| import OAI.SetTheory.PartitionPrinciple.Definability.DecodedFunctionGraphs |
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| namespace OAI |
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| section |
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| namespace PartitionPilot.Forcing.GroupCode |
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| open SetRealization |
| universe u |
| variable {G m i e : ZFSet.{u}} (h : GroupCode G m i e) |
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| 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) |
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| noncomputable def inv (x : SetMembers G) : SetMembers G := |
| (SetMembers.presentation G).decodeFunction (SetMembers.presentation G) i h.inv_function x |
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| noncomputable def one : SetMembers G := SetMembers.ofMem e h.identity_mem |
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| theorem raw_one : SetMembers.raw h.one = e := SetMembers.raw_ofMem e h.identity_mem |
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| 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) |
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| 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 |
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| 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 |
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| 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) |
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| 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) |
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| 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) |
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| 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) |
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| @[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 |
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| 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 |
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| 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 |
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| theorem identity_code_eq : |
| letI := h.toGroup |
| (SetMembers.presentation G).encode 1 = e := h.raw_one |
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| end PartitionPilot.Forcing.GroupCode |
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| namespace PartitionPilot.Forcing.TransitiveGround |
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| open SetRealization SetFormula |
| universe u |
| variable (M : TransitiveGround.{u}) |
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| @[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 |
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| 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⟩ |
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| end PartitionPilot.Forcing.TransitiveGround |
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| end |
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| end OAI |
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