File size: 2,958 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 | import OAI.SetTheory.PartitionPrinciple.Geometry.AssignmentNestedSplices
namespace OAI
section
namespace PartitionPilot.Forcing
universe u
def FlatHistoryCuts (stages history levels row A : ZFSet.{u}) : Prop :=
∀ beta ∈ stages, ∃ Q, ZFSet.pair beta Q ∈ history ∧
ZFSet.pair row (assignmentCut levels beta A) ∈ Q
def FlatLimitConditionCode (stages history levels z : ZFSet.{u}) : Prop :=
∃ row A, z = ZFSet.pair row A ∧ FlatHistoryCuts stages history levels row A
theorem flatLimitConditionCode_pair (stages history levels row A : ZFSet.{u}) :
FlatLimitConditionCode stages history levels (ZFSet.pair row A) ↔
FlatHistoryCuts stages history levels row A := by
constructor
· rintro ⟨row', A', equal, valid⟩
obtain ⟨rfl, rfl⟩ := ZFSet.pair_injective equal
exact valid
· intro valid
exact ⟨row, A, rfl, valid⟩
theorem flatHistoryCuts_lookup_iff {stages history levels row A : ZFSet.{u}}
(function : FunctionGraphOn stages history) :
FlatHistoryCuts stages history levels row A ↔
∀ beta ∈ stages, ∀ Q, ZFSet.pair beta Q ∈ history →
ZFSet.pair row (assignmentCut levels beta A) ∈ Q := by
constructor
· intro valid beta hbeta Q entry
obtain ⟨Q', entry', member⟩ := valid beta hbeta
exact function.unique entry' entry ▸ member
· intro valid beta hbeta
obtain ⟨Q, entry⟩ := function.total hbeta
exact ⟨Q, entry, valid beta hbeta Q entry⟩
theorem flatHistoryCuts_restrict {stages earlier history levels row A : ZFSet.{u}}
(valid : FlatHistoryCuts stages history levels row A) (subset : earlier ⊆ stages) :
FlatHistoryCuts earlier (graphRestriction earlier history) levels row A := by
intro beta hbeta
obtain ⟨Q, entry, member⟩ := valid beta (subset hbeta)
exact ⟨Q, pair_mem_graphRestriction.2 ⟨entry, hbeta⟩, member⟩
theorem flatHistoryCuts_cut {stages earlier history levels alpha row A : ZFSet.{u}}
(valid : FlatHistoryCuts stages history levels row A) (subset : earlier ⊆ stages)
(nested : ∀ beta ∈ earlier, assignmentCut levels beta (assignmentCut levels alpha A) =
assignmentCut levels beta A) :
FlatHistoryCuts earlier (graphRestriction earlier history) levels row
(assignmentCut levels alpha A) := by
intro beta hbeta
obtain ⟨Q, entry, member⟩ := flatHistoryCuts_restrict valid subset beta hbeta
exact ⟨Q, entry, (nested beta hbeta).symm ▸ member⟩
theorem flatHistoryCuts_cut_of_subset {stages earlier history levels alpha row A : ZFSet.{u}}
(valid : FlatHistoryCuts stages history levels row A) (subset : earlier ⊆ stages)
(below : ∀ beta ∈ earlier, beta ⊆ alpha) :
FlatHistoryCuts earlier (graphRestriction earlier history) levels row
(assignmentCut levels alpha A) :=
flatHistoryCuts_cut valid subset (fun beta hbeta => assignmentCut_nested (below beta hbeta))
end PartitionPilot.Forcing
end
end OAI
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