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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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | import OAI.SetTheory.PartitionPrinciple.Transfinite.FinalRectangleStages
namespace OAI
namespace PartitionPilot.Compression.RecursiveMaps.MainStage.OrdinalRun
variable (Label : Type 1)
theorem rectangle_data_in_stage
(r : ActionRectangle (finalOrder Label) (finalAction Label).iso) (X : FinalPoint Label) :
∃ a, ∃ t : ActionRectangle (stage Label a).order (stage Label a).action.iso,
∃ Y : (stage Label a).Point,
(finalInclusion Label a).toFun t.C = r.C ∧
(finalInclusion Label a).toFun t.B = r.B ∧
(finalInclusion Label a).toFun t.D = r.D ∧
finalGroupInclusion Label a t.first = r.first ∧
finalGroupInclusion Label a t.second = r.second ∧
(finalInclusion Label a).toFun Y = X := by
let points (j : Bool × Bool) : FinalPoint Label :=
if j.1 then (if j.2 then X else r.D) else (if j.2 then r.B else r.C)
let actors (j : Bool) : FinalActor Label := if j then r.second else r.first
rcases countable_data_in_stage Label points actors with ⟨a, hp, hg⟩
obtain ⟨C, hC⟩ : ∃ C, (finalInclusion Label a).toFun C = r.C := hp (false, false)
obtain ⟨B, hB⟩ : ∃ B, (finalInclusion Label a).toFun B = r.B := hp (false, true)
obtain ⟨D, hD⟩ : ∃ D, (finalInclusion Label a).toFun D = r.D := hp (true, false)
obtain ⟨Y, hY⟩ : ∃ Y, (finalInclusion Label a).toFun Y = X := hp (true, true)
obtain ⟨g1, hg1⟩ : ∃ g, finalGroupInclusion Label a g = r.first := hg false
obtain ⟨g2, hg2⟩ : ∃ g, finalGroupInclusion Label a g = r.second := hg true
have cd : (stage Label a).order.le C D := by
apply ((finalInclusion Label a).le_iff _ _).mp
rw [hC, hD]
exact r.C_le_D
have bd : (stage Label a).order.le B D := by
apply ((finalInclusion Label a).le_iff _ _).mp
rw [hB, hD]
exact r.B_le_D
have agree : ∀ x, (stage Label a).order.le x ((stage Label a).order.meet C B) →
((stage Label a).action.iso g1).toFun x = ((stage Label a).action.iso g2).toFun x := by
intro x lower
apply embedding_injective (finalInclusion Label a)
rw [← final_action_stage Label a g1 x, ← final_action_stage Label a g2 x, hg1, hg2]
apply r.agree
have h := ((finalInclusion Label a).le_iff _ _).mpr lower
rw [(finalInclusion Label a).map_meet, hC, hB] at h
exact h
have separated : (stage Label a).order.meet
(((stage Label a).action.iso g1).toFun D) (((stage Label a).action.iso g2).toFun D) =
((stage Label a).action.iso g1).toFun ((stage Label a).order.meet C B) := by
apply embedding_injective (finalInclusion Label a)
rw [(finalInclusion Label a).map_meet, ← final_action_stage Label a g1 D, ← final_action_stage Label a g2 D, ← final_action_stage Label a g1 ((stage Label a).order.meet C B), (finalInclusion Label a).map_meet, hC, hB, hD, hg1, hg2]
exact r.separated
exact ⟨a, ⟨C, B, D, g1, g2, cd, bd, agree, separated⟩, Y, hC, hB, hD, hg1, hg2, hY⟩
theorem final_rectangles
(r : ActionRectangle (finalOrder Label) (finalAction Label).iso) (X : FinalPoint Label)
(first_upper : (finalOrder Label).le (((finalAction Label).iso r.first).toFun r.D) X)
(second_upper : (finalOrder Label).le (((finalAction Label).iso r.second).toFun r.D) X) :
∃ h1 h2 : FinalActor Label,
(∀ z, (finalOrder Label).le z r.C →
((finalAction Label).iso h1).toFun z = ((finalAction Label).iso r.first).toFun z) ∧
(∀ z, (finalOrder Label).le z r.C →
((finalAction Label).iso h2).toFun z = ((finalAction Label).iso r.second).toFun z) ∧
(∀ z, (finalOrder Label).le z r.B →
((finalAction Label).iso h1).toFun z = ((finalAction Label).iso h2).toFun z) ∧
(finalOrder Label).meet (((finalAction Label).iso h1).toFun r.D) X =
((finalAction Label).iso r.first).toFun r.C ∧
(finalOrder Label).meet (((finalAction Label).iso h2).toFun r.D) X =
((finalAction Label).iso r.second).toFun r.C ∧
(finalOrder Label).meet (((finalAction Label).iso h1).toFun r.D)
(((finalAction Label).iso h2).toFun r.D) = ((finalAction Label).iso h1).toFun r.B := by
rcases rectangle_data_in_stage Label r X with ⟨a, t, Y, hC, hB, hD, hg1, hg2, hY⟩
rcases final_rectangles_at Label a t Y with ⟨h, agreeC, agreeB, rootMeet, pairMeet⟩
have actors (j : Bool) : finalGroupInclusion Label a ((stage Label a).rectangleOldActor t j) =
if j then r.second else r.first := by
cases j with
| false => exact hg1
| true => exact hg2
have cAgree (j : Bool) (z : FinalPoint Label) (lower : (finalOrder Label).le z r.C) :
((finalAction Label).iso (h j)).toFun z =
((finalAction Label).iso (if j then r.second else r.first)).toFun z := by
have eq := agreeC j z (hC.symm ▸ lower)
rw [actors] at eq
exact eq
have mappedRoot (j : Bool) :
(finalInclusion Label a).toFun ((stage Label a).order.meet
(((stage Label a).action.iso ((stage Label a).rectangleOldActor t j)).toFun t.C) Y) =
((finalAction Label).iso (if j then r.second else r.first)).toFun r.C := by
rw [(finalInclusion Label a).map_meet, ← final_action_stage, actors, hC, hY]
apply meet_eq_left_of_le
cases j with
| false =>
exact (finalOrder Label).trans
((((finalAction Label).iso r.first).le_iff _ _).mpr r.C_le_D) first_upper
| true =>
exact (finalOrder Label).trans
((((finalAction Label).iso r.second).le_iff _ _).mpr r.C_le_D) second_upper
refine ⟨h false, h true, cAgree false, cAgree true, ?_, ?_, ?_, ?_⟩
· simpa only [hB] using agreeB
· simpa only [hD, hY, Bool.false_eq_true, ↓reduceIte] using
(rootMeet false).trans (mappedRoot false)
· simpa only [hD, hY, ↓reduceIte] using (rootMeet true).trans (mappedRoot true)
· simpa only [hD, hB] using pairMeet
end PartitionPilot.Compression.RecursiveMaps.MainStage.OrdinalRun
end OAI
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