n D t.nu t.nv i) (names L R n D t.nu t.nv j))).val := by
have add_ne (a b : ℕ) : a+b≠0 ↔ a≠0 ∨ b≠0 := by omega
simp only [costExpr,Expr.cond_eval,Expr.eval,add_ne,
forwardExpr_nonzero L R n D t x y a i j hi hj,
forwardExpr_nonzero L R n D t y x a j i hj hi,
copyCostExpr_eval L R n D t x y a i j hi hj,linkCost,ProjectionSystem.ZeroLink]
split_ifs <;> simp_all [Capped.clip]
end
end LargeIndependentSets.GeometryNames
end OAI
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