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9dbc64e | 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 | import OAI.Computability.DepthThree.LanguageBitLoopInvariants
import Mathlib.Data.List.GetD
namespace OAI
namespace DepthThreeLowerBound
def xorListAt (c : List Bool) (k : ℕ) (v : Bool) : List Bool :=
c.set k (Bool.xor (c.getD k false) v)
@[simp] theorem xorListAt_length (c : List Bool) (k : ℕ) (v : Bool) :
(xorListAt c k v).length = c.length := by
simp [xorListAt]
theorem xorListAt_getD (c : List Bool) (j : ℕ) (v : Bool)
(hj : j < c.length) (k : ℕ) :
(xorListAt c j v).getD k false =
Bool.xor (c.getD k false) (if j = k then v else false) := by
by_cases h : j = k
· subst k
simp [xorListAt, List.getElem?_set_self hj]
· simp [xorListAt, h, List.getElem?_set_ne h]
theorem xorListAt_split («prefix» suffix : List Bool) (old v : Bool) :
xorListAt («prefix» ++ old :: suffix) «prefix».length v =
«prefix» ++ Bool.xor old v :: suffix := by
unfold xorListAt
rw [List.getD_append_right «prefix» (old :: suffix) false «prefix».length le_rfl]
simp
def convolutionUpdateRow (a b : List Bool) (i : ℕ) :
List ℕ → List Bool → List Bool
| [], c => c
| j :: js, c => convolutionUpdateRow a b i js
(xorListAt c (i + j) (a.getD i false && b.getD j false))
@[simp] theorem convolutionUpdateRow_nil (a b : List Bool) (i : ℕ) (c : List Bool) :
convolutionUpdateRow a b i [] c = c := rfl
theorem convolutionUpdateRow_step (a b : List Bool) (i j : ℕ)
(js : List ℕ) (c : List Bool) :
convolutionUpdateRow a b i (j :: js) c =
convolutionUpdateRow a b i js
(xorListAt c (i + j) (a.getD i false && b.getD j false)) := rfl
@[simp] theorem convolutionUpdateRow_length (a b : List Bool) (i : ℕ)
(js : List ℕ) (c : List Bool) :
(convolutionUpdateRow a b i js c).length = c.length := by
induction js generalizing c with
| nil => rfl
| cons j js ih =>
rw [convolutionUpdateRow_step, ih, xorListAt_length]
theorem convolutionUpdateRow_getD (a b : List Bool) (i : ℕ)
(js : List ℕ) (c : List Bool) (k : ℕ)
(hbound : ∀ j ∈ js, i + j < c.length) :
(convolutionUpdateRow a b i js c).getD k false =
Bool.xor (c.getD k false)
(xorBits (js.map fun j =>
if i + j = k then a.getD i false && b.getD j false else false)) := by
induction js generalizing c with
| nil => simp
| cons j js ih =>
have hj : i + j < c.length := hbound j (by simp)
have hrest : ∀ j' ∈ js, i + j' <
(xorListAt c (i + j) (a.getD i false && b.getD j false)).length := by
intro j' hj'
rw [xorListAt_length]
exact hbound j' (List.mem_cons_of_mem j hj')
rw [convolutionUpdateRow_step, ih _ hrest,
xorListAt_getD c (i + j) (a.getD i false && b.getD j false) hj k]
simp only [List.map_cons, xorBits_cons, Bool.xor_assoc]
def convolutionUpdateRows (a b : List Bool) : List ℕ → List Bool → List Bool
| [], c => c
| i :: is, c => convolutionUpdateRows a b is
(convolutionUpdateRow a b i (List.range b.length) c)
@[simp] theorem convolutionUpdateRows_nil (a b : List Bool) (c : List Bool) :
convolutionUpdateRows a b [] c = c := rfl
theorem convolutionUpdateRows_step (a b : List Bool) (i : ℕ)
(is : List ℕ) (c : List Bool) :
convolutionUpdateRows a b (i :: is) c =
convolutionUpdateRows a b is
(convolutionUpdateRow a b i (List.range b.length) c) := rfl
@[simp] theorem convolutionUpdateRows_length (a b : List Bool)
(is : List ℕ) (c : List Bool) :
(convolutionUpdateRows a b is c).length = c.length := by
induction is generalizing c with
| nil => rfl
| cons i is ih =>
rw [convolutionUpdateRows_step, ih, convolutionUpdateRow_length]
theorem convolutionUpdateRows_getD (a b : List Bool)
(is : List ℕ) (c : List Bool) (k : ℕ)
(hbound : ∀ i ∈ is, ∀ j < b.length, i + j < c.length) :
(convolutionUpdateRows a b is c).getD k false =
Bool.xor (c.getD k false)
(xorBits (is.map fun i => xorBits (convolutionRow a b i k))) := by
induction is generalizing c with
| nil => simp
| cons i is ih =>
have hcols : ∀ j ∈ List.range b.length, i + j < c.length := by
intro j hj
exact hbound i (by simp) j (List.mem_range.mp hj)
have hrest : ∀ i' ∈ is, ∀ j < b.length, i' + j <
(convolutionUpdateRow a b i (List.range b.length) c).length := by
intro i' hi' j hj
rw [convolutionUpdateRow_length]
exact hbound i' (List.mem_cons_of_mem i hi') j hj
rw [convolutionUpdateRows_step, ih _ hrest,
convolutionUpdateRow_getD a b i (List.range b.length) c k hcols]
simp only [List.map_cons, xorBits_cons, convolutionRow, Bool.xor_assoc]
def convolveByUpdates (a b : List Bool) : List Bool :=
convolutionUpdateRows a b (List.range a.length)
(List.replicate (a.length + b.length) false)
@[simp] theorem convolveByUpdates_length (a b : List Bool) :
(convolveByUpdates a b).length = a.length + b.length := by
simp [convolveByUpdates]
theorem convolutionUpdate_index_lt (a b : List Bool) {i j : ℕ}
(hi : i < a.length) (hj : j < b.length) :
i + j < a.length + b.length :=
Nat.add_lt_add hi hj
theorem convolveByUpdates_getD (a b : List Bool) (k : ℕ) :
(convolveByUpdates a b).getD k false = convolutionBit a b k := by
have hbound : ∀ i ∈ List.range a.length, ∀ j < b.length,
i + j < (List.replicate (a.length + b.length) false).length := by
intro i hi j hj
simpa only [List.length_replicate] using
convolutionUpdate_index_lt a b (List.mem_range.mp hi) hj
unfold convolveByUpdates
rw [convolutionUpdateRows_getD a b (List.range a.length)
(List.replicate (a.length + b.length) false) k hbound]
simp [convolutionBit, convolutionRows]
theorem convolveByUpdates_eq (a b : List Bool) :
convolveByUpdates a b = convolveBitLists a b := by
apply List.ext_getElem
· simp
· intro k hk hl
have hk' : k < a.length + b.length := by
simpa only [convolveBitLists_length] using hl
have hleft := convolveByUpdates_getD a b k
have hright : (convolveBitLists a b).getD k false = convolutionBit a b k := by
simp [convolveBitLists, List.getElem?_range hk']
rw [List.getD_eq_getElem _ _ hk] at hleft
rw [List.getD_eq_getElem _ _ hl] at hright
exact hleft.trans hright.symm
end DepthThreeLowerBound
end OAI
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