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namespace OAI
section
namespace PartitionPilot.Forcing.GroupCode
open SetRealization
universe u
variable {G m i e : ZFSet.{u}} (h : GroupCode G m i e)
noncomputable def mul (x y : SetMembers G) : SetMembers G :=
((SetMembers.presentation G).product (SetMembers.presentation G)).decodeFunction
(SetMembers.presentation G) m h.mul_function (x, y)
noncomputable def inv (x : SetMembers G) : SetMembers G :=
(SetMembers.presentation G).decodeFunction (SetMembers.presentation G) i h.inv_function x
noncomputable def one : SetMembers G := SetMembers.ofMem e h.identity_mem
theorem raw_one : SetMembers.raw h.one = e := SetMembers.raw_ofMem e h.identity_mem
theorem mul_mem (x y : SetMembers G) :
ZFSet.pair (ZFSet.pair (SetMembers.raw x) (SetMembers.raw y)) (SetMembers.raw (h.mul x y)) ∈ m :=
((SetMembers.presentation G).product (SetMembers.presentation G)).decodeFunction_mem
(SetMembers.presentation G) m h.mul_function (x, y)
theorem inv_mem (x : SetMembers G) :
ZFSet.pair (SetMembers.raw x) (SetMembers.raw (h.inv x)) ∈ i :=
(SetMembers.presentation G).decodeFunction_mem (SetMembers.presentation G) i h.inv_function x
theorem mul_eq_iff (x y z : SetMembers G) : h.mul x y = z ↔
ZFSet.pair (ZFSet.pair (SetMembers.raw x) (SetMembers.raw y)) (SetMembers.raw z) ∈ m :=
((SetMembers.presentation G).product (SetMembers.presentation G)).decodeFunction_iff
(SetMembers.presentation G) m h.mul_function (x, y) z
theorem decoded_one_mul (x : SetMembers G) : h.mul h.one x = x := by
apply (h.mul_eq_iff _ _ _).2
rw [h.raw_one]
exact h.one_mul _ (SetMembers.raw_mem x)
theorem decoded_mul_one (x : SetMembers G) : h.mul x h.one = x := by
apply (h.mul_eq_iff _ _ _).2
rw [h.raw_one]
exact h.mul_one _ (SetMembers.raw_mem x)
theorem decoded_inv_mul (x : SetMembers G) : h.mul (h.inv x) x = h.one := by
apply (h.mul_eq_iff _ _ _).2
rw [h.raw_one]
exact h.inv_mul _ (SetMembers.raw_mem x) _ (SetMembers.raw_mem (h.inv x)) (h.inv_mem x)
theorem decoded_mul_assoc (x y z : SetMembers G) :
h.mul (h.mul x y) z = h.mul x (h.mul y z) := by
apply Eq.symm
apply (h.mul_eq_iff _ _ _).2
exact h.mul_assoc _ (SetMembers.raw_mem x) _ (SetMembers.raw_mem y) _ (SetMembers.raw_mem z)
_ (SetMembers.raw_mem (h.mul x y)) _ (SetMembers.raw_mem (h.mul y z))
_ (SetMembers.raw_mem (h.mul (h.mul x y) z)) (h.mul_mem x y) (h.mul_mem y z)
(h.mul_mem (h.mul x y) z)
@[instance_reducible] noncomputable def toGroup : Group (SetMembers G) where
mul := h.mul
one := h.one
inv := h.inv
mul_assoc := h.decoded_mul_assoc
one_mul := h.decoded_one_mul
mul_one := h.decoded_mul_one
inv_mul_cancel := h.decoded_inv_mul
theorem multiplication_graph_eq :
letI := h.toGroup
((SetMembers.presentation G).product (SetMembers.presentation G)).graph
(SetMembers.presentation G) (fun x => x.1 * x.2) = m := by
let := h.toGroup
exact ((SetMembers.presentation G).product (SetMembers.presentation G)).graph_decodeFunction
(SetMembers.presentation G) m h.mul_function
theorem inverse_graph_eq :
letI := h.toGroup
(SetMembers.presentation G).graph (SetMembers.presentation G) (fun x => x⁻¹) = i := by
let := h.toGroup
exact (SetMembers.presentation G).graph_decodeFunction (SetMembers.presentation G) i h.inv_function
theorem identity_code_eq :
letI := h.toGroup
(SetMembers.presentation G).encode 1 = e := h.raw_one
end PartitionPilot.Forcing.GroupCode
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u})
@[instance_reducible] noncomputable def decodedGroup {G m i e : ZFSet.{u}}
(hG : G ∈ M.sets) (hm : m ∈ M.sets) (hi : i ∈ M.sets) (he : e ∈ M.sets)
(laws : Realize M.sets (BoundedFormula.groupFormula 0 1 2 3).toFormula
(extend G (extend m (extend i (extend e Fin.elim0))))) : Group (SetMembers G) :=
((M.realize_groupCode hG hm hi he).1 laws).toGroup
theorem decoded_group_graphs_mem {G m i e : ZFSet.{u}} (h : GroupCode G m i e)
(hm : m ∈ M.sets) (hi : i ∈ M.sets) :
letI := h.toGroup
((SetMembers.presentation G).product (SetMembers.presentation G)).graph
(SetMembers.presentation G) (fun x => x.1 * x.2) ∈ M.sets ∧
(SetMembers.presentation G).graph (SetMembers.presentation G) (fun x => x⁻¹) ∈ M.sets := by
let := h.toGroup
rw [h.multiplication_graph_eq, h.inverse_graph_eq]
exact ⟨hm, hi⟩
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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