File size: 2,997 Bytes
3277efa | 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 62 63 64 65 66 67 68 69 70 71 | import OAI.SetTheory.PartitionPrinciple.Definability.TaskCapFamilyCodes
namespace OAI
section
namespace PartitionPilot.Forcing
universe u
theorem relationImage_retarget_eq {A Q V F e : ZFSet.{u}}
(shape : e ⊆ ZFSet.prod A Q) (subset : Q ⊆ V) :
relationImage F V e = relationImage F Q e := by
apply ZFSet.ext
intro x
rw [mem_relationImage, mem_relationImage]
constructor
· rintro ⟨_, f, hf, entry⟩
exact ⟨(ZFSet.pair_mem_prod.1 (shape entry)).2, f, hf, entry⟩
· rintro ⟨hx, f, hf, entry⟩
exact ⟨subset hx, f, hf, entry⟩
namespace CodedCapProfile
variable {P r μ c E K F : ZFSet.{u}}
theorem fixedTop (profile : CodedCapProfile P r μ c E K F) (meet : MeetOrderCode P r μ) :
CodedCapProfile (fixedTopCarrier P) (fixedTopOrder P r) (fixedTopMeet P r) c E K F := by
refine ⟨fixedTopCarrier_old profile.target_mem, fixedTopCarrier_old profile.separator_mem,
fixedTopCarrier_old profile.cap_mem, fun _ hf => fixedTopCarrier_old (profile.filter_subset hf),
profile.nonempty, ?_, fixedTopOrder_subset P r profile.target_below_cap,
fun lower => profile.cap_not_below_target ((fixedTopOrder_old_iff profile.target_mem).1 lower),
fun lower => profile.separator_not_below_cap ((fixedTopOrder_old_iff profile.cap_mem).1 lower)⟩
intro x hx above
rcases (mem_fixedTopCarrier P x).1 hx with equal | oldx
· subst x
have positive : ∃ f ∈ F, ZFSet.pair f P ∈ fixedTopOrder P r := by
obtain ⟨f, hf⟩ := profile.nonempty
exact ⟨f, hf, fixedTopOrder_greatest (fixedTopCarrier_old (profile.filter_subset hf))⟩
exact ⟨fun _ => fixedTopOrder_greatest (fixedTopCarrier_old profile.cap_mem),
fun negative => False.elim (negative positive)⟩
· have old := profile.profile x oldx ((fixedTopOrder_old_iff oldx).1 above)
have positive : (∃ f ∈ F, ZFSet.pair f x ∈ fixedTopOrder P r) ↔
∃ f ∈ F, ZFSet.pair f x ∈ r := by
simp only [fixedTopOrder_old_iff oldx]
exact ⟨fun hp => fixedTopOrder_subset P r (old.1 (positive.1 hp)),
fun hn => fixedTopMeet_subset meet (old.2 (fun hp => hn (positive.2 hp)))⟩
end CodedCapProfile
namespace OldProfilePreservationCode
variable {P r μ Q R ν e Filters : ZFSet.{u}}
theorem fixedTop (preserved : OldProfilePreservationCode P r μ Q R ν e Filters)
(target : MeetOrderCode Q R ν) (shape : e ⊆ ZFSet.prod P Q) :
OldProfilePreservationCode P r μ (fixedTopCarrier Q) (fixedTopOrder Q R)
(fixedTopMeet Q R) e Filters := by
intro F hF c hc E hE K hK profile c' _ E' _ K' _ imageC imageE imageK
have inherited := preserved F hF c hc E hE K hK profile
c' (ZFSet.pair_mem_prod.1 (shape imageC)).2
E' (ZFSet.pair_mem_prod.1 (shape imageE)).2
K' (ZFSet.pair_mem_prod.1 (shape imageK)).2 imageC imageE imageK
rw [relationImage_retarget_eq shape (fixedTopCarrier_subset Q)]
exact inherited.fixedTop target
end OldProfilePreservationCode
end PartitionPilot.Forcing
end
end OAI
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