File size: 9,113 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 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 | import OAI.SetTheory.PartitionPrinciple.Definability.DepthSlotCodes
import OAI.SetTheory.PartitionPrinciple.Definability.FiniteWordMapCodes
namespace OAI
namespace PartitionPilot.Forcing
open SetRealization
universe u
def FiniteWordAppendCode (A omega add f g h : ZFSet.{u}) : Prop :=
∃ m ∈ omega, ∃ n ∈ omega, ∃ p ∈ omega,
ZFSet.IsFunc m A f ∧ ZFSet.IsFunc n A g ∧ ZFSet.IsFunc p A h ∧
ZFSet.pair (ZFSet.pair m n) p ∈ add ∧
∀ i ∈ p, ∀ a ∈ A, ZFSet.pair i a ∈ h ↔
ZFSet.pair i a ∈ f ∨ ∃ j ∈ n,
ZFSet.pair (ZFSet.pair m j) i ∈ add ∧ ZFSet.pair j a ∈ g
def FiniteWordReverseCode (A omega add f h : ZFSet.{u}) : Prop :=
∃ n ∈ omega, ZFSet.IsFunc n A f ∧ ZFSet.IsFunc n A h ∧
∀ i ∈ n, ∀ a ∈ A, ZFSet.pair i a ∈ h ↔
∃ j ∈ n, ∃ k ∈ n, ZFSet.pair (ZFSet.pair i j) k ∈ add ∧
n = insert k k ∧ ZFSet.pair j a ∈ f
theorem finiteWordGraph_append_entry {B : Type u} (s : Presentation B)
(xs ys : List B) (k : Nat) (a : ZFSet.{u}) :
ZFSet.pair (natSetCode k) a ∈ finiteWordGraph s (xs ++ ys) ↔
ZFSet.pair (natSetCode k) a ∈ finiteWordGraph s xs ∨
∃ j, k = xs.length + j ∧ ZFSet.pair (natSetCode j) a ∈ finiteWordGraph s ys := by
constructor
· intro entry
obtain ⟨hk, value⟩ := (pair_mem_finiteWordGraph s (xs ++ ys) k a).1 entry
by_cases left : k < xs.length
· apply Or.inl
apply (pair_mem_finiteWordGraph s xs k a).2
refine ⟨left, ?_⟩
simpa only [List.get_eq_getElem, List.getElem_append_left left] using value
· have right : xs.length ≤ k := Nat.le_of_not_gt left
have hj : k - xs.length < ys.length := by simp only [List.length_append] at hk; omega
refine Or.inr ⟨k - xs.length, by omega, (pair_mem_finiteWordGraph s ys _ a).2 ⟨hj, ?_⟩⟩
simpa only [List.get_eq_getElem, List.getElem_append_right right] using value
· rintro (entry | ⟨j, equal, entry⟩)
· obtain ⟨hk, value⟩ := (pair_mem_finiteWordGraph s xs k a).1 entry
apply (pair_mem_finiteWordGraph s (xs ++ ys) k a).2
have bound : k < (xs ++ ys).length := by simp only [List.length_append]; omega
refine ⟨bound, ?_⟩
simpa only [List.get_eq_getElem, List.getElem_append_left hk] using value
· subst k
obtain ⟨hj, value⟩ := (pair_mem_finiteWordGraph s ys j a).1 entry
apply (pair_mem_finiteWordGraph s (xs ++ ys) (xs.length + j) a).2
have bound : xs.length + j < (xs ++ ys).length := by simp only [List.length_append]; omega
refine ⟨bound, ?_⟩
simpa only [List.get_eq_getElem,
List.getElem_append_right (Nat.le_add_right xs.length j), Nat.add_sub_cancel_left] using value
theorem finiteWordGraph_reverse_entry {B : Type u} (s : Presentation B)
(xs : List B) (i : Nat) (a : ZFSet.{u}) :
ZFSet.pair (natSetCode i) a ∈ finiteWordGraph s xs.reverse ↔
∃ j, i + j + 1 = xs.length ∧ ZFSet.pair (natSetCode j) a ∈ finiteWordGraph s xs := by
