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0dd3af4 | 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 | import Mathlib
import OAI.GroupTheory.SimpleOvergroups.ActionMonoid
namespace OAI
noncomputable section
namespace SimpleFInftyOvergroups.RingAssembly
open SourceActionMonoid
open scoped Classical
variable {V Y R : Type*} [Group V] [MulAction V Y] [Ring R]
local instance actionCornerMonoidWithZero : MonoidWithZero (M V Y) where
__ := (inferInstance : Monoid (M V Y))
zero := omega
zero_mul := omega_mul
mul_zero := mul_omega
variable (δ : M V Y →*₀ R)
def actionUnits : V →* Rˣ := (δ.toMonoidHom.comp groupMap).toHomUnits
theorem actionUnits_val (w : V) : (actionUnits δ w).val=δ (groupMap w) := rfl
theorem actionUnits_injective (hδ : Function.Injective δ) : Function.Injective (actionUnits δ) := by
intro w v hwv
apply groupMap_injective (Z:=Y)
exact hδ (congrArg Units.val hwv)
theorem corner_TS (y z : Y) : δ (delete y) * δ (prepend z)=if y=z then 1 else 0 := by
rw [←map_mul]
by_cases hyz : y=z
· subst z
rw [ite_eq_left rfl,delete_prefix,map_one]
· rw [ite_eq_right hyz,delete_prefix_ne y z hyz]
exact map_zero δ
theorem corner_ρS (w : V) (y : Y) : (actionUnits δ w).val * δ (prepend y)=δ (prepend (w • y)) := by
rw [actionUnits_val,←map_mul,group_prefix]
theorem corner_Tρ (w : V) (y : Y) : δ (delete y) * (actionUnits δ w).val=δ (delete (w⁻¹ • y)) := by
rw [actionUnits_val,←map_mul,delete_group]
end SimpleFInftyOvergroups.RingAssembly
end
end OAI
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