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2fca478 | 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 | import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Abel
namespace OAI
universe uG uA uι
namespace PeriodicTilingThree
variable {G : Type uG} {A : Type uA} [AddCommGroup G] [AddCommGroup A]
def configDiff (v : G) (f : G → A) : G → A :=
fun x => f (x + v) - f x
@[simp] theorem configDiff_apply (v : G) (f : G → A) (x : G) :
configDiff v f x = f (x + v) - f x := rfl
@[simp] theorem configDiff_zero (v : G) : configDiff v (0 : G → A) = 0 := by
ext x
simp [configDiff]
@[simp] theorem configDiff_const (v : G) (a : A) :
configDiff v (fun _ : G => a) = 0 := by
ext x
simp [configDiff]
theorem configDiff_add (v : G) (f g : G → A) :
configDiff v (f + g) = configDiff v f + configDiff v g := by
ext x
simp only [configDiff_apply, Pi.add_apply]
abel
theorem configDiff_sub (v : G) (f g : G → A) :
configDiff v (f - g) = configDiff v f - configDiff v g := by
ext x
simp only [configDiff_apply, Pi.sub_apply]
abel
theorem configDiff_comm (v w : G) (f : G → A) :
configDiff v (configDiff w f) = configDiff w (configDiff v f) := by
ext x
have h : x + v + w = x + w + v := by abel
simp only [configDiff_apply, h]
abel
theorem configDiff_eq_zero_iff (v : G) (f : G → A) :
configDiff v f = 0 ↔ Function.Periodic f v := by
constructor
· intro h x
have hx := congrFun h x
exact sub_eq_zero.mp hx
· intro h
ext x
exact sub_eq_zero.mpr (h x)
theorem configDiff_add_period (v p : G) (f : G → A)
(hp : Function.Periodic f p) : configDiff (v + p) f = configDiff v f := by
ext x
simp only [configDiff_apply, ← add_assoc, hp (x + v)]
theorem configDiff_preserves_period (v p : G) (f : G → A)
(hp : Function.Periodic f p) : Function.Periodic (configDiff v f) p := by
apply (configDiff_eq_zero_iff p _).mp
rw [configDiff_comm, (configDiff_eq_zero_iff p f).mpr hp, configDiff_zero]
theorem configDiff_sum {ι : Type uι} (s : Finset ι) (v : G) (f : ι → G → A) :
configDiff v (∑ i ∈ s, f i) = ∑ i ∈ s, configDiff v (f i) := by
classical
induction s using Finset.induction_on with
| empty => simp
| @insert i s hi ih =>
simp only [Finset.sum_insert hi, configDiff_add, ih]
def configDiffs : List G → (G → A) → (G → A)
| [], f => f
| v :: vs, f => configDiff v (configDiffs vs f)
@[simp] theorem configDiffs_nil (f : G → A) : configDiffs [] f = f := rfl
@[simp] theorem configDiffs_cons (v : G) (vs : List G) (f : G → A) :
configDiffs (v :: vs) f = configDiff v (configDiffs vs f) := rfl
@[simp] theorem configDiffs_zero (vs : List G) :
configDiffs vs (0 : G → A) = 0 := by
induction vs with
| nil => rfl
| cons v vs ih => simp only [configDiffs_cons, ih, configDiff_zero]
theorem configDiff_configDiffs (v : G) (vs : List G) (f : G → A) :
configDiff v (configDiffs vs f) = configDiffs vs (configDiff v f) := by
induction vs with
| nil => rfl
| cons w vs ih =>
simp only [configDiffs_cons, configDiff_comm v w, ih]
theorem configDiffs_add (vs : List G) (f g : G → A) :
configDiffs vs (f + g) = configDiffs vs f + configDiffs vs g := by
induction vs with
| nil => rfl
| cons v vs ih => simp only [configDiffs_cons, ih, configDiff_add]
theorem configDiffs_sub (vs : List G) (f g : G → A) :
configDiffs vs (f - g) = configDiffs vs f - configDiffs vs g := by
induction vs with
| nil => rfl
| cons v vs ih => simp only [configDiffs_cons, ih, configDiff_sub]
theorem configDiffs_sum {ι : Type uι} (s : Finset ι) (vs : List G)
(f : ι → G → A) :
configDiffs vs (∑ i ∈ s, f i) = ∑ i ∈ s, configDiffs vs (f i) := by
classical
induction s using Finset.induction_on with
| empty => simp
| @insert i s hi ih =>
simp only [Finset.sum_insert hi, configDiffs_add, ih]
theorem configDiffs_eq_zero_of_mem {v : G} {vs : List G} (hv : v ∈ vs)
(f : G → A) (hf : configDiff v f = 0) : configDiffs vs f = 0 := by
induction vs with
| nil => simp at hv
| cons w vs ih =>
rcases List.mem_cons.mp hv with h | h
· subst w
rw [configDiffs_cons, configDiff_configDiffs, hf, configDiffs_zero]
· rw [configDiffs_cons, ih h, configDiff_zero]
theorem configDiffs_const {vs : List G} (hvs : vs ≠ []) (a : A) :
configDiffs vs (fun _ : G => a) = 0 := by
cases vs with
| nil => exact (hvs rfl).elim
| cons v vs =>
rw [configDiffs_cons, configDiff_configDiffs, configDiff_const,
configDiffs_zero]
theorem configDiffs_preserves_period (vs : List G) (p : G) (f : G → A)
(hp : Function.Periodic f p) : Function.Periodic (configDiffs vs f) p := by
apply (configDiff_eq_zero_iff p _).mp
rw [configDiff_configDiffs, (configDiff_eq_zero_iff p f).mpr hp,
configDiffs_zero]
theorem configDiffs_replicate (n : ℕ) (v : G) (f : G → A) :
configDiffs (List.replicate n v) f = (configDiff v)^[n] f := by
induction n with
| zero => rfl
| succ n ih =>
simp only [List.replicate_succ, configDiffs_cons, ih,
Function.iterate_succ_apply']
end PeriodicTilingThree
end OAI
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