constructor
· intro entry
obtain ⟨hi, value⟩ := (pair_mem_finiteWordGraph s xs.reverse i a).1 entry
have bound : i < xs.length := by simpa only [List.length_reverse] using hi
have hj : xs.length - 1 - i < xs.length := by omega
refine ⟨xs.length - 1 - i, by omega, (pair_mem_finiteWordGraph s xs _ a).2 ⟨hj, ?_⟩⟩
simpa only [List.get_eq_getElem, List.getElem_reverse] using value
· rintro ⟨j, sum, entry⟩
obtain ⟨hj, value⟩ := (pair_mem_finiteWordGraph s xs j a).1 entry
have hi : i < xs.reverse.length := by simp only [List.length_reverse]; omega
have equal : xs.length - 1 - i = j := by omega
apply (pair_mem_finiteWordGraph s xs.reverse i a).2
refine ⟨hi, ?_⟩
simpa only [List.get_eq_getElem, List.getElem_reverse, equal] using value
theorem finiteWordGraph_append_code {B : Type u} (s : Presentation B) (xs ys : List B) :
FiniteWordAppendCode s.carrier ZFSet.omega natAddGraph
(finiteWordGraph s xs) (finiteWordGraph s ys) (finiteWordGraph s (xs ++ ys)) := by
refine ⟨natSetCode xs.length, natSetCode_mem_omega _, natSetCode ys.length, natSetCode_mem_omega _,
natSetCode (xs.length + ys.length), natSetCode_mem_omega _, finiteWordGraph_function s xs,
finiteWordGraph_function s ys, ?_, (pair_mem_natAddGraph _ _ _).2 rfl, ?_⟩
· simpa only [List.length_append] using finiteWordGraph_function s (xs ++ ys)
· intro i hi a _
obtain ⟨k, _, rfl⟩ := (mem_natSetCode i (xs.length + ys.length)).1 hi
rw [finiteWordGraph_append_entry]
apply or_congr Iff.rfl
constructor
· rintro ⟨j, equal, entry⟩
have hj := (ZFSet.pair_mem_prod.1 ((finiteWordGraph_function s ys).1 entry)).1
exact ⟨natSetCode j, hj, (pair_mem_natAddGraph _ _ _).2 equal, entry⟩
· rintro ⟨j, hj, sum, entry⟩
obtain ⟨l, _, rfl⟩ := (mem_natSetCode j ys.length).1 hj
exact ⟨l, (pair_mem_natAddGraph _ _ _).1 sum, entry⟩
theorem finiteWordGraph_reverse_code {B : Type u} (s : Presentation B) (xs : List B) :
FiniteWordReverseCode s.carrier ZFSet.omega natAddGraph
(finiteWordGraph s xs) (finiteWordGraph s xs.reverse) := by
refine ⟨natSetCode xs.length, natSetCode_mem_omega _, finiteWordGraph_function s xs, ?_, ?_⟩
· simpa only [List.length_reverse] using finiteWordGraph_function s xs.reverse
· intro i hi a _
obtain ⟨k, _, rfl⟩ := (mem_natSetCode i xs.length).1 hi
rw [finiteWordGraph_reverse_entry]
constructor
· rintro ⟨j, sum, entry⟩
have hj := (ZFSet.pair_mem_prod.1 ((finiteWordGraph_function s xs).1 entry)).1
refine ⟨natSetCode j, hj, natSetCode (k + j), (natSetCode_mem_iff _ _).2 (by omega),
(pair_mem_natAddGraph _ _ _).2 rfl, ?_, entry⟩
exact (congrArg natSetCode sum).symm
· rintro ⟨j, hj, l, hl, sum, successor, entry⟩
obtain ⟨j, _, rfl⟩ := (mem_natSetCode j xs.length).1 hj
obtain ⟨l, _, rfl⟩ := (mem_natSetCode l xs.length).1 hl
have added := (pair_mem_natAddGraph _ _ _).1 sum
