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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.CellGeometry
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section ConditionalIntervalDifference
variable {a m : ℕ}
theorem conditionalAlternatingHom_union (U V : polygonAlgebra a)
(hd : Disjoint U.val V.val) (s : alternatingGroup (Fin m)) :
conditionalAlternatingHom (U ⊔ V) s = conditionalAlternatingHom U s*conditionalAlternatingHom V s := by
apply Subtype.ext
apply Subtype.ext
apply Equiv.ext
intro p
change conditionalPerm (U ⊔ V) s.val p = conditionalPerm U s.val (conditionalPerm V s.val p)
by_cases hu : p.2 ∈ U.val <;> by_cases hv : p.2 ∈ V.val
· exact False.elim (Set.disjoint_left.mp hd hu hv)
all_goals simp [conditionalPerm,hu,hv]
theorem conditionalAlternatingHom_interval_difference (d : Fin 2) (u v w : CutRing)
(huv : ordinary u ≤ ordinary v) (hvw : ordinary v ≤ ordinary w)
(hlen : ordinary w-ordinary u < 1) (s : alternatingGroup (Fin m)) :
(conditionalAlternatingHom (coordinateInterval a d u v) s)⁻¹ *
conditionalAlternatingHom (coordinateInterval a d u w) s =
conditionalAlternatingHom (coordinateInterval a d v w) s := by
rw [coordinateInterval_union d u v w huv hvw hlen,
conditionalAlternatingHom_union _ _ (coordinateInterval_disjoint d u v v w huv le_rfl hvw hlen)]
group
end ConditionalIntervalDifference
end SimpleAmenable
end
end
end OAI
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