| 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 | |