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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 | import OAI.SetTheory.PartitionPrinciple.Geometry.CompressionExternalTileOverlaps
namespace OAI
section
namespace PartitionPilot.Forcing.GroundCompressionCompletion
open SetRealization
universe u
variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}}
variable {I Sources Reps G m inv one T τ P0 r0 μ0 A0 : ZFSet.{u}}
variable (completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ)
variable (group : GroupCode G m inv one) (oldAction : PosetActionCode G P0 m one r0 A0)
variable (compression : CompressionOriginalData P0 r0 μ0 A0 G
(completion.algebra.union.fields 0) (completion.algebra.union.fields 1)
(completion.algebra.union.fields 2) (completion.algebra.union.fields 12))
theorem externalFamilyVertexCode_injective
(r : ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0)) :
Function.Injective (completion.externalFamilyVertexCode group oldAction compression r) := by
rcases r with i | (x | z)
all_goals
intro v w equal
cases v with
| none =>
cases w with
| none => rfl
| some w =>
cases w with | up w =>
cases w <;> simp only [externalFamilyVertexCode_root, externalFamilyVertexCode_compression,
externalFamilyVertexCode_separation, externalFamilyVertexCode_rectangle,
natSetCode_injective.eq_iff] at equal <;> contradiction
| some v =>
cases w with
| none =>
cases v with | up v =>
cases v <;> simp only [externalFamilyVertexCode_root, externalFamilyVertexCode_compression,
externalFamilyVertexCode_separation, externalFamilyVertexCode_rectangle,
natSetCode_injective.eq_iff] at equal <;> contradiction
| some w =>
cases v with | up v =>
cases w with | up w =>
cases v <;> cases w <;> simp_all only [externalFamilyVertexCode_compression,
externalFamilyVertexCode_separation, externalFamilyVertexCode_rectangle,
natSetCode_injective.eq_iff, Bool.false_eq_true] <;> simp_all
theorem externalFamilyVertexCode_range
(r : ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0)) (v : ZFSet.{u}) :
(v = natSetCode 0 ∨ v = natSetCode 1 ∨
(externalRequestIndexCode r ∈ ZFSet.prod {natSetCode 2} (externalRectangles P0 G r0 μ0 A0) ∧
v = natSetCode 2)) ↔
∃ i, completion.externalFamilyVertexCode group oldAction compression r i = v := by
rcases r with j | (x | z)
· simp only [externalRequestIndexCode, ZFSet.pair_mem_prod, ZFSet.mem_singleton,
natSetCode_injective.eq_iff, show ¬ (0 : Nat) = 2 by decide, false_and, or_false]
constructor
· rintro (rfl | rfl)
· exact ⟨none, rfl⟩
· exact ⟨some ⟨()⟩, rfl⟩
· rintro ⟨i, rfl⟩
cases i with
| none => exact Or.inl rfl
| some i => exact Or.inr rfl
· simp only [externalRequestIndexCode, ZFSet.pair_mem_prod, ZFSet.mem_singleton,
natSetCode_injective.eq_iff, show ¬ (1 : Nat) = 2 by decide, false_and, or_false]
constructor
· rintro (rfl | rfl)
· exact ⟨none, rfl⟩
· exact ⟨some ⟨()⟩, rfl⟩
· rintro ⟨i, rfl⟩
cases i with
| none => exact Or.inl rfl
| some i => exact Or.inr rfl
· simp only [externalRequestIndexCode, ZFSet.pair_mem_prod, ZFSet.mem_singleton,
SetMembers.raw_mem, and_self, true_and]
constructor
· rintro (rfl | rfl | rfl)
· exact ⟨none, rfl⟩
· exact ⟨some ⟨false⟩, rfl⟩