have length : xs.length = l + 1 := natSetCode_injective successor
exact ⟨j, by omega, entry⟩
theorem FiniteWordAppendCode.unique {A f g h h' : ZFSet.{u}}
(first : FiniteWordAppendCode A ZFSet.omega natAddGraph f g h)
(second : FiniteWordAppendCode A ZFSet.omega natAddGraph f g h') : h = h' := by
obtain ⟨m, hm, n, hn, p, hp, fm, gn, funH, sum, entries⟩ := first
obtain ⟨m', hm', n', hn', p', hp', fm', gn', funH', sum', entries'⟩ := second
obtain ⟨m, rfl⟩ := (mem_omega_iff_natSetCode _).1 hm
obtain ⟨n, rfl⟩ := (mem_omega_iff_natSetCode _).1 hn
obtain ⟨p, rfl⟩ := (mem_omega_iff_natSetCode _).1 hp
obtain ⟨m', rfl⟩ := (mem_omega_iff_natSetCode _).1 hm'
obtain ⟨n', rfl⟩ := (mem_omega_iff_natSetCode _).1 hn'
obtain ⟨p', rfl⟩ := (mem_omega_iff_natSetCode _).1 hp'
have mEqual := finiteWord_domain_unique fm' fm
have nEqual := finiteWord_domain_unique gn' gn
subst m' n'
have pEqual := ((pair_mem_natAddGraph _ _ _).1 sum').trans ((pair_mem_natAddGraph _ _ _).1 sum).symm
subst p'
apply ZFSet.ext
intro z
constructor
· intro hz
obtain ⟨i, hi, a, ha, rfl⟩ := ZFSet.mem_prod.1 (funH.1 hz)
exact (entries' i hi a ha).2 ((entries i hi a ha).1 hz)
· intro hz
obtain ⟨i, hi, a, ha, rfl⟩ := ZFSet.mem_prod.1 (funH'.1 hz)
exact (entries i hi a ha).2 ((entries' i hi a ha).1 hz)
theorem FiniteWordReverseCode.unique {A f h h' : ZFSet.{u}}
(first : FiniteWordReverseCode A ZFSet.omega natAddGraph f h)
(second : FiniteWordReverseCode A ZFSet.omega natAddGraph f h') : h = h' := by
obtain ⟨n, hn, funF, funH, entries⟩ := first
obtain ⟨n', hn', funF', funH', entries'⟩ := second
obtain ⟨n, rfl⟩ := (mem_omega_iff_natSetCode _).1 hn
obtain ⟨n', rfl⟩ := (mem_omega_iff_natSetCode _).1 hn'
have same := finiteWord_domain_unique funF' funF
subst n'
apply ZFSet.ext
intro z
constructor
· intro hz
obtain ⟨i, hi, a, ha, rfl⟩ := ZFSet.mem_prod.1 (funH.1 hz)
exact (entries' i hi a ha).2 ((entries i hi a ha).1 hz)
· intro hz
obtain ⟨i, hi, a, ha, rfl⟩ := ZFSet.mem_prod.1 (funH'.1 hz)
exact (entries i hi a ha).2 ((entries' i hi a ha).1 hz)
theorem finiteWordAppendCode_iff {B : Type u} (s : Presentation B) (xs ys : List B) (h : ZFSet.{u}) :
FiniteWordAppendCode s.carrier ZFSet.omega natAddGraph
(finiteWordGraph s xs) (finiteWordGraph s ys) h ↔ h = finiteWordGraph s (xs ++ ys) :=
⟨fun code => code.unique (finiteWordGraph_append_code s xs ys),
fun equal => equal.symm ▸ finiteWordGraph_append_code s xs ys⟩
theorem finiteWordReverseCode_iff {B : Type u} (s : Presentation B) (xs : List B) (h : ZFSet.{u}) :
FiniteWordReverseCode s.carrier ZFSet.omega natAddGraph (finiteWordGraph s xs) h ↔
h = finiteWordGraph s xs.reverse :=
⟨fun code => code.unique (finiteWordGraph_reverse_code s xs),
fun equal => equal.symm ▸ finiteWordGraph_reverse_code s xs⟩
end PartitionPilot.Forcing
end OAI
|