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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 | import OAI.Geometry.PeriodicTiling.ConfigurationDifferences
namespace OAI
universe uG uA uB uι
namespace PeriodicTilingThree
variable {G : Type uG} {A : Type uA} {B : Type uB}
[AddCommGroup G] [AddCommGroup A] [AddCommGroup B]
theorem configDiffs_eq_iterate_of_period (vs : List G) (v p : G) (f : G → A)
(hp : Function.Periodic f p)
(hvs : ∀ u ∈ vs, ∃ k : ℤ, u = v + k • p) :
configDiffs vs f = (configDiff v)^[vs.length] f := by
induction vs with
| nil => rfl
| cons u vs ih =>
obtain ⟨k, hk⟩ := hvs u (List.mem_cons_self ..)
have htail : ∀ w ∈ vs, ∃ k : ℤ, w = v + k • p := by
intro w hw
exact hvs w (List.mem_cons_of_mem _ hw)
have hper := (configDiffs_preserves_period vs p f hp).zsmul k
rw [configDiffs_cons, hk, configDiff_add_period v (k • p) _ hper,
ih htail, List.length_cons, Function.iterate_succ_apply']
theorem configDiffs_component_zero {ι : Type uι} [Fintype ι]
(j : ι) (vs : List G) (hvs : vs ≠ []) (f : ι → G → A) (a : A)
(hsum : (∑ i, f i) = fun _ => a)
(hkill : ∀ i, i ≠ j → ∃ v ∈ vs, Function.Periodic (f i) v) :
configDiffs vs (f j) = 0 := by
classical
have hsum' := congrArg (configDiffs vs) hsum
rw [configDiffs_sum, configDiffs_const hvs] at hsum'
have hone : (∑ i, configDiffs vs (f i)) = configDiffs vs (f j) := by
apply Finset.sum_eq_single j
· intro i _hi hij
obtain ⟨v, hv, hp⟩ := hkill i hij
exact configDiffs_eq_zero_of_mem hv (f i)
((configDiff_eq_zero_iff v (f i)).mpr hp)
· intro hj
exact (hj (Finset.mem_univ j)).elim
rw [hone] at hsum'
exact hsum'
theorem map_configDiff (φ : A →+ B) (v : G) (f : G → A) :
(fun x => φ (configDiff v f x)) = configDiff v (fun x => φ (f x)) := by
ext x
exact map_sub φ _ _
theorem map_configDiffs (φ : A →+ B) (vs : List G) (f : G → A) :
(fun x => φ (configDiffs vs f x)) = configDiffs vs (fun x => φ (f x)) := by
induction vs with
| nil => rfl
| cons v vs ih =>
rw [configDiffs_cons, map_configDiff, ih, configDiffs_cons]
end PeriodicTilingThree
end OAI
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