· exact ⟨some ⟨true⟩, rfl⟩
· rintro ⟨i, rfl⟩
cases i with
| none => exact Or.inl rfl
| some i =>
cases i with | up b =>
cases b
· exact Or.inr (Or.inl rfl)
· exact Or.inr (Or.inr rfl)
noncomputable def externalFamilyTileVertexPresentation :
Presentation (Sigma fun r : ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0) =>
Option ((completion.externalFamily group oldAction compression).Child r) ×
SetMembers (completion.algebra.union.fields 0)) where
carrier := externalTileVertices (externalRequestCarrier I P0 (externalRectangles P0 G r0 μ0 A0))
(externalRectangles P0 G r0 μ0 A0) (completion.algebra.union.fields 0)
encode v := ZFSet.pair (ZFSet.pair (externalRequestIndexCode v.1)
(completion.externalFamilyVertexCode group oldAction compression v.1 v.2.1)) (SetMembers.raw v.2.2)
injective := by
rintro ⟨r, i, x⟩ ⟨s, j, y⟩ equal
have outer := ZFSet.pair_inj.1 equal
have inner := ZFSet.pair_inj.1 outer.1
have same := externalRequestIndexCode_injective I P0 (externalRectangles P0 G r0 μ0 A0) inner.1
cases same
have vertex := completion.externalFamilyVertexCode_injective group oldAction compression r inner.2
have point := SetMembers.raw_injective _ outer.2
cases vertex
cases point
rfl
mem_carrier z := by
constructor
· intro hz
have shape : ∃ q i x, z = ZFSet.pair (ZFSet.pair q i) x := by
rcases ZFSet.mem_union.1 hz with h | h
all_goals
obtain ⟨qi, hqi, x, _, rfl⟩ := ZFSet.mem_prod.1 h
obtain ⟨q, _, i, _, rfl⟩ := ZFSet.mem_prod.1 hqi
exact ⟨q, i, x, rfl⟩
obtain ⟨q, i, x, rfl⟩ := shape
obtain ⟨hqi, hx⟩ := (triple_mem_externalTileVertices _ _ _ _ _ _).1 hz
have hq : q ∈ externalRequestCarrier I P0 (externalRectangles P0 G r0 μ0 A0) := by
rcases hqi with ⟨hq, _⟩ | ⟨hq, _⟩
· exact hq
· exact ZFSet.mem_union.2 (Or.inr hq)
obtain ⟨r, rfl⟩ := ((externalRequestPresentation I P0 (externalRectangles P0 G r0 μ0 A0)).mem_carrier q).1 hq
have hv : i = natSetCode 0 ∨ i = natSetCode 1 ∨
(externalRequestIndexCode r ∈ ZFSet.prod {natSetCode 2} (externalRectangles P0 G r0 μ0 A0) ∧
i = natSetCode 2) := by tauto
obtain ⟨v, rfl⟩ := (completion.externalFamilyVertexCode_range group oldAction compression r i).1 hv
exact ⟨⟨r, v, SetMembers.ofMem x hx⟩, by simp only [SetMembers.raw_ofMem]; rfl⟩
· rintro ⟨⟨r, i, x⟩, rfl⟩
apply (triple_mem_externalTileVertices _ _ _ _ _ _).2
refine ⟨?_, SetMembers.raw_mem x⟩
have hv := (completion.externalFamilyVertexCode_range group oldAction compression r _).2 ⟨i, rfl⟩
have hq := (externalRequestPresentation I P0 (externalRectangles P0 G r0 μ0 A0)).encode_mem r
tauto
theorem externalFamilyTileVertexPresentation_encode
(r : ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0))
(i : Option ((completion.externalFamily group oldAction compression).Child r))
(x : SetMembers (completion.algebra.union.fields 0)) :
(completion.externalFamilyTileVertexPresentation group oldAction compression).encode ⟨r, i, x⟩ =
ZFSet.pair (ZFSet.pair (externalRequestIndexCode r)
(completion.externalFamilyVertexCode group oldAction compression r i)) (SetMembers.raw x) := rfl
end PartitionPilot.Forcing.GroundCompressionCompletion
end
end OAI
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