File size: 5,223 Bytes
6fe30b6 | 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 | import OAI.Computability.FourierTransform.NetworkLabels
namespace OAI
namespace PowerSaving.Cluster
open Matrix Binary Finset
noncomputable section
/-- Conveniently prove the shear and inverse-transpose shear together. -/
def DPath (j : J) (F : StartAxes) (s t : Lab) (M : Q) (d : ℕ) : Prop :=
let A := basisBox j F
∃ N : Q, M * N = 1 ∧ Path A s t (actPoint M) d ∧ Path A s t (actPoint Nᵀ) d
lemma rat_if (P : Prop) [Decidable P] :
(((if P then 1 else 0):ℚ):ℂ) = if P then 1 else 0 := by split <;> simp_all
lemma ap_one : actPoint (1:Q) = id := by
funext x i
unfold actPoint
rw [Finset.sum_eq_single i]
· simp
· intro j _ h
simp [one_apply,Ne.symm h]
· simp
lemma ap_mul (M N : Q) :
actPoint (M*N) = actPoint M ∘ actPoint N := by
funext x i
simp only [actPoint, Function.comp_apply, Matrix.mul_apply, Rat.cast_sum, Rat.cast_mul,
sum_mul,mul_sum,_root_.mul_assoc]
rw [sum_comm]
namespace DPath
variable {j : J} {F : StartAxes} {s t u : Lab} {M N : Q} {d e : ℕ}
lemma of_id (h : Path (basisBox j F) s t id d) : DPath j F s t 1 d := by
refine ⟨1,mul_one _,?_,?_⟩ <;> simpa [ap_one]
lemma trans (a : DPath j F s t M d) (b : DPath j F t u N e) :
DPath j F s u (N*M) (d+e) := by
obtain ⟨x,hx,hm,hn⟩ := a
obtain ⟨y,hy,hl,hk⟩ := b
refine ⟨x*y,?_,?_,?_⟩
· rwa [Matrix.mul_assoc, ← Matrix.mul_assoc M, hx, one_mul]
· rw [ap_mul]
exact hm.trans hl
· rw [transpose_mul, ap_mul]
exact hn.trans hk
lemma paired (h : M*M=0) (g : syncBox j F s M) :
DPath j F s s (1+M) 0 ∧ DPath j F s s (1-M) 0 := by
have cond (i k) (h : M i k≠0) :
frame (basisBox j F i) (s i) = frame (basisBox j F k) (s k) :=
frame_congr _ _ _ _ (g i k h)
constructor
· refine ⟨1-M,inv_one_add h,Path.gate _ ?_,Path.gate _ ?_⟩
· intro i k h
by_cases hik : i=k
· subst k; rfl
· apply cond i k; simpa [one_apply,hik] using h
· intro i k h
by_cases hik : k=i
· subst k; rfl
· exact (cond k i (by simpa [one_apply,hik] using h)).symm
· refine ⟨1+M,inv_one_sub h,Path.gate _ ?_,Path.gate _ ?_⟩
· intro i k h
by_cases hik : i=k
· subst k; rfl
· apply cond i k; simpa [one_apply,hik] using h
· intro i k h
by_cases hik : k=i
· subst k; rfl
· exact (cond k i (by simpa [one_apply,hik] using h)).symm
lemma rise (g : ∀ i, s i ⊆ t i) : DPath j F s t 1 0 :=
of_id (Path.inc _ g)
end DPath
theorem invocation_path (j : J) (F : StartAxes) :
DPath j F (bp j) (bend j) (1+eXY) (33*32) := by
have h0 := DPath.paired zero_JJ (sync0 j F) |>.2
have h1 := DPath.paired zero_RR (sync1 j F) |>.2
have h2 := DPath.paired zero_VV (sync2 j F) |>.1
have h3 := DPath.paired zero_GG (sync3 j F) |>.1
have h4 := DPath.paired zero_RR (sync4 j F) |>.1
have h5 := DPath.paired zero_JJ (sync5 j F) |>.1
have h6 := DPath.paired zero_GG (sync6 j F) |>.2
have h7 := DPath.paired zero_VV (sync7 j F) |>.2
have rise : ∀ {u v : Lab}, (∀ i, u i ⊆ v i) → DPath j F u v 1 0 := fun h => DPath.rise h
have down : DPath j F (b3 j) (b4 j) 1 (33*32) := by
rw [← loss34 j]; exact DPath.of_id (Path.reframe ..)
have H0 := (rise (inc_bp_b0 j)).trans h0
have H1 := H0.trans (rise (inc_b0_b1 j)) |>.trans h1
have H2 := H1.trans (rise (inc_b1_b2 j)) |>.trans h2
have H3 := H2.trans (rise (inc_b2_b3 j)) |>.trans h3
have H4 := H3.trans down |>.trans h4
have H5 := H4.trans (rise (inc_b4_b5 j)) |>.trans h5
have H6 := H5.trans (rise (inc_b5_b6 j)) |>.trans h6
have H7 := H6.trans (rise (inc_b6_b7 j)) |>.trans h7
have Hl := H7.trans (rise (inc_b7_bend j))
rw [← net_eq, net]
simpa only [← Matrix.mul_assoc,mul_one,one_mul,Nat.zero_add,Nat.add_zero] using Hl
/-- Induced logical actions. -/
def forward (x : Box→ℂ) : Box→ℂ
| .inl (.inr i) => x (byy i) + x (bx i)
| r => x r
def backward (x : Box→ℂ) : Box→ℂ
| .inl (.inl i) => x (bx i) - x (byy i)
| r => x r
lemma ap_add (M N : Q) (x : Box→ℂ) (r : Box) :
actPoint (M+N) x r = actPoint M x r + actPoint N x r := by
simp [actPoint, _root_.add_mul,sum_add_distrib]
lemma ap_sub (M N : Q) (x : Box→ℂ) (r : Box) :
actPoint (M-N) x r = actPoint M x r - actPoint N x r := by
simp [actPoint, _root_.sub_mul,sum_sub_distrib]
lemma ap_f : actPoint (1+eXY)=forward := by
funext x i
rw [ap_add, ap_one]
rcases i with ((i|i)|(i|i)) <;>
simp [actPoint, Fintype.sum_sum_type, eXY, one_apply, rat_if, ite_mul, forward]
lemma ap_b : actPoint (1-eXY)ᵀ=backward := by
funext x i
rw [transpose_sub, transpose_one, ap_sub, ap_one]
rcases i with ((i|i)|(i|i)) <;>
simp [actPoint, Fintype.sum_sum_type, eXY, one_apply, rat_if, ite_mul, backward]
theorem invocation (j : J) (F : StartAxes) :
Path (basisBox j F) (bp j) (bend j) forward (33*32) ∧
Path (basisBox j F) (bp j) (bend j) backward (33*32) := by
obtain ⟨N, hN, H, G⟩ := invocation_path j F
have h : N=1-eXY := by
calc
N = ((1-eXY)*(1+eXY))*N := by rw [inv_one_sub zero_XX, one_mul]
_ = _ := by rw [Matrix.mul_assoc,hN,mul_one]
subst N
exact ⟨ap_f ▸ H,ap_b ▸ G⟩
end
end PowerSaving.Cluster
end OAI